Nonlinear Frame Research Articles

Wind resistant size optimization of geometrically nonlinear lattice structures using a modified optimality criterion method

Lattice structures normally have slender members and light weight, high flexibility and small damping ratios. Hence, their structural behavior may be affected by the geometric nonlinearity and they are prone to large displacements and safety issues due to wind loads. Optimal design of such nonlinear and flexible structures is difficult and complicated. This paper presents an automatic optimizing flow for the integrated wind-induced response analysis and wind-resistant optimal design of linear and non-linear lattice structures based on the modified Optimality Criterion (OC) algorithm. The method for structural dynamic analysis and wind-induced response of lattice structure is developed by combining Proper Orthogonal Decomposition (POD) with Load Dependent Ritz (LDR) vector method, together with the Harmonic Excitation Method (HEM) and Load Response Correlation (LRC) method. Meanwhile the quadratic programming method is used to evaluate the Lagrange multipliers in the modified optimality criterion method. Based on the derived design sensitivity analysis results for the nodal displacements, element stresses and nonlinear critical load factor of linear and nonlinear space frame structures in a companion paper, an automatic optimizing flow is then established by adopting the Application Programming Interface (API) of SAP2000 to integrate the wind-induced response, design sensitivity analysis and wind-resistant optimal design of space frame structural system. The proposed automatic optimizing flow is then applied to the optimized process of a lattice structure under dynamic wind loading. It is found that for the lattice structure, the geometric non-linear analysis and the constraint of nonlinear critical load factor need to be considered in their optimization, meanwhile updating on the time-history of wind loads and Equivalent Static Wind Loads (ESWL) is necessary as result of the variations in the design member size during the optimization process. It is also found that the maximum displacement constraint is the most important factor in the optimization of the lattice structure and maximum stress of structural elements are normally found to be inactive constraints during the optimization procedure. In the case that a low limit for the maximum displacement is pre-set, the nonlinear critical load factor becomes the important constraint. Optimized results from this study would be helpful to those involved in wind resistant analysis and optimization design research for such lattice structures.

Finite-element implementation for nonlinear static and dynamic frame analysis of tapered members

Non-prismatic members are popular for civil engineering structures. This paper derives a set of unified beam-column formulations for nonlinear static and dynamic analyses of the structures made of members with tapered sections, addressing the problems in engineering design practices. The element shape-functions are established upon the local-axes by extracting the rigid-body movements for simplifying mathematical expressions. To represent the variations in the stiffness gradients of tapered sections, the tapered-variability indexes are proposed. The generalized tangent stiffness and consistent mass matrices are developed based on the indexes. When analyzing non-prismatic members, the conventional method for handling member loads is inapplicable because it is derived for prismatic sections. Therefore, a new approach for converting the member loads acting on tapered members into the equivalent nodal forces is proposed based on the energy conservation principle. To consider the offsets in section axes, the eccentricity matrices are employed. For allowing large deflections, the incremental secant-stiffness method (ISM) based on Updated-Lagrangian (UL) description is proposed. Finally, extensive examples are provided for validating the accuracy and efficiency of the proposed element formulations in solving both the static and dynamic nonlinear problems.

Nonlinear finite element simulation of unbonded prestressed concrete beams

Prestressed concrete with internal unbonded tendons has been recognized as an excellent structural option for beams and slabs and is employed worlwide. Numerical solutions for the analysis of such structures are still an active field of research. This work presents a finite element model for the physical and geometrical nonlinear analysis of prestressed concrete beams with unbonded internal tendons, under short-term loading. The reinforced concrete beam is modeled by Euler-Bernoulli nonlinear plane frame elements and a total Lagrangian approach. The prestressing tendon is modeled by a single polygonal element embedded in a specified subset of the frame elements. Due to lack of strain compatibility between the concrete and the tendon at a given cross-section, the cable strain is computed from the displacements of all associated frame elements. Geometric and material nonlinearities are considered for both the reinforced concrete beam and the prestressing tendons. The internal force vector and corresponding tangent stiffness matrix of each element under large displacements are derived consistently, and novel expressions for the tangent stiffness operator which ensure the convergence rates of the Newton-Raphson scheme are developed. The accuracy of the formulation is assessed by comparison with experimental tests, with very good results.

Life-cycle cost evaluation of steel structures retrofitted with steel slit damper and shape memory alloy–based hybrid damper

In this study, the seismic capacity of a hybrid damper, composed of a steel slit plate damper and two shape memory alloy bars, is investigated through fragility analysis and life-cycle cost evaluation of a steel frame retrofitted with the damper. The nonlinear time history analysis model frames show that the seismic responses of the frames equipped with hybrid damper are significantly lesser than the frames retrofitted with conventional slit dampers. The enhancement in the seismic performance of frames retrofitted with hybrid damper is because of extra stiffness, energy dissipation, and self-centering capability provided by the shape memory alloy bars. It is also observed that the life-cycle cost of the frames equipped with hybrid dampers is smallest compared with the life-cycle cost of the bare frames and the frames equipped with slit dampers, even though the initial cost is of the hybrid damper is higher than that of the conventional slit damper.

Graphic dynamic prediction of polarized earthquake incidence response for plan-irregular single story buildings

A graphical dynamic model is presented to predict the directional earthquake response of two-ways plan-asymmetric buildings. The theoretical principles inherent to torsional dynamics and vibrations are investigated and the dynamic directional response is rationally explained based on modal rotational kinematics about modal torsional pivots. Seismic forces and response decomposition are handled through geometric modal torsional trends and the earthquake incidence response envelopes are described through directional modal participation radii and graphic spectrum-based ‘8-shaped’ directional influence circles. The graphic approach provides good predictions of the maximum response and of the critical angle computed through directional combination methods. A nonlinear 3D frame model with eccentric infills is analyzed through linear and nonlinear response history analyses (RHA, NLRHA) changing the earthquake incidence angle. Upon confirmation of weak modal coupling, graphic-dynamic modal torsional trends and directional inelastic response envelopes are used to predict the nonlinear response. Directional incremental dynamic analyses and uncoupled modal response history analyses confirm the prediction of polarization.

Combined state and parameter identification of nonlinear structural dynamical systems based on Rao-Blackwellization and Markov chain Monte Carlo simulations

The problem of combined state and parameter estimation in nonlinear state space models, based on Bayesian filtering methods, is considered. A novel approach, which combines Rao-Blacicwellized particle filters for state estimation with Markov chain Monte Carlo (MCMC) simulations for parameter identification, is proposed. In order to ensure successful performance of the MCMC samplers, in situations involving large amount of dynamic measurement data and (or) low measurement noise, the study employs a modified measurement model combined with an importance sampling based correction. The parameters of the process noise covariance matrix are also included as quantities to be identified. The study employs the Rao-Blackwellization step at two stages: one, associated with the state estimation problem in the particle filtering step, and, secondly, in the evaluation of the ratio of likelihoods in the MCMC run. The satisfactory performance of the proposed method is illustrated on three dynamical systems: (a) a computational model of a nonlinear beam moving oscillator system, (b) a laboratory scale beam traversed by a loaded trolley, and (c) an earthquake shake table study on a bending-torsion coupled nonlinear frame subjected to uniaxial support motion. (C) 2017 Elsevier Ltd. All rights reserved.

A dynamic load estimation method for nonlinear structures with unscented Kalman filter

Abstract A force estimation method is proposed for hysteretic nonlinear structures. The equation of motion for the nonlinear structure is represented in state space and the state variable is augmented by the unknown the time history of external force. Unscented Kalman filter (UKF) is improved for the force identification in state space considering the ill-condition characteristic in the computation of square roots for the covariance matrix. The proposed method is firstly validated by a numerical simulation study of a 3-storey nonlinear hysteretic frame excited by periodic force. Each storey is supposed to follow a nonlinear hysteretic model. The external force is identified and the measurement noise is considered in this case. Then a case of a seismically isolated building subjected to earthquake excitation and impact force is studied. The isolation layer performs nonlinearly during the earthquake excitation. Impact force between the seismically isolated structure and the retaining wall is estimated with the proposed method. Uncertainties such as measurement noise, model error in storey stiffness and unexpected environmental disturbances are considered. A real-time substructure testing of an isolated structure is conducted to verify the proposed method. In the experimental study, the linear main structure is taken as numerical substructure while the one of the isolations with additional mass is taken as the nonlinear physical substructure. The force applied by the actuator on the physical substructure is identified and compared with the measured value from the force transducer. The method proposed in this paper is also validated by shaking table test of a seismically isolated steel frame. The acceleration of the ground motion as the unknowns is identified by the proposed method. Results from both numerical simulation and experimental studies indicate that the UKF based force identification method can be used to identify external excitations effectively for the nonlinear structure with accurate results even with measurement noise, model error and environmental disturbances.

Modelling and probabilistic study of the residual stress of cold-formed hollow steel sections

Cold-formed Hollow Steel Sections (HSS) are widely used in the construction industry. The distribution of residual stress in non-stress-relieved cold-formed HSS is complex due to the highly non-uniform variation around the cross-section and through the plate thickness. Modeling residual stress in frame analysis is therefore a difficult task. This paper presents a practical method for approximating the effects of different components of residual stress on the behavior of cold-formed HSS by modifying the steel stress-strain curve. The method proposes a convenient means of including residual stress in beam element-based nonlinear frame analysis. A probabilistic study is then carried out to study the effect of the uncertainty in residual stress on frame strength.

Precast industrial buildings in Southern Europe: loss of support at frictional beam-to-column connections under seismic actions

This paper presents the evaluation of the loss-of-support conditions in frictional beam-to-column connections of industrial precast concrete buildings under seismic actions. This type of connection is widespread throughout Southern Europe in non-seismically designed industrial precast buildings. First, geometric properties of industrial precast buildings and of the frictional beam-to-column connections, together with reference values for the friction coefficient, are reviewed. Then, earthquake time histories taken from the European Strong-Motion sets and recordings of the two major shocks of the 2012 Emilia-Romagna events are presented and discussed showing the importance of the vertical component. Two dynamic models of increasing complexity are used to ascertain loss-of-support conditions under seismic action. The first model is an elastic one, representing a single frame of the industrial buildings. Results are obtained according to: (1) 2D analyses, disregarding the time correlation between the response peaks along the horizontal and vertical directions, (2) 2D analyses taking into account time correlation, and (3) 3D analyses to evaluate also directionality effects. The second model is a 2D non-linear planar frame developed within the OpenSees framework. Results show that simplified (linear) models are a good proxy to more refined (non-linear) ones. However, one must resort to non-linear models if differential displacements between beam and column are of interest. The non-linear numerical investigations show that friction coefficient, horizontal and vertical periods and damping, and column reinforcement ratio are the key variables in estimating the loss-of-support conditions.

An explicit asymptotic preserving low Froude scheme for the multilayer shallow water model with density stratification

We present an explicit scheme for a two-dimensional multilayer shallow water model with density stratification, for general meshes and collocated variables. The proposed strategy is based on a regularized model where the transport velocity in the advective fluxes is shifted proportionally to the pressure potential gradient. Using a similar strategy for the potential forces, we show the stability of the method in the sense of a discrete dissipation of the mechanical energy, in general multilayer and non-linear frames. These results are obtained at first-order in space and time and extended using a second-order MUSCL extension in space and a Heun's method in time. With the objective of minimizing the diffusive losses in realistic contexts, sufficient conditions are exhibited on the regularizing terms to ensure the scheme's linear stability at first and second-order in time and space. The other main result stands in the consistency with respect to the asymptotics reached at small and large time scales in low Froude regimes, which governs large-scale oceanic circulation. Additionally, robustness and well-balanced results for motionless steady states are also ensured. These stability properties tend to provide a very robust and efficient approach, easy to implement and particularly well suited for large-scale simulations. Some numerical experiments are proposed to highlight the scheme efficiency: an experiment of fast gravitational modes, a smooth surface wave propagation, an initial propagating surface water elevation jump considering a non-trivial topography, and a last experiment of slow Rossby modes simulating the displacement of a baroclinic vortex subject to the Coriolis force.

Comparison between simplified and FE models for short-term deflection in continuous RC beams

Abstract In this work, a comparative study of different simplified methods and nonlinear finite element (FE) models used for calculating short-term deflections (vertical displacements of the longitudinal axis) in continuous reinforced concrete (RC) beams, under service loads, is performed. The simplified methods employed are the one proposed by Branson and the bilinear method recommended by the European Code CEB - Design Manual on Cracking and Deformations. Two finite element models are utilized: the first one with frame elements in which material nonlinearities are considered along the element and its cross section divided into layers, by using of constitutive relationships for steel and concrete, while the second one utilizes beam elements, with physical nonlinearity considered by means of moment-curvature diagrams, obtained from Branson equation. Several examples of continuous RC beams under service loads are analysed and the results obtained by the different models are compared taking as reference the nonlinear frame element model. A few conclusions and recommendations regarding the use of the different methods are drawn at the end of the work.

Effect of Response Related Weighting Matrices on Performance of Active Control Systems for Nonlinear Frames

In this paper, the effect of various arrangements of displacement, velocity and acceleration related weighting matrices on the performance of active control systems on nonlinear frames has been studied. Different arrangements of weighting matrices and feedback combinations of the response have been considered to design the active controllers using a single actuator for reducing the response of an eight-storey bilinear hysteretic frame under white noise excitations. The nonlinear Newmark-based instantaneous optimal control algorithm has been used, where the distributed genetic algorithm (DGA) is employed to determine the proper set of weighting matrices. For each set of feedback and weighting matrices, the active control system has been designed with the optimal weights determined. Here, the objective is to minimize the maximum control force required to reduce the maximum structural drift to a value below the desired level. The numerical results of simulation show that, for the cases studied, the use of different arrangements of weighting matrices in the proposed method for the performance index of the active control law has no significant impact on the performance of the active control system. However, the type of response feedback combination included in the control law considerably affects the performance, and the controllers designed based on velocity feedback have been found to be more effective. It was also shown that for all the weight-cases, using the full feedback of response can lead to design controllers that require minimum control force to reduce the structural response with more online measurements. The robustness of the designed controllers for different weighting matrices arrangements and feedback combinations has also been tested under a number of real earthquake excitations with the results discussed.

An efficient method for reliability-based design optimization of nonlinear inelastic steel space frames

This paper proposes an effective numerical procedure for reliability-based design optimization (RBDO) of nonlinear inelastic steel frames by integrating a harmony search technique (HS) for optimization and a robust method for failure probability analysis. The practical advanced analysis using the beam-column approach is used for capturing the nonlinear inelastic behaviors of frames, while a detail implement of HS for discrete optimization of steel frames is introduced. The failure probability of structures is evaluated by using the combination of the improved Latin Hypercube (IHS) and a new effective importance sampling (EIS). The efficiency and accuracy of the proposed procedure are demonstrated through three mathematical examples and five steel frames. The results obtained in this paper prove that the proposed procedure is computationally efficient and can be applied in practical design. Furthermore, it is shown that the use of nonlinear inelastic analysis in the optimization of steel frames yields more realistic results.

On the existence of non-linear frames

A stronger version of the notion of frame in Banach space called Strong Retro Banach frame (SRBF) is defined and studied. It has been proved that if $\mathcal{X}$ is a Banach space such that $\mathcal{X^*}$ has a SRBF, then $\mathcal{X}$ has a Bi-Banach frame with some geometric property. Also, it has been proved that if a Banach space $\mathcal{X}$ has an approximative Schauder frame, then $\mathcal{X^*}$ has a SRBF. Finally, the existence of a non-linear SRBF in the conjugate of a separable Banach space has been proved.

Experimental evaluation of four ground-motion scaling methods for dynamic response-history analysis of nonlinear structures

This paper experimentally evaluates four methods to scale earthquake ground-motions within an ensemble of records to minimize the statistical dispersion and maximize the accuracy in the dynamic peak roof drift demand and peak inter-story drift demand estimates from response-history analyses of nonlinear building structures. The scaling methods that are investigated are based on: (1) ASCE/SEI 7–10 guidelines; (2) spectral acceleration at the fundamental (first mode) period of the structure, S a (T 1); (3) maximum incremental velocity, MIV; and (4) modal pushover analysis. A total of 720 shake-table tests of four small-scale nonlinear building frame specimens with different static and dynamic characteristics are conducted. The peak displacement demands from full suites of 36 near-fault ground-motion records as well as from smaller “unbiased” and “biased” design subsets (bins) of ground-motions are included. Out of the four scaling methods, ground-motions scaled to the median MIV of the ensemble resulted in the smallest dispersion in the peak roof and inter-story drift demands. Scaling based on MIV also provided the most accurate median demands as compared with the “benchmark” demands for structures with greater nonlinearity; however, this accuracy was reduced for structures exhibiting reduced nonlinearity. The modal pushover-based scaling (MPS) procedure was the only method to conservatively overestimate the median drift demands.

A numerical study on the symmetrization of tangent stiffness matrix in non-linear analysis using corotational approach

In the corotational kinematics (EIRC), the movement is decomposed in deformational and rigid body components using projection operators. Large rigid body translations and rotations, and infinitesimal deformation tensors are adopted. In this way, a non-symmetric stiffness matrix is obtained for the finite element. Literature points that this matrix may be symmetrized in the usual way and the original non-symmetric matrix tends to be in a symmetric form as equilibrium is approached. This paper performed symmetrization studies using Frobenius norm and Absolute Maximum Coefficient of the anti-symmetric part of stiffness matrix in several numerical analyses with high non-linearity solved with an incremental-iterative scheme. An element independent corotational (EICR) kinematics is used to analyze non-linear spatial frames with 3D Euler–Bernoulli beam elements. The results show that this symmetrization not always occurs and depends significantly on the magnitude of displacements and finite-element mesh refinement. The authors demonstrated that, in some cases, the symmetrization would only take place with very refined meshes when the discretized structure approaches the continuum. In this way, the current literature and general mathematical proof of that symmetrization have to be restated.

Post-divergence and post-flutter analysis of sub-tangentially loaded and damped Beck column using a nonlinear FE procedure

Both post-divergence and post-flutter behaviors of damped Beck columns subjected to a sub-tangentially follower force are rigorously explored using an exact co-rotational frame element. First a linear stability theory of the damped Beck column is summarized using a stability map. A geometrically nonlinear frame element based on the co-rotational formulation is then formulated including mass matrix, Rayleigh damping matrix, and load-correction stiffness matrix due to circulatory forces. The dynamic FE analysis is performed using Newmark integration method. Finally a Beck column model is parametrically analyzed in order to investigate non-linear stability characteristics of the internally and externally damped non-conservative system. In particular, interesting nonlinear behaviors of Beck column quite different from those predicted by the linear theory are reported through static and dynamic nonlinear analysis with variation of sub-tangentiality of the follower force.

Erratum: Defining the frame of minimum non-linear Hubble expansion variation

We characterize a cosmic rest frame in which the monopole variation of the spherically averaged nonlinear Hubble expansion is most uniform, under arbitrary local Lorentz boosts of the central observer. Using the COMPOSITE sample of 4534 galaxies, we identify a degenerate set of candidate minimum nonlinear variation frames, which includes the rest frame of the Local Group (LG) of galaxies, but excludes the standard Cosmic Microwave Background (CMB) frame. Candidate rest frames defined by a boost from the LG frame close to the plane of the galaxy have a statistical likelihood similar to the LG frame. This may result from a lack of constraining data in the Zone of Avoidance. We extend our analysis to the Cosmicflows-2 (CF2) sample of 8162 galaxies. While the signature of a systematic boost offset between the CMB and LG frame averages is still detected, the spherically averaged nonlinear expansion variation in all rest frames is significantly larger in the CF2 sample than would be reasonably expected. We trace this to the CF2 distances being reported without a correction for inhomogeneous distribution Malmquist bias. Systematic differences in the inclusion of the large SFI++ subsample into the COMPOSITE and CF2 catalogues are analysed. Our results highlight the importance of a careful treatment of Malmquist biases for future peculiar velocities studies, including tests of the hypothesis of Wiltshire et al [Phys. Rev. D 88 (2013) 083529; arXiv:1201.5371] that a significant fraction of the CMB temperature dipole may be nonkinematic in origin.

Towards a Reconceptualization of “(Post)Transitional” South African Cultural Expression

In this paper, I problematize the notion of the “post-transitional” (Samuelson 2008; Frenkel and MacKenzie 2010) as a way of theorizing and delineating recent South African cultural expression. I argue that this idea relies – whether intentionally or not – on a limiting conception of South Africa’s contemporary socio-political history and transition in terms that are inevitably linear and teleological. I propose that contemporary cultural expression in South Africa (and indeed across Africa) may be productively considered, in broad terms, as literatures of transition. This is an overarching non-linear and non-teleological continuum embracing multiple transitions that are not necessarily discrete or mutually exclusive, and that can be delineated in connection with specific contexts and moments. Without rejecting the general features and trends of recent South African cultural expression identified under the problematical rubric of “post-transitional,” I propose that the emergent post-2000 literatures be theorized as “post-TRC.” This highlights the significance of the Truth and Reconciliation Commission as an influential point of reference without ignoring the wider, non-linear, and non-teleological frame of on-going multiple transitions – social, political, economic, and cultural.

Identification of model-free structural nonlinear restoring forces using partial measurements of structural responses

Identification of nonlinear structural system is an important but challenging task for structural health monitoring. Due to the complexities of structural nonlinearities, it is hard to establish proper mathematical models for some structural nonlinear behaviors. Moreover, only partial structural responses can be measured in practice; it is essential to conduct identification of nonlinear structural systems using only partial measurements of structural responses. To cope with these issues, an algorithm is proposed in this article for the identification of some model-free structural nonlinear restoring forces using only partial measurements of structural responses. First, an equivalent linear structural system is introduced for the identification of the locations of structural nonlinearities. Then, a model-free structural nonlinear restoring force is approximated by a power series polynomial. The unknown coefficients of the power series polynomials together with other structural parameters are identified by the extended Kalman filter so that the characteristics of the behaviors of the model-free of nonlinear restoring forces can be identified. Some numerical examples including the identification of two nonlinear multi-story shear frames and a planar nonlinear truss with different structural nonlinear restoring forces are used to validate the proposed algorithm.