Stiffness Evaluation and Damage Detection of Ceramic Candle Filters

This paper presents a nondestructive evaluation method to identify the structural stiffness of ceramic candle filters. A ceramic candle filter is a hollow cylindrical structure made of a porous ceramic material used in advanced, coal-fired power generation systems. The candle filters need to sustain an extreme thermal and chemical environment over a great period of time to protect the gas turbine components from exposure to particulate matter. A total of 92 new candle filters and 29 used candle filters have been tested nondestructively using a dynamic characterization technique. All filters were subjected to an excitation force, and the response was picked up by an accelerometer in a free-free boundary condition. The frequency response function and vibration mode shapes of each filter were evaluated. Beam vibration equations and finite-element models were built to calculate the filter's dynamic response. Results indicate that the vibration signatures can be used as an index to quantify the structural properties of ceramic candle filters. The results also show estimations of the overall bending stiffness values for four different types of candle filters. The used filters show a trend of stiffness degradation, which was related to the filter's exposure time. Damage detection procedures using modal strain energy and finite-element simulation were studied for detection of a localized damage in the candle filter. The location and the size of the damaged section can be identified using the measured model strain energy.     

Computation of Eigenvector Derivatives Using HPDF Expression

The computation of many eigenvector derivatives is required in the fields of structural optimal design, and update and localization of model error. Existing methods, except for the incomplete truncated modal methods and a dynamic flexibility method proposed by the writers, are not efficient for the calculation of many eigenvector derivatives. The precision of the incomplete truncated modal methods sometimes is poor. For this reason, a higher-precision dynamic flexibility (HPDF) expression is first proposed. Then on the basis of this HPDF expression, both a higher-precision truncated modal method and an HPDF method have been analyzed. The HPDF method is a different form of the early dynamic flexibility method. The HPDF expression is based on a geometrical series expansion. That is, the first several terms of the series can allow the solution to converge to an exact value of the dynamic flexibility expression when the difference between each eigenvalue at the frequency axis is large. Once the concentrated eigenroots exist, the convergence speed slows. This limitation can be overcome using a shifting frequency technique.     

Dynamic Monitoring of Steel Girder Highway Bridge

Currently there are different monitoring techniques that have been considered for use in the structural evaluation of bridges. These include approaches based on both static and dynamic behavior. The use of dynamic properties has advantages over static properties, since components of the dynamic properties are only marginally influenced by variations in the loading. When dynamic properties are used, field studies have shown that it is not always sufficient to use only natural frequencies and modal displacements. Some research for structural evaluation of bridges indicates that techniques based on use of derivatives of the natural frequencies and the modal displacements may be more effectively used to generate effective diagnostic parameters for structural identification. This paper presents the results of applying one of these methods, the modal flexibility approach, to a field study of a bridge in which the bearings were partially restrained in colder weather. While others have used impact methods with the modal flexibility method, in this study the approach is modified so that excitation is provided by vehicular traffic. The results show that the modified modal flexibility method provides a clear indication that there have been changes in the bridge’s structural behavior.     

Research on mode localization of reticulated shell structures

Reticulated shell structures (RSSs) are characterized as cyclically periodic structures. Mistuning of RSSs will induce structural mode localization. Mode localization has the following two features: some modal vectors of the structure change remarkably when the values of its physical parameters (mass or stiffness) have a slight change; and the vibration of some modes is mainly restricted in some local areas of the structure. In this paper, two quantitative assessment indexes are introduced that correspond to these two features. The first feature is studied through a numerical example of a RSS, and its induced causes are analyzed by using the perturbation theory. The analysis showed that internally, mode localization is closely related to structural frequencies and externally, slight changes of the physical parameters of the structure cause instability to the RSS. A scaled model experiment to examine mode localization was carried out on a Kiewit single-layer spherical RSS,and both features of mode localization are studied. Eight tests that measured the changes of the physical parameters were carried out in the experiment. Since many modes make their contribution in structural dynamic response, six strong vibration modes were tested at random in the experimental analysis. The change and localization of the six modes are analyzed for each test. The results show that slight changes to the physical parameters are likely to induce remarkable changes and localization of some modal vectors in the RSSs.     

Cable Modal Parameter Identification. II: Modal Tests

The cable dynamic stiffness describes the load–deformation behavior that reflects the cable intrinsic dynamic characteristics. It is defined as a ratio of response to excitation and represents a very similar frequency response property to the frequency response function (FRF). Therefore, by fitting both analytical cable dynamic stiffness and measured frequency response function, the modal parameters of cables can be identified. Based on the simplified cable dynamic stiffness proposed in the first part of the two-part paper, this paper presents a cable dynamic stiffness based procedure to identify the cable modal parameters (natural frequencies and damping ratios) by modal tests. To carry out the curve fitting, a nonlinear least-squares approach is used. A numerical simulation example is first introduced to illustrate the feasibility of the proposed method. Further, a series of cable modal tests are conducted in the laboratory with different cable tensions and the frequency response functions are measured accordingly. A number of issues related to the cable modal tests have been discussed, such as accelerometer arrangement and excitation placement, frequency resolution, windowing, and averaging. It is demonstrated that the cable modal parameters can be effectively identified by using the proposed method through the cable modal tests.     

Cable Modal Parameter Identification. I: Theory

Cable modal parameters (natural frequencies and damping ratios) that represent the cable inherent dynamic characteristics play an important role in the construction, vibration control, condition assessment, and long-term health monitoring of cable-supported structures. The existing options to identify cable modal parameters through vibration measurements are somewhat limited. For this purpose, a cable dynamic stiffness based method is presented to effectively identify the cable modal parameters. In the first part of this two-part paper, the cable dynamic stiffness is analytically discussed for a viscously damped, uniform, inclined sagging cable supported at the lower end and subjected to a harmonically varying arbitrary angle displacement excitation in an arbitrary angle at the upper end when the cable is assumed to have a parabolic profile at its position of static equilibrium. Special attention is paid to the physical meaning and significance of every part of the frequency-dependent closed-form cable dynamic stiffness. Comprehensive numerical analyses have been carried out and a simplified cable dynamic stiffness is proposed for the purpose of identifying the cable modal parameters with a good accuracy over a wide range of frequencies.     

Three-Dimensional Indirect Boundary Element Method Formulation for Dynamic Analysis of Frames Buried in Semiinfinite Elastic Media

This paper presents a formulation of the indirect boundary element method based on the principle of virtual work for the dynamic analysis of frame structures buried in semi-infinite elastic media. The present formulation, which falls in the category of symmetric Galerkin boundary element methods, leads to symmetric stiffness matrices for the continuum that may be defined in terms of conventional structural analysis variables (i.e., generalized displacements and lumped forces). It is shown that, in the context of the present formulation, rotation degrees of freedom may readily be introduced in the interpolation scheme with very little additional computational effort. The consistency of the present formulation with well-established results is assessed by comparing the predictions for the static and dynamic stiffness of single piles with other results from the literature. Finally, the dynamic stiffness of a single buried frame under vertical and horizontal loading is studied. The analysis shows that the stiffness of the full frame may not always be accurately estimated by means of results for single piles, even when dynamic interaction factors are used.     

Stiffness Assessment through Modal Analysis of an RC Slab Bridge before and after Strengthening

As part of Main Roads Western Australia’s (MRWA) bridge management and bridge upgrading program, MRWA bridge no. 3014 was assessed to evaluate its condition before and after strengthening works with carbon-fiber-reinforced-polymers (CFRP). The assessment process coupled analytical results with field observations and dynamic testing of the structure. Vibration-based structural assessment of the bridge was conducted before and after the completion of the upgrading works. This paper presents the results of the vibration tests and modal analysis performed before and after the structure upgrading. In particular, the change in the structural properties and stiffness, before and after the strengthening, based on the analyses of the updated models of the bridge, is presented and discussed. The results demonstrate the effectiveness of using the dynamic assessment method to determine the elastic flexural stiffness of bridge structures retrofitted with CFRP.     

Improved Damage Quantification from Elemental Modal Strain Energy Change

An improved structural damage quantification algorithm is presented based on the elemental modal strain energy change before and after the occurrence of damage in a structure. The algorithm includes the analytical stiffness and mass matrices of the system in the damage quantification. It reduces significantly the modal truncation error and the finite-element modeling error from higher analytical modes in the computation, and it improves the convergence properties of the existing algorithm by Shi et al. (2000). “Structural damage detection from elemental modal strain energy change.” J. Eng. Mech., 126(12), 1216–1223]. The effectiveness of the proposed algorithm is demonstrated via a numerical example and experimental results from a two-storey steel portal frame, and it is demonstrated to be an efficient and robust method for damage quantification.     

Effects of Edge-Stiffening Elements and Diaphragms on Bridge Resistance and Load Distribution

This research investigates the effects of barriers, sidewalks, and diaphragms (secondary elements) on bridge structure ultimate capacity and load distribution. Simple-span, two-lane highway girder bridges with composite steel and prestressed concrete girders are considered. The finite-element method is used for structural analysis. For the elastic range, typical secondary elements can reduce girder distribution factors (GDF) between 10 and 40%, depending on stiffness and bridge geometry. For the inelastic response, steel is modeled using von Mises yield criterion and isotropic (work) hardening. Concrete is modeled with a softening curve in compression with the ability to crack in tension. At ultimate capacity, typical secondary elements can reduce GDF an additional 5 to 20%, and bridge system ultimate capacity can be increased from 1.1 to 2.2 times that of the base bridge without secondary elements, depending on bridge geometry and secondary-element dimensions.     

Bayesian Probabilistic Approach to Structural Health Monitoring

A Bayesian probabilistic methodology for structural health monitoring is presented. The method uses a sequence of identified modal parameter data sets to compute the probability that continually updated model stiffness parameters are less than a specified fraction of the corresponding initial model stiffness parameters. In this approach, a high likelihood of reduction in model stiffness at a location is taken as a proxy for damage at the corresponding structural location. The concept extends the idea of using as indicators of damage the changes in structural model parameters that are identified from modal parameter data sets when the structure is initially in an undamaged state and then later in a possibly damaged state. The extension is needed, since effects such as variation in the identified modal parameters in the absence of damage, as well as unavoidable model error, lead to uncertainties in the updated model parameters that in practice obscure health assessment. The method is illustrated by simulating on-line monitoring, wherein specified modal parameters are identified on a regular basis and the probability of damage for each substructure is continually updated.     

Discretized Subregion Variational Principle for Dynamic Substructuring

The purpose of this paper is to derive and apply the relationships of the dynamic substructure synthesis method, based on the subregion variational principle of elastodynamics in discrete form. Two theorems are derived for discretized systems and are used to demonstrate the interface compatibility condition of dynamic substructure synthesis. This enables the dynamic substructure synthesis method to be derived from the subregion variational principle. A complete modal expansion is used to derive the residual modes for both the free and fixed interface methods. The alternative formulations of the hybrid, free interface, and fixed interface methods, as well as Guyan-Irons reduction, are shown by coordinate transformations to be degenerate cases. These cases are all reasonable representations of the Rayleigh-Ritz method, and if they use constraint modes and∕or attachment modes, good convergence can be assured because the static modes are included in the substructure reduction. The relationships between these cases are explored, and illustrative examples are solved.     

Dynamic Analysis of Rigid Walls Considering Flexible Foundation

The dynamic behavior of a rigid wall is studied by the elastic approach. The analyses presented by Veletsos and Younan are extended to include foundation flexibility and damping. A closed-form analytical solution is obtained by assuming a simple backfill-foundation interface condition. The wall foundation effect is included by using well-known dynamic stiffness and radiation dashpot for a rigid strip foundation so that the wall rotation stiffness, proposed in previous works, is replaced by measurable foundation properties. It is shown that both “static” and dynamic base shears may be reduced by the foundation effects such that the base shears computed by the elastic approach may be of the same order as that estimated by the Okabe-Mononobe equation, even for a rigid gravity wall.     

Dynamic Modal Analysis and Stability of Cantilever Shear Buildings: Importance of Moment Equilibrium

The dynamic modal analysis (i.e., the natural frequencies, modes of vibration, generalized masses, and modal participation factors) and static stability (i.e., critical loads and buckling modes) of two-dimensional (2D) cantilever shear buildings with semirigid flexural restraint and lateral bracing at the base support as well as lumped masses at both ends and subjected to a linearly distributed axial load along its span are presented using an approach that fulfills both the lateral and moment equilibrium conditions along the member. The proposed model includes the simultaneous effects and couplings of shear deformations, translational and rotational inertias of all masses considered, a linearly applied axial load along the span, the shear force component induced by the applied axial force as the member deforms and the cross section rotates, and the rotational and lateral restraints at the base support. The proposed model shows that the stability and dynamic behavior of 2D cantilever shear buildings are highly sensitive to the coupling effects just mentioned, particularly in members with limited rotational restraint and lateral bracing at the base support. Analytical results indicate that except for members with a perfectly clamped base (i.e., zero rotation of the cross sections), the stability and dynamic behavior of shear buildings are governed by the flexural moment equation, rather than the second-order differential equation of transverse equilibrium or shear-wave equation. This equation is formulated in the technical literature by simply applying transverse equilibrium “ignoring” the flexural moment equilibrium equation. This causes erroneous results in the stability and dynamic analyses of shear buildings with base support that is not perfectly clamped. The proposed equations reproduce, as special cases: (1) the nonclassical vibration modes of shear buildings including the inversion of modes of vibration when higher modes cross lower modes in shear buildings with soft conditions at the base, and the phenomena of double frequencies at certain values of beam slenderness (L/r); and (2) the phenomena of tension buckling in shear buildings. These phenomena have been discussed recently by the writer (2005) in columns made of elastomeric materials.     

Efficient Model Correction Method with Modal Measurement

An efficient model correction method is proposed by using the modal measurement from a structural system. The method corrects/updates the mass and stiffness matrix without imposing any parameterization. It considers the information from both the nominal finite-element model and the measurement of modal frequencies and mode shapes. The method is computationally very efficient and it does not require computation of the complete set of eigenvalues and eigenvectors of the nominal model. Instead, only the nominal eigenvalues and eigenvectors of the modes to be corrected are needed. The Gram-Schmidt orthogonalization process is used to construct a basis that satisfies the mass orthogonality condition. This basis is used to transform the eigenvectors of the nominal model so that the corrected model is compatible with the measurement. A thousand-degree-of-freedom chainlike system and a 1,440-degree-of-freedom structural frame are used to illustrate the proposed method.     

Damage Localization and Finite-Element Model Updating Using Multiresponse NDT Data

New techniques for both finite-element model updating and damage localization are presented using multiresponse nondestructive test (NDT) data. A new protocol for combining multiple parameter estimation algorithms for model updating is presented along with an illustrative example. This approach allows for the simultaneous use of both static and modal NDT data to perform model updating at the element level. A new damage index based on multiresponse NDT data is presented for damage localization of structures. This index is based on static and modal strain energy changes in a structure as a result of damage. This method depicts changes in physical properties of each structural element compared to its initial state using NDT data. Deficient or potentially damaged structural elements are then selected as the unknown parameters to be updated by parameter estimation. Error function normalization, error function stacking, and multiresponse parameter estimation methods are proposed for using multiple data types for simultaneous stiffness and mass parameter estimation. Also, multiple sets of measurements with various sizes and missing data points can be utilized. This paper uses a laboratory grid model of a bridge deck built at the University of Cincinnati Infrastructure Institute and the corresponding NDT data for validation of the above damage localization and model updating methods. Multiresponse parameter estimation has been utilized to update the stiffness of bearing pads, and both the stiffness and mass of the connections, using static and dynamic NDT data. The static and modal responses of the updated grid model presented a closer match with the NDT data than the responses from the initial model.     

Ambient Vibration Data Analysis for Structural Identification and Global Condition Assessment

System identification is an area which deals with developing mathematical models to characterize the input-output behavior of an unknown system by means of experimental data. Structural health monitoring (SHM) provides the tools and technologies to collect and analyze input and output data to track the structural behavior. One of the most commonly used SHM technologies is dynamic testing. Ambient vibration testing is a practical dynamic testing method especially for large civil structures where input excitation cannot be directly measured. This paper presents a conceptual and reliable methodology for system identification and structural condition assessment using ambient vibration data where input data are not available. The system identification methodology presented in this study is based on the use of complex mode indicator functions (CMIFs) coupled with the random decrement (RD) method to identify the modal parameters from the output only data sets. CMIF is employed for parameter identification from the unscaled multiple-input multiple-output data sets generated using the RD method. For condition assessment, unscaled flexibility and the deflection profiles obtained from the dynamic tests are presented as a conceptual indicator. Laboratory tests on a steel grid and field tests on a long-span bridge were conducted and the dynamic properties identified from these tests are presented. For demonstrating condition assessment, deflected shapes obtained from unscaled flexibility are compared for undamaged and damaged laboratory grid structures. It is shown that structural changes on the steel grid structure are identified by using the unscaled deflected shapes.     

Nonlinear Dynamics of Structures Via Residual Flexibility of Components

An approach for the analysis of systems comprising multiple components subjected to dynamic loading is presented. It allows for an efficient treatment, stepwise in time, of linear and nonlinear connections between components. The constraint forces at the junctions of the components are computed directly without the synthesis of component modes of the determination of system modes. This is accomplished by expressing the displacements at the junction coordinates of the components in terms of the retained unconstrained normal modes and the residual flexibility of the unretained modes, in conjunction with a Newmark algorithm representation of nodal kinematics within a time step. This leads to a set of junction-sized equations, similar in form to that of the flexibility formulation in statics, in terms of the unknown junction forces. For the linear problem, the connection forces are solved for directly. For the nonlinear problem, the connection forces are determined in an iterative manner. The approach is applied to a problem involving the dynamic response of a Mini-Pressurized Logistic Module (MPLM) rack in a Space Shuttle liftoff event. The results of the proposed approach are compared with pertinent results derived by relying on component-mode synthesis.     

Effects of Size and Slenderness on Ductility of Fracturing Structures

The ductility of an elastic structure with a growing crack may be defined as the ratio of the additional load-point displacement that is caused by the crack at the moment of loss of stability under displacement control to the elastic displacement at no crack at the moment of peak load. The stability loss at displacement control is known to occur when the load-deflection curve of the whole structural system with the loading device (characterized by a spring) reaches a snapback point. Based on the known stress intensity factor as a function of crack length, the well-known method of linear elastic fracture mechanics is used to calculate the load-deflection curve and determine the states of snapback and maximum loads. An example of a notched three-point bend beam with a growing crack is analyzed numerically. The ductility is determined and its dependence of the structure size, slenderness, and stiffness of the loading device is clarified. The family of the curves of ductility versus structure size at various loading device stiffnesses is found to exhibit at a certain critical stiffness a transition from bounded single-valued functions of D to unbounded two-valued functions of D. The method of solution is general and is applicable to cracked structures of any shape. The flexibility (force) method can be adapted to extend the ductility analysis to structural assemblages provided that the stress intensity factor of the cracked structural part considered alone is known. This study leads to an improved understanding of ductility, which should be useful mainly for design against dynamic loads.     

Enhanced Realization∕Identification of Physical Modes

Physical structures are often sufficiently complicated to preclude constructing an accurate mathematical model of the system dynamics from simple analysis using the laws of physics. Consequently, determination of an accurate model requires utilization of (generally noisy) output measurements from dynamic tests. In this paper, we present a robust method for constructing accurate, structural‐dynamic models from discrete time‐domain measurements. The method processes the measurements in order to determine the number of modes present, the damping and frequency of each mode, and the mode shape. The structure may be highly damped. Although the mode‐shape identification is more sensitive to measurement noise than the order, frequency, and damping identification, the method is considerably less sensitive to noise than other leading methods. Accurate detection of the modal parameters and mode shapes is demonstrated for modes with damping ratios exceeding 15%.