Plane Frame Structure Research Articles

Free vibration analysis of classical single-story multi-bay planar frames

This paper concerns free vibration analysis of single-story multi-bay planar frame structures. An exact analytical solution is obtained using a wave vibration approach, in which vibrations are described as waves propagating along uniform structural elements and are reflected and transmitted at structural discontinuities. The coupling effects between bending and longitudinal vibrations in the multi-bay frames are taken into account. Both natural frequencies and modeshapes are obtained. Numerical examples are presented along with comparisons to results available in the literature.

Simplified collapse capacity assessment of earthquake excited regular frame structures vulnerable to P-delta

In this paper the prediction of the global collapse capacity of earthquake excited regular moment-resisting frame structures is discussed. The addressed flexible planar frame structures with non-deteriorating inelastic component behavior are vulnerable to the destabilizing effect of gravity loads. A simplified collapse assessment methodology based on pushover analyses, equivalent single-degree-of-freedom systems, and a collapse capacity spectrum is proposed. This methodology, which is referred to as collapse capacity spectrum methodology, is examined for a series of frame structures covering a wide range of structural parameters. Therefore, the global collapse capacity of these structures derived with this methodology and with the computationally expensive incremental dynamic analysis procedure is set in contrast. From the outcomes it can be concluded that in the initial design process the collapse capacity spectrum methodology is an appropriate tool to assess reliably the median and the dispersion of collapse capacity of P-delta sensitive regular moment-resisting frame structures subjected to severe earthquake excitation.

DETECTION OF DAMAGE LOCATION USING A NOVEL SUBSTRUCTURE-BASED FREQUENCY RESPONSE FUNCTION APPROACH WITH A WIRELESS SENSING SYSTEM

This paper intends to detect the damage locations for building structures under an earthquake excitation using a novel substructure-based FRF approach with a damage location index (SubFRFDI). An Imote2.NET-based wireless structural health monitoring system was developed and employed in the experimental studies for the sake of deployment flexibility, low maintenance cost, low power consumption, self-organization capability, and wireless communication capability. The feasibility of the proposed approach for damage detection was examined using the numerical response of a six-storey shear plane frame structure subjected to a base excitation. The results demonstrate that the SubFRFDI can be successfully used to identify the damage of different levels at a single site or multiple sites. The SubFRFDI is independent of the responses to various input earthquake excitations. Even with the addition of noises, the SubFRFDI still functions well. The feasibility and robustness of the proposed Imote2.NET-based wireless structural health monitoring system were assessed using a 1/8-scale three-storey steel-frame model. Following this, the proposed SubFRFDI was further applied to identifying the damage locations in a 1/4-scale six-storey steel structure with the proposed Imote2.NET-based wireless monitoring system. It was confirmed experimentally that good data transportation quality can be achieved via reliable data transmission and sensing protocol in identifying the structural dynamic properties, and the proposed SubFRFDI can be used to identify the damage locations effectively.

Substructure damage identification based on response reconstruction in frequency domain and model updating

A substructural damage identification approach based on dynamic response reconstruction in frequency domain is proposed with numerical and experimental verifications. The response reconstruction is based on transforming the measured responses into responses at other locations with the transmissibility matrix. Damage identification is conducted by minimizing the difference between a measured response vector and the reconstructed response vector. Measured acceleration responses from the damaged substructure and the finite element model of the intact substructure only are required in the identification algorithm. A dynamic response sensitivity-based method is used for the damage identification. Numerical studies on a seven-storey plane frame structure are conducted to validate the correctness and effectiveness of the proposed approach. The simulated damage can be identified efficiently with 10% noise effect in the measurements and random initial model errors in the finite element model of the structure. Experimental studies on a steel plane frame structure in the laboratory are performed to further verify the response reconstruction in frequency domain and the proposed substructural damage identification approach. The locations of the introduced damage are detected accurately and the stiffness reductions in the damaged elements of the target substructure are identified close to the theoretical values.

Influence of Reinforcing Steel Post-Capping Strength Deterioration on Collapse Resistance Capacity of a Single-Storey RC Planar Frame

Seismic analysis and design based on collapse probability under intense earthquakes is the main trend in the field of earthquake engineering. Accurate evaluation of collapse resistance capacity is necessary to achieve this target. Due to the accumulated deformation during seismic excitations, structural components would have different stiffness and strength deterioration characteristics. For evaluating the collapse capacity of the structure, it is important to take into account the deterioration characteristics in the analytical model. In this paper, the constitutive models for concrete and reinforcing steel, especially those adopted in OpenSees analytical software, are reviewed first. Then, a reinforced concrete single-storey planar frame structure with one bay is taken as case study and the effect of post-capping strength deterioration of reinforcing steel on evaluation results of global collapse resistance capacity is mainly investigated by incremental dynamic analysis (IDA) via OpenSees software. Analytical results show that global collapse resistance capacity will be significantly overestimated without considering the effect of post-capping strength deterioration of reinforcing steel.

Wave Analysis of In-Plane Vibrations of L-Shaped and Portal Planar Frame Structures

This paper concerns in-plane vibration analysis of coupled bending and longitudinal vibrations in L-shaped and portal planar frame structures. An exact analytical solution is obtained using wave vibration approach. The classical Euler-Bernoulli as well as the advanced Timoshenko bending theories are applied in modeling the flexural vibrations in planar frames. Reflection and transmission matrices corresponding to incident waves arriving at the “L” joint from various directions are obtained. A concise and systematic approach to both free and forced vibration analysis of coupled bending and longitudinal vibrations in L-shaped and portal planar frame structures is presented. Results are compared to the Euler-Bernoulli model results available in the literature. Good agreements have been reached.

Strength of graphenes containing randomly dispersed vacancies

In the present work, the tensile strength of graphenes containing randomly dispersed vacancies is predicted using an atomistic-based continuum progressive fracture model. The concept of the model is based on the assumption that graphene, when loaded, behaves like a plane-frame structure. The finite element method is used to model the structure of graphene and the modified Morse interatomic potential to simulate the nonlinear behavior of the C–C bonds. Randomly dispersed vacancies (1 missing atom) are introduced into graphene using a random numbers algorithm. Graphenes are subjected to incremental uniaxial tension. The model is capable of simulating fracture evolution considering defect interaction. The effects of size, chirality, defect density and defect topology on the Young’s modulus, strength and failure strain of graphenes are examined. Computed results reveal that vacancies may counterbalance the extraordinary mechanical properties of graphene, since 4.4% of missing atoms, corresponding to 13.2% of missing bonds, result in a 50% reduction in Young’s modulus and tensile strength of the material. Also found is a secondary effect of defect topology.

Difference and Application of Matrix Displacement Method and Finite Element Method in the Truss Structure Analysis

This article analyses the theories of two methods of matrix displacement and finite element, and the application of calculation in the actual structure. Then it explores the subtle differences between them: Matrix displacement method can only be used for analysis with known unit nodes force-unit nodes of the relationship between the displacement of the truss structure, and not the continuous body structure. The major differences between both are the establishment of the basic principle and solution of the emphasis. Establishing total stiffness matrix according to the geometric property of the structure and then forming integral force vector via loading conditions when matrix displacement method is used to solve the unit fixed-end force. It’s necessary to stack fixed end force; While finite element method mainly bases on the known load and the material nature of structure after structure discretization. Node force or node stress is directly got by the derivation of displacement, but the fixed end force is lost, which dropping the accuracy of the stress. To solve plane frame structure and get structural internal force, the distinguish is checked on the example.

Wave control of vibrations in multi-story planar frame structures based on classical vibration theories

An active control approach to vibrations in multi-story planar frame structures is presented in this paper. The controller is designed from a wave vibration standpoint, in which vibrations are described as waves that propagate along uniform waveguides, and are reflected and transmitted upon structural discontinuities. Regardless of the complexity of a structure, from the wave point of view it consists of only two basic types of structural components, namely, structural elements and joints. In this paper, vibrations in a multi-story planar frame structure are controlled through controlling its structural elements and its joints.

Internal Force Analyzing of Plane Frame Structure Based on Interaction Theory

Based on researchers of interaction theory and its application, mechanism of interaction among superstructure, base and foundation was analyzed and a design method was proposed. Combining with engineering construction cases, change law of internal force of superstructure in interaction system was researched through comparing results between interaction analyzing method and custom computation method. Some meaningful engineering design devices were concluded finally.

Discussion on the Method of Pushover with Cyclic Loading

The cyclic loading pushover method which can simulate the real ground motion are suggested, and the target displacements of cyclic loading are discussed to overcome the drawbacks of traditional Pushover method which cannot consider the important seismic response of plane frames under earthquake, such as cyclic loading history, duration of strong motion, and the cumulative damage of structures. Through the comparison analysis of the story drift and hysteretic energy of plane framed structures gotten by adopting cyclic loading pushover method, traditional pushover method, and dynamic elastic-plastic time-history analysis method respectively, the conclusion that cyclic loading pushover method can better reflect the loading process with ground motion and the cumulative disruptive effect caused by hysteretic energy of structure can be obtained ,which provides some references for performance evaluation of elastic-plastic analysis of seismic structures.

Substructural Response Reconstruction in Wavelet Domain

An approach for structural response reconstruction in a substructure using the unit impulse response function in the wavelet domain is presented. The forces at the interface degrees-of-freedom are taken as input excitations on the substructure. The response reconstruction is conducted by transforming the measured responses into responses at other selected locations. The response reconstruction using the proposed wavelet domain method is then compared with that from an existing frequency domain method. Numerical studies on a seven-story plane frame structure are carried out to investigate the accuracy and effectiveness of the proposed method, and the effects of influencing parameters such as the number of sampled data, sampling rate, sensor numbers, and measurement noise are studied.

Erratum for “Equilibrium-Based Finite-Element Formulation for the Geometrically Exact Analysis of Planar Framed Structures” by H. A. F. A. Santos and J. P. Moitinho de Almeida

Erratum for “Equilibrium-Based Finite-Element Formulation for the Geometrically Exact Analysis of Planar Framed Structures” by H. A. F. A. Santos and J. P. Moitinho de Almeida

Solution for Geometrically Non-Linear Elastic Deformation of Simple Frames by a Shooting Method

Based on an exact geometric nonlinear theory for plane curved beams, geometrically nonlinear equilibrium equations and boundary conditions governing the nonlinear bending of a simple plane frame structure subjected distributed loads were derived. By using the shooting method to numerically solve the boundary value problem of nonlinear ordinary differential equations, large deformation equilibrium configurations of a simple frame with both straight and the curved beam elements subjected uniformly distributed load were obtained. The theory and methodology presented can be used to analyze large deformation of plane simple frames with a variety of geometries and loadings.

Structural response reconstruction with transmissibility concept in frequency domain

This paper presents a method for structural response reconstruction in the full structure or in a substructure, using the generalized transmissibility concept in frequency domain. The response reconstruction is based on transforming the measured responses into responses at other selected locations with the transmissibility matrix. The use of transmissibility concept in a substructure for response reconstruction is introduced by taking the interface forces at the interface degrees-of-freedom as input excitations. The First-Order-Hold input approximation is used in the forward response calculation to improve the accuracy of the dynamic response analysis. Numerical studies on a seven-storey plane frame structure are conducted to investigate the accuracy and efficiency of the proposed method and the effects of influencing parameters, such as the system sampling duration, sampling rate, sensor numbers and measurement noise are studied. Accurate reconstruction is achieved in all studies when there is significant vibration in the measured responses.

Finite Element Model Updating Using Frequency Response Function of Incomplete Strain Data

A method is presented to detect changes in stiffness and mass parameters of a structure using strain data in the frequency domain. Sensitivity of the strain-based frequency response function is characterized as a function of the changes in stiffness, mass, and damping at the element level. The transfer function of a damaged structure is approximated using the measured natural frequencies of the damaged structure and the analytical mode shapes of the intact structure. These approximations result in a set of quasi-linear sensitivity equations. A sensitivity-based algorithm is proposed for finite element model updating. These sensitivities are used to select the measured frequency points for model updating and to investigate the weighting of sensitivity equations. The proposed method was successfully applied to a plane truss and a plane frame structure using numerically simulated, error-contaminated strain data.

Equilibrium-Based Finite-Element Formulation for the Geometrically Exact Analysis of Planar Framed Structures

This paper addresses the development of a hybrid-mixed finite-element formulation for the geometrically exact quasi-static analysis of elastic planar framed structures, modeled using the two-dimensional Reissner beam theory. The proposed formulation relies on a modified principle of complementary energy, which involves, as independent variables, the generalized vectors of stress resultants and displacements and, in addition, a set of Lagrange multipliers used to enforce the stress continuity between elements. The adopted finite-element discretization produces numerical solutions that strongly satisfy the equilibrium differential equations in the elements, as well as the static boundary conditions. It consists, therefore, in a true equilibrium formulation for arbitrarily large displacements and rotations. Furthermore, as it does not suffer from shear locking or any other artificial stiffening phenomena, it may be regarded as an alternative to the standard displacement-based formulation. To validate and assess the accuracy of the proposed formulation, some benchmark problems are analyzed and their solutions are compared with those obtained using the standard two-node displacement-based formulation. Numerical analyses of convergence of the proposed finite-element formulation are also included.

Differentiating damage effects in a structural component from the time response

This paper proposes an approach to differentiate the effect of local anomalies on the different load-resisting capacities of a structural component made of elastic isotropic material. The sensitivities of structural dynamic response with respect to different damage indices are derived. They are then used in a time-domain sensitivity-based algorithm for model updating using a gradient-based approach. A planar frame structure is studied in the numerical simulation. Results show that the proposed method can effectively locate and quantify the different types of damage effects, and the proposed method is insensitive to measurement noise. Laboratory verification was performed with a three-dimensional truss structure with the identified results clearly showing how much a local physical damage can change the different physical parameters that lead to the load-resisting capacities of a structural component.

In-plane Vibrations of Classical Planar Frame Structures — an Exact Wave-based Analytical Solution

In this paper, classical theories are applied in modeling in-plane vibrations in planar frame structures. An exact analytical solution has been obtained using a wave vibration approach, in which vibrations are described as waves that propagate along a uniform structural element (waveguide) and are reflected and transmitted at discontinuities (such as structural joints). The propagation, reflection, and transmission matrices, as well as the matrix relations between the injected waves and externally applied forces and moments are obtained using classical vibration theories. The coupling effect between bending and longitudinal vibrations is considered in the derivations. Assembling these matrices provides a concise and systematic approach to vibration analysis of complex in-plane coupled bending and longitudinal vibrations of planar frame structures. Numerical examples are presented, with comparisons to results available in the literature.

Consistent Regularization for Damage Detection with Noise and Model Errors

NUMBER of methods have been developed for damage detection in the last two decades. Many of the methods are based on the observation of dynamic behavior of a structure (Doebling et al. [1], Brincker et al. [2], Maeck and DeRoeck [3]), among which the sensitivity approach via a model-updating technique is commonly accepted and applied extensively in the engineering industry because of its clear mathematical background andquantitativeindications.However,thistypeofmethodisweakat accommodating the influence of measurement errors, leading to illconditioned problems, as demonstrated by Friswell et al. [4] and Humar et al. [5]. This shortcoming means that the existence and uniqueness of the solution is not ensured and numerical instability is likely to take place in the course of the solution process [6,7]. Investigations have since been conducted to deal with the illconditioned problems in model updating. Hansen [8,9] and Vogel [10] proposed regularization methods for obtaining a solution of the inverse problem. It is recognized in regularization theory that the conventional output error, which is usually the vector of differences between thecomputedandmeasuredresponses,canbemadeunrealistically small if the process of damage identification is allowed to behave badly, such that the variable has arbitrarily large deviations from the true set of parameter change, or there may be infinite sets of ill-posed solutions. A stable solution scheme can be achieved by imposing certain constraints in the form of added penalty terms with adjustable weighting parameters based on posterior knowledge. Recently,TiturusandFriswell[11]presentedthesensitivity-based model-updating method with an additional regularization criterion and computed the solutions based on the generalized singular value decomposition.Specificfeaturesoftheparameterandresponsepaths were explored when the regularization parameter varies. The four different types of spaces that arise in the solution were discussed, together with the characteristics of the families of the regularized solution. Weber et al. [12] applied the Tikhonov regularization and truncated singular value decomposition consistently to a nonlinear updating problem. Line-search and stopping criteria known from numericaloptimizationwereadaptedtotheregularizedproblem.The optimal regularization parameter was determined by generalized cross-validation. From experiences gained in model updating with laboratory test structures, the authors found that Tikhonov regularization can give the optimal solution when there is no noise or very small noise in the measurement. But in the updating procedure for a nonlinear inverse problem with the inclusion of noise and model error, the signal-tonoise ratio is getting smaller [13,14] with each iteration, and the solution obtained from a poor regularization parameter is usually unacceptable. The algorithm does not accurately converge, and the results depend strongly on the convergence criteria and tolerances. In this paper, two techniques are proposed in a new regularization method for the identification of local damages in a structure. One technique proposed a new side condition that classified the elements aspossiblydamagedandundamagedelements,whichwillbetreated differently later. The other technique restricts the range of variation of the regularization parameter, such that the regularization solution will be in a realistic range, and the correct optimal point from the curvature of the L-curve is consistently chosen to ensure a continuousconvergingprocess.Bothtechniquesmakefulluseofthe information from results obtained in previous iteration steps. A plane frame structure is studied with two damaged elements and different levels of noise and model errors to illustrate the application of the proposed method. Numerical results show that the proposed consistent regularization method is very effective at improving the results in the inverse problem with ill-conditioning, compared with the conventional Tikhonov regularization.