Flexibility Matrices Research Articles

Efficient dynamic topology optimization of 2D metamaterials based on a complementary energy formulation

The advent of additive manufacturing has revolutionized the design and development of hierarchical structures, with potential applications in compliant, auxetic, and band-gap structures. This paper presents an innovative approach to developing a dynamic Topology Optimization (TO) framework for designing printable lattice structures that exhibit specific dynamic properties. Utilizing parametrically defined filament-based unit cell structures for topology optimization, we achieve desired natural frequency bandgaps in the structures composed of these unit cells. To enhance computational efficiency, we employ a complementary energy-based formulation to (semi)analytically derive the flexibility and stiffness matrices of the unit cell structure, thus, eliminating extensive finite element discretization. Consequently, a wide variety of parametrically defined filament-based meso-structures can be mathematically explored. We apply this innovative framework specifically for band-gap maximization of 2D lattice structures. By tuning the geometry within each cell using TO, we maximize the band gap. Our results show the potential of this approach to create more efficient and effective hierarchical structures with desired band-gap properties.

Cycle Bases of Graphs for the Analysis of Frames Structures Using Force Method

The formation of suitable cycle bases corresponding to sparse flexibility matrices for the force method of frame analysis has always been an interesting problem in structural mechanics. These cycle bases are needed for the formation of static bases for efficient force method of structural analysis. Similarly, such bases are required in the mesh analysis of other networks. This paper reviews methods for the cycle basis selection by utilizing different embeddings on higher dimensional topological spaces, and using the ideas and concept from this study, graph theory algorithms are developed for efficient computational algorithms for the formation of subminimal, and minimal cycle bases.

Design and Analyses of a Novel Reconfigurable Logging Branch Mechanism

A reconfigurable logging and branch-lopping mechanism has been designed to shorten the production cycle, reduce work intensity, and improve labor survival rate in the logging industry. The mechanism integrating logging and branch-lopping functions can flexibly morph into three con-figurations, namely, series, parallel and mixed, as well as eight structure states to satiate different requirements. Not only does it possess sufficient stiffness to fell large trees but it also offers high flexibility to trim and crop small branches which superior performance has relatively wide engineering application prospects. The kinematics and static stiffness of the mechanism are the key techniques for its engineering application. With the mechanism as the research object, the working principles were analysed and the adjacency matrix of each structure state was presented. The Screw Theory and CGK formula were deployed to calculate the degrees of freedom of some of the structure mechanisms. The kinematics of some of the structure states was analysed and their flexibility matrices were obtained with vector algebra. The static stiffness of two structure states was compared indirectly by contrasting their flexibility matrices before their kinematics was simulated numerically. Engineering application of the mechanism in welding robots, robots that pick up objects, cleaning robots and other fields is demonstrated towards the end of this paper.

RITZ METHOD IN THE DISCRETE APPROXIMATION OF DISPLACEMENTS FOR SLAB CALCULATION

Introduction: The FEM reduces the problem of structural analysis for various building structures to the formation and solution of a system of linear algebraic equations. For this purpose, there are techniques available for obtaining FE stiffness and flexibility matrices where the main structural deformation characteristics are taken into account. However, the FEM can also be considered as a special case of the Ritz method in the discrete approximation of the required functions. In the functional of full potential deformation energy with regard to the considered structure, all adopted stress-strain state characteristics are taken into account. Since it is difficult or impossible to find continuous approximation functions both in the classic version of the Ritz method and in the Bubnov–Galerkin method for some types of edge restraint in such building structures as beams, slabs, or shells, it is possible to use the Ritz method in the discrete approximation of the required functions (by analogy with the FEM). This paper presents a method of such calculations using slab calculations as an example. It is shown that, due to introducing some notations (operators), the process of finding the coefficients of the system of linear algebraic equations creates no difficulties and is easily programmable. The proposed method is not an alternative to the FEM, which is the most effective numerical method for the calculation of complex three-dimensional building structures. Purpose of the study: We aimed to create a method for calculating slabs by the Ritz method in the discrete approximation of the deflection function for edge restraint cases when it is difficult or impossible to find continuous approximation functions in the classic version of the Ritz method and the Bubnov–Galerkin method. Methods: Based on the application of the Ritz variational method in the discrete approximation of displacements for slab calculation, all the basic relations for rectangular finite elements with 12 degrees of freedom are obtained, and an algorithm for forming the coefficients of the system of linear algebraic equations is developed. Results: For the first time, the solution by the Ritz method in the discrete approximation of slab displacements is obtained for the case when two edges of the slab are rigidly restrained and other two edges are free. In this case, the correct solution of the above problem is possible only with the use of the proposed method and FEM. For the test problem, we performed a comparison of the results of the calculation using the proposed method with the results using the classic Ritz method, which showed their very close agreement. The accuracy of the obtained results was assessed.

A damage-informed neural network framework for structural damage identification

In this work, an effective Damage-Informed Neural Network (DINN) is first developed to pinpoint the position and extent of structural damage. Instead of resolving the damage identification problem by conventional numerical methods, a Deep Neural Network (DNN) is employed to minimize the loss function which is designed by combining multiple damage location assurance criterion and flexibility matrices to guide the training process. In our computational framework, the parameters of the network, which include both weights and biases, are treated as new design variables instead of damage ratios. Therein, the training data consists only of a set of spatial coordinates of elements, whilst corresponding the damaged ratios of elements unknown to the network are factored into the output. To achieve the goal, the loss value is calculated relying on the predicted damage ratios with supporting Finite Element Analysis (FEA). Additionally, Bayesian Optimization (BO) algorithm is used to automatically tune hyperparameters of the network for enhancing reliability in damage identification. Several numerical examples for damage localization of truss and frame structures are investigated to evaluate the effectiveness and reliability of the suggested methodology. The obtained results point out that our model not only correctly locates the actual damage sites but also requires the least number of structural analyses and faster convergence rate compared with other algorithms.

CHARACTERIZING SPINAL LOAD-DISPLACEMENT BEHAVIOUR USING REPRESENTATIVE COMPONENTS

AbstractObjectivesSpinal stiffness and flexibility terms are typically evaluated from linear regression of experimental data and are then assembled into 36-element matrices. Summarising in vitro test results in this manner is quick, computationally cheap and has the distinct advantage of outputting simple characteristic values which make it easy to compare results. However, this method disregards many important experimental features such as stiffening effects, neutral and elastic zones magnitudes, extent of asymmetry and energy dissipation (hysteresis). Alternatives to the linear least squares method include polynomials, separation of the load-displacement behaviour into the neutral and elastic zones using various deterministic methods and variations on the double sigmoid and Boltzmann curve fits. While all these methods have their advantages, none provide a comprehensive and complete characterisation of the load-displacement behaviour of spine specimens. In 1991, Panjabi demonstrated that the flexion-extension and mediolateral bending behaviour of functional spinal units could be approximated using the viscoelastic model consisting of a nonlinear spring in series with a linear Kelvin element. Nowadays viscoelastic models are mainly used to describe creep and stress relaxation, rather than for cyclic loading. The aim of this study was to conclusively prove the viscoelastic nature of spinal behaviour subject to cyclic loading. Being able to describe the behaviour of spine specimens using springs and dampers would yield characterising coefficients with recognisable physical meaning, thus providing an advantage over existing techniques.MethodsSix porcine isolated spinal disc specimens (ISDs) were tested under position and load control. Visual inspection of the load-displacement graphs from which the principal terms of the stiffness and flexibility matrices are derived suggest that the load-displacement behaviour could be idealised by a nonlinear spring system with damping. It was hypothesised that the contributions arising from non-linear spring-like behaviour and damping could be separated for each of the principal load-displacement graphs.ResultsThe principal elements from position and load control tests were plotted with load on the vertical axis and displacement on the horizontal axis, and a polynomial representing spring force was fitted to the data. This polynomial was subtracted from the experimental data. The remainder - representing damping force - was plotted against displacement and velocity and compared to idealised plots for different types of damping behaviour. Applying this reasoning to the six principal load-displacement plots obtained from stiffness and flexibility tests revealed that four of six exhibit behaviour which is a combination of a nonlinear hardening spring and viscous damper. However, for flexion-extension and mediolateral bending, the damping behaviour is more akin to coulomb damping with viscous influences, characterised by a steep central linear region and near horizontal regions at the extremes. These trends were common amongst all six specimens within the study sample size.ConclusionsThe unique approach described here has allowed the characterisation of the load-displacement behaviour of ISDs in terms of characteristic mass-spring-damper systems. This is an important development which allows experimental findings to be framed in terms of mechanical contributions to a given load-displacement behaviour.Declaration of Interest(b) declare that there is no conflict of interest that could be perceived as prejudicing the impartiality of the research reported:I declare that there is no conflict of interest that could be perceived as prejudicing the impartiality of the research project.

An FDD-based modal parameter-less proportional flexibility-resembling matrix for response-only damage detection

Modal flexibility-based methods are effective tools for vibration-based structural damage detection, including in the output-only case. These methods are typically characterized by two stages: first, the modal parameters are identified, thus obtaining a certain number of modes; second, these modal parameters are used to assemble the modal flexibility matrix. This paper proposes a method for estimating a matrix that approximates a proportional flexibility matrix, termed proportional flexibility-resembling (PFR) matrix, and shows that this matrix can be used for damage detection and localization purposes. This matrix is obtained through signal processing operations to be executed after applying the first steps of the frequency-domain decomposition (FDD) technique—i.e., after the singular value decomposition of the spectral density matrix. The defining aspect of the PFR matrix is that, differently from the traditional formulation of modal flexibility and proportional flexibility matrices, it can be assembled without the need of an explicit identification of the modal parameters. In fact, the matrix is estimated by processing all first singular vectors and also all first singular values in a selected frequency range. In the proposed method, the typical two stage approach of traditional modal flexibility methods is avoided, and the intervention of an operator is limited to setting the values of a few parameters in the initial phase of the process. Numerical simulations and experimental data from a testbed structure were used to show the effectiveness of the proposed approach, and the analyses were performed by considering structures with different damage scenarios and damping properties.

Lateral response of a pile group embedded in the transversely isotropic saturated soil due to surcharge loads

This paper investigates the behavior of pile groups in the transversely isotropic saturated soil under an adjacent surcharge load using a coupling method. The analytical layer-element theory is utilized to the transversely isotropic saturated soil model, and the fundamental solutions of the soil are applied to derive the soil flexibility matrices. And the FEM is used for the piles modeled as Bernoulli–Euler beams considering pile–pile interaction. Then, the dynamic response equations of pile groups are derived by using the FEM–BEM coupled method and considering pile-soil boundary coordination condition. The results of the present model compared well with those in existing solutions. Numerical examples are presented to examine the influences of the soils’ transverse isotropy, the magnitude of the surcharge load, the pile's physical properties and the distance between the load and the pile on the behavior of pile groups. It is observed that as the surcharge load grows, the position of the pile maximum displacement response changes from the middle of the soft soil layer to the top of the pile. The impact of the surcharge load is not significant when the loading action distance is greater than twice the thickness of soft soil layer. Furthermore, relatively large bending moments appear in the middle of the soft soil layer and at the interface of the soil layers.

Numerical analysis for isolation of vibrations by pile rows in transversely isotropic saturated media due to moving loads

An analytical layer-element–boundary-element coupling approach is presented to explore the isolation effect of pile rows in stratified transverse isotropic saturated media for moving-load induced vibration. The analytical layer-element basic solution for the soils is derived. The soil flexibility matrices are obtained using boundary integral equations for pile-soil. Then, the pile is modeled as a Bernoulli–Euler beam utilizing the finite difference approach. The dynamic equations for the piles are achieved by considering the pile-soil displacement coordination condition. Finally, to obtain the amplitude reduction ratio, the displacement of an observation point is calculated with and without pile isolation, respectively. The results are in good consistency with the existing solutions. The impacts of the soils’ transverse isotropy, the pile’s size and physical properties on the isolation effect are evaluated using numerical examples. It is observed that when the pile length exceeds a certain value, the increment in the pile length shows no obvious improvement in the vibration isolation effect. A stiffer pile has a greater impact on vibration reduction. The material anisotropy is also an important parameter for the pile isolation problem.

Vibration isolation of row of piles due to moving harmonic loads embedded in the layered isotropic saturated medium

AbstractThis paper develops an analytical layer element‐boundary element coupling method to study the vibration isolation effect of pile rows in multi‐layered isotropic saturated soils subjected to a moving harmonic load. The isotropic saturated soil is represented using an analytical layer‐element solution as a kernel function, and the pile‐soil boundary integral equation is applied to obtain the soil flexibility matrices. The finite difference method is used for the piles modeled as a Bernoulli–Euler beam. The dynamic response equations of the pile rows under a harmonic moving load are achieved by considering pile‐soil displacement coordination condition. Then, the displacements of the observation point with and without the pile effect are calculated, respectively. Several numerical examples are presented to examine the influences of the soils’ stratification characteristics, the pile length, pile spacing, pile‐soil Young's modulus ratio, and the load moving speed on the isolation effect of the pile rows.

A COMPREHENSIVE STUDY OF LIMITATIONS ASSOCIATED WITH SIX DEGREES OF FREEDOM SPINE TESTING

Lower back pain (LBP) is a worldwide clinical problem and a prominent area for research. Numerous in vitro biomechanical studies on spine specimens have been undertaken, attempting to understand spinal response to loading and possible factors contributing to LBP. However, despite employing similar testing protocols, there are challenges in replicating in vivo conditions and significant variations in published results. The aim of this study was to use the University of Bath (UoB) spine simulator to perform tests to highlight the major limitations associated with six degree of freedom (DOF) dynamic spine testing.A steel helical spring was used as a validation model and was potted in Wood's metal. Six porcine lumbar spinal motion segments were harvested and dissected to produce isolated spinal disc specimens. These were potted in Wood's metal, ensuring the midplane of the disc remained horizontal and then sprayed with 0.9% saline and wrapped in saline-soaked tissue and plastic wrap to prevent dehydration. A 400N axial preload was used for spinal specimens. Specimens were tested under the stiffness and flexibility protocols.Tests were performed using the UoB custom 6-axis spine simulator with coordinate axes. Tests comprised five cycles with data acquired at 100Hz. Stiffness and flexibility matrices were evaluated from the last three motion cycles using the linear least squares method.According to theory, inverted flexibility matrices should equal stiffness matrices. In the case of the spring, the matrices matched analytical solutions and inverted flexibility matrices were equivalent to stiffness matrices. Matrices from the spinal tests demonstrated some symmetry, with similarities between inverted flexibility- and stiffness matrices, though these were unequal overall. Matrix element values were significantly affected by displacements assumed to occur at disc centre.Spring tests proved that for linear, elastic specimens, the spine simulator functioned as expected. However, multiple factors limit the confidence in spine test results. Centre of rotation, displacement assumptions and rigid body transformations are known to impact the results from spinal testing, and these should be addressed going forward to improve the replication of in vivo conditions.

A coupled FETI-BDNM for solving 3D elastic frictional contact problem

Based on the B-differentiable Newton method (BDNM) and finite element tearing and interconnecting (FETI) algorithm, a coupled approach named FETI-BDNM is developed to solve the 3D elastic contact problem with friction. In this method, the contact bodies are decomposed into a number of subdomains. The interfaces between the two adjacent substructures will be skillfully considered as the special contact surfaces that are always in the stick state. By resorting to the generalized inverse of stiffness matrices of the subdomains, the relationships between the relative displacements and forces on contact/interface, i.e., contact flexibility matrices will be computed efficiently. The contact equations, displacement continuity equations and equilibrium equations for each subdomain are expressed in the form of B-differentiable equations (BDEs) uniformly, which can be solved by the damped Newton-Raphson method with 1D Armijo type linear search. The proposed method makes it possible to solve large-scale contact problems without compromising the solution accuracy compared to the original BDNM, and the good convergence of BDEs and parallel scalability of the FETI algorithm will be preserved. Numerical examples validate the accuracy, efficiency, robustness and effects of the number of subdomains on the computational time of the proposed method.

Exact Winkler Solution for Laterally Loaded Piles in Inhomogeneous Soil

A novel exact analytical solution is derived for the equation y(4)+xny=0 in the region x≥0, which is important for the analysis of piles in soil with stiffness varying with depth. To date, exact solutions for long piles are available only for the cases where n=−4, 0, and 1. For other values of the exponent n, solutions have formerly been obtained numerically, mainly by the finite-difference method or approximate analytical solutions. An inherent difficulty in obtaining solutions for long beams (which are used to model flexible piles) lies in the inability to isolate the regular, converging part of the solution over the singular part that diverges with increasing x. In this paper, an exact solution is derived for n>−4, focusing on the important case of semi-infinite beams. Key aspects of the problem such as the stiffness and flexibility matrices at the pile head, and the peak bending moments due to eccentrically acting lateral loads, are discussed. A novel approach for deriving Winkler spring moduli for combined force and moment loading is proposed and shown to provide good agreement with rigorous numerical continuum results.

A numerical strategy for predicting the coupling-strength of acoustic fluid-structure interaction

The coupling-strength analysis can guide the proper modeling and efficient response analysis for acoustic fluid–structure problems. In this paper, a coupling-strength factor is proposed to quantify the dynamic interaction between acoustic fluid and structure, and its efficient computational strategy is established via a fast algorithm. This dimensionless factor expresses the average difference between the uncoupled and the coupled structural displacement (acoustic pressure) fields. The factor expression is related to the coupling matrix and the dynamic flexibility matrices of each physical field for the discrete system. Then a fast dynamic flexibility matrix algorithm of the non-positive definite system is deduced to ensure the feasibility of numerically calculating the factor, and the effects of the unconstrained structure and the acoustic rigid wall on the coupling-strength can be observed. Additionally, the strongly decoupling eigen-equations of each physical field are derived by the dynamic flexibility matrix expansion. Solving the resonance frequencies is adopted to verify the correctness of the proposed coupling-strength analysis strategy. In numerical examples, the coupling-strength factor is high-efficiently calculated with acceptable accuracy, the coupling-strength is investigated for the various acoustic fluid mediums and different boundary conditions, and the applicability and effectiveness of the factor are verified.

Molecular-dynamic modeling applied for analysis of composite wear resistance increasing as compared with the original polymer matrix

The influence of filling on the mechanical properties of polytetrafluoroethylene (PTFE) was inves- tigated by molecular dynamic modeling. Molecular models of PTFE and its composite F4K20 were built. Energy values of intermolecular interaction were determined, stiffness and flexibility matrices of PTFE and F4K20 were obtained. It was shown that energy of intermolecular interaction of F4K20 is approximately 15 times higher in comparison with energy of intermolecular interaction of PTFE. Calculation based on model- ing showed that the introduction of the filler leads to a significant increase in the composite shear modulus in comparison with the initial matrix, which may be the reason of wear resistance increasing of polymer compo- sites.

Proportional flexibility-based damage detection for buildings in unknown mass scenarios: The case of severely truncated modal spaces

Modal flexibility-based methods are recognized in the literature as effective methods for detecting damage in structures using vibration measurements. However, the application of such methods with output-only data (as is the case of civil structures tested under ambient excitations) is challenging, because the mass normalization factors required for assembling the modal flexibility matrix cannot be estimated directly from the data. To address this issue, a method for output-only damage detection and localization in building structures that can be applied with minimal or no a-priori information about the structural masses has been proposed in a previous work by the authors. In the method, to estimate the mass distribution and to assemble the proportional flexibility matrices, the modal orthogonality relationships that involve the mode shapes and the structural masses are used to form a system of equations, and through that system of equations, an inverse problem is solved for the estimation of the mass distribution. Reducing the number of available modes, however, reduces the number of available equations, while the number of unknowns (i.e., the number of degrees of freedom) stay the same; hence, the approach cannot be applied when the modal space is severely truncated. This paper proposes a technique for estimating the mass distribution and assembling proportional flexibility matrices from output-only vibration data that overcomes the mentioned limitations and that can be applied for any number of identified modes, including the case in which the modal space is severely truncated. The proposed technique is then integrated into the above-mentioned method for output-only damage detection in building structures, to verify the applicability of the whole procedure when considering the case of severely truncated modal spaces. Numerical simulations and experimental vibration tests were used to demonstrate the validity and effectiveness of the approaches proposed in this paper.

Novel computational mathematical algorithms for structural optimization using graph-theoretical methods

PurposeThe purpose is to reduce round-off errors in numerical simulations. In the numerical simulation, different kinds of errors may be created during analysis. Round-off error is one of the sources of errors. In numerical analysis, sometimes handling numerical errors is challenging. However, by applying appropriate algorithms, these errors are manageable and can be reduced. In this study, five novel topological algorithms were proposed in setting up a structural flexibility matrix, and five different examples were used in applying the proposed algorithms. In doing so round-off errors were reduced remarkably.Design/methodology/approachFive new algorithms were proposed in order to optimize the conditioning of structural matrices. Along with decreasing the size and duration of analyses, minimizing analytical errors is a critical factor in the optimal computer analysis of skeletal structures. Appropriate matrices with a greater number of zeros (sparse), a well structure and a well condition are advantageous for this objective. As a result, a problem of optimization with various goals will be addressed. This study seeks to minimize analytical errors such as rounding errors in skeletal structural flexibility matrixes via the use of more consistent and appropriate mathematical methods. These errors become more pronounced in particular designs with ill-suited flexibility matrixes; structures with varying stiffness are a frequent example of this. Due to the usage of weak elements, the flexibility matrix has a large number of non-diagonal terms, resulting in analytical errors. In numerical analysis, the ill-condition of a matrix may be resolved by moving or substituting rows; this study examined the definition and execution of these modifications prior to creating the flexibility matrix. Simple topological and algebraic features have been mostly utilized in this study to find fundamental cycle bases with particular characteristics. In conclusion, appropriately conditioned flexibility matrices are obtained, and analytical errors are reduced accordingly.Findings(1) Five new algorithms were proposed in order to optimize the conditioning of structural flexibility matrices. (2) A JAVA programming language was written for all five algorithms and a friendly GUI software tool is developed to visualize sub-optimal cycle bases. (3) Topological and algebraic features of the structures were utilized in this study.Research limitations/implicationsThis is a multi-objective optimization problem which means that sparsity and well conditioning of a matrix cannot be optimized simultaneously. In conclusion, well-conditioned flexibility matrices are obtained, and analytical errors are reduced accordingly.Practical implicationsEngineers always finding mathematical modeling of real-world problems and make them as simple as possible. In doing so, lots of errors will be created and these errors could cause the mathematical models useless. Applying decent algorithms could make the mathematical model as precise as possible.Social implicationsErrors in numerical simulations should reduce due to the fact that they are toxic for real-world applications and problems.Originality/valueThis is an original research. This paper proposes five novel topological mathematical algorithms in order to optimize the structural flexibility matrix.

Bridge Flexibility Identification through a Reference-Free Substructuring Integration Method Driven by Mode Fitting

Flexibility plays an important role in bridge load-carrying capacity assessment. The flexibility of bridges has been extracted from the measured data of numerical impact testing and in laboratory experiments. Recently, substructuring impact testing has been increasingly investigated and developed because it avoids deploying sensors throughout bridges while obtaining the same results as full-structure testing. However, most substructuring testing methods require overlapping transition areas between the divided substructures, thereby affecting the operability and efficiency of the test. This study proposes a novel substructuring flexibility integration method that uses limited instruments to sequentially test different substructures subdivided from the whole girder bridge structure without requiring the overlapping reference points of adjacent substructures. Using the basis that the structural mode shapes satisfy the orthogonality and mode node theorem, the parameters of each substructure can be efficiently integrated and the entire flexibility matrix can be identified from the sparse flexibility matrices of the substructures. Case studies using experimental data from a cantilever beam and a continuous girder bridge are considered to verify the availability and effectiveness of the proposed method.

An implementation for 2nd-order M-N coupling and geometric stiffness adaptation in tapered beam-column elements

The present work outlines a computational approach, based on the so-called Finite Analytic Method (FAM), to deal with second-order effects in the structural analysis of tapered members accounting for geometrical nonlinearity by using only one element per physical member. The proposed modelization considers the equilibrium in the deformed configuration of a “beam-column” element, by addressing the resulting mechanical coupling between bending moment and axial force (M-N interaction). A flexibility-based method is developed, to derive the force-displacement relations of the finite element, by obtaining its flexibility and stiffness matrices. Through Gauss-Lobatto integration, the stiffness coefficients capturing second-order geometrical effects are consistently evaluated, to achieve the complete stiffness matrix. Examples and comparisons with reference studies are provided, in order to prove the effectiveness of the proposed methodology. Results on the critical load evaluation of sample tapered members are delivered.

Locating and Quantifying Damage in Beam-like Structures Using Modal Flexibility-based Deflection Changes

This paper presents an enhanced method to locate and quantify damage in beam-like structures using changes in deflections estimated from modal flexibility (MF) matrices. The method is developed from explicit relationship between a series of MF-based deflection change vectors and the damage characteristics. Based on this, three damage locating criteria are defined and used to detect and locate damage. Once the damage is located, its severity is estimated conveniently from a closed-form function. The capability of the proposed method is examined through numerical and experimental verifications on a steel beam model. The result shows that the method accurately locates and quantifies damage under various scenarios using a few modes of vibration, with satisfactory or even better results compared to those obtained from traditional static deflection-based method. The performance of the proposed method is also compared with three well-known vibration-based damage detection methods using changes in MF and modal strain energy. It is found that the proposed method outperformed the other three methods, especially for multiple damage cases. As beams can represent various structural components, the proposed method provides a promising damage identification tool targeting the application to real-life structures.