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Iso-parametric finite elements with linear shape functions show in general a too stiff element behavior, called locking. By the investigation of structural parts under bending loading the so-called shear locking appears, because these elements can not reproduce pure bending modes. Many studies dealt with the locking problem and a number of methods to avoid the undesirable effects have been developed. Two well known methods are the >Assumed Natural Strain< (ANS) method and the >Enhanced Assumed Strain< (EAS) method. In this study the EAS method is applied to a four-node plane element with four EAS-parameters. The paper will describe the well-known linear formulation, its extension to nonlinear materials and the modeling of material uncertainties with random fields. For nonlinear material behavior the EAS parameters can not be determined directly. Here the problem is solved by using an internal iteration at the element level, which is much more efficient and stable than the determination via a global iteration. To verify the deterministic element behavior the results of common test examples are presented for linear and nonlinear materials. The modeling of material uncertainties is done by point-discretized random fields. To show the applicability of the element for stochastic finite element calculations Latin Hypercube Sampling was applied to investigate the stochastic hardening behavior of a cantilever beam with nonlinear material. The enhanced linear element can be applied as an alternative to higher-order finite elements where more nodes are necessary. The presented element formulation can be used in a similar manner to improve stochastic linear solid elements.
In the context of finite element model updating using output-only vibration test data, natural frequencies and mode shapes are used as validation criteria. Consequently, the correct pairing of experimentally obtained and numerically derived natural frequencies and mode shapes is important. In many cases, only limited spatial information is available and noise is present in the measurements. Therefore, the automatic selection of the most likely numerical mode shape corresponding to a particular experimentally identified mode shape can be a difficult task. The most common criterion for indicating corresponding mode shapes is the modal assurance criterion. Unfortunately, this criterion fails in certain cases and is not reliable for automatic approaches. In this paper, the purely mathematical modal assurance criterion will be enhanced by additional physical information from the numerical model in terms of modal strain energies. A numerical example and a benchmark study with experimental data are presented to show the advantages of the proposed energy-based criterion in comparison to the traditional modal assurance criterion.
In the context of finite element model updating using vibration test data, natural frequencies and mode shapes are used as validation criteria. Consequently, the order of natural frequencies and mode shapes is important. As only limited spatial information is available and noise is present in the measurements, the automatic selection of the most likely numerical mode shape corresponding to a measured mode shape is a difficult task. The most common criterion to indicate corresponding mode shapes is the modal assurance criterion. Unfortunately, this criterion fails in certain cases. In this paper, the pure mathematical modal assurance criterion will be enhanced by additional physical information of the numerical model in terms of modal strain energies. A numerical example and a benchmark study with real measured data are presented to show the advantages of the enhanced energy based criterion in comparison to the traditional modal assurance criterion.
Numerical models and their combination with advanced solution strategies are standard tools for many engineering disciplines to design or redesign structures and to optimize designs with the purpose to improve specific requirements. As the successful application of numerical models depends on their suitability to represent the behavior related to the intended use, they should be validated by experimentally obtained results. If the discrepancy between numerically derived and experimentally obtained results is not acceptable, a model revision or a revision of the experiment need to be considered. Model revision is divided into two classes, the model updating and the basic revision of the numerical model. The presented thesis is related to a special branch of model updating, the vibration-based model updating. Vibration-based model updating is a tool to improve the correlation of the numerical model by adjusting uncertain model input parameters by means of results extracted from vibration tests. Evidently, uncertainties related to the experiment, the numerical model, or the applied numerical solving strategies can influence the correctness of the identified model input parameters. The reduction of uncertainties for two critical problems and the quantification of uncertainties related to the investigation of several nominally identical structures are the main emphases of this thesis. First, the reduction of uncertainties by optimizing reference sensor positions is considered. The presented approach relies on predicted power spectral amplitudes and an initial finite element model as a basis to define the assessment criterion for predefined sensor positions. In combination with geometry-based design variables, which represent the sensor positions, genetic and particle swarm optimization algorithms are applied. The applicability of the proposed approach is demonstrated on a numerical benchmark study of a simply supported beam and a case study of a real test specimen. Furthermore, the theory of determining the predicted power spectral amplitudes is validated with results from vibration tests. Second, the possibility to reduce uncertainties related to an inappropriate assignment for numerically derived and experimentally obtained modes is investigated. In the context of vibration-based model updating, the correct pairing is essential. The most common criterion for indicating corresponding mode shapes is the modal assurance criterion. Unfortunately, this criterion fails in certain cases and is not reliable for automatic approaches. Hence, an alternative criterion, the energy-based modal assurance criterion, is proposed. This criterion combines the mathematical characteristic of orthogonality with the physical properties of the structure by modal strain energies. A numerical example and a case study with experimental data are presented to show the advantages of the proposed energy-based modal assurance criterion in comparison to the traditional modal assurance criterion. Third, the application of optimization strategies combined with information theory based objective functions is analyzed for the purpose of stochastic model updating. This approach serves as an alternative to the common sensitivity-based stochastic model updating strategies. Their success depends strongly on the defined initial model input parameters. In contrast, approaches based on optimization strategies can be more flexible. It can be demonstrated, that the investigated nature inspired optimization strategies in combination with Bhattacharyya distance and Kullback-Leibler divergence are appropriate. The obtained accuracies and the respective computational effort are comparable with sensitivity-based stochastic model updating strategies. The application of model updating procedures to improve the quality and suitability of a numerical model is always related to additional costs. The presented innovative approaches will contribute to reduce and quantify uncertainties within a vibration-based model updating process. Therefore, the increased benefit can compensate the additional effort, which is necessary to apply model updating procedures.