Stress Mode Shapes Research Articles

A framework for rapid fatigue hotspot localization and damage assessment of plate with multiple holes based on the fatigue damage response spectrum method

AbstractThis paper proposes a novel framework for the random vibration fatigue assessment of thin‐walled plate with multiple holes based on the fatigue damage response spectrum method. Compared with other frequency‐domain evaluation methods, this framework fully exploits the dynamic characteristics of complex structures, decoupling the external excitation characteristics from the spatial characteristics of the structural response. This approach significantly enhances evaluation efficiency by avoiding the complex calculations associated with stress response spectral moments. The proposed method is employed to evaluate the contribution of each mode to the overall damage, additionally, the stress mode shapes are used to identify and refine the mesh around fatigue hotspots. Modal damage contribution factors are proposed to identify the key modes. By leveraging both structural dimension reduction and modal reduction techniques, the proposed framework can swiftly and accurately locate fatigue hotspots within complex structures and conduct precise fatigue assessments using the Absolute Sum ‐ Square Root of the Sum of Squares hybrid method. Finite element simulation analysis is conducted on a single‐lap structure containing numerous circular openings, validating the accuracy and efficiency of the proposed stochastic vibration fatigue assessment.

Shear horizontal wave in a piezoelectric semiconductor substrate covered with a metal layer with consideration of Schottky junction effects

In this paper, the shear horizontal wave in a transversely isotropic piezoelectric semiconductor substrate covered with a metal layer are obtained. The Schottky junction, which is created by the n-type piezoelectric semiconductor and the metal with a lower Fermi level, is considered as an electrically gradient layer between the layer and the substrate. Making use of the state transfer equation method, the transfer matrix of the state vector of the space charge region is derived mathematically. Basing on the numerical calculations, we investigate the physical influences of the doping density, Schottky junction and covered layer thickness on the dispersion and attenuation and the effects of Schottky junction on the wave mode shapes. Our results show that the Schottky junction has negligible influences on the dispersion curve and displacement and stress mode shapes but evident influences on the attenuation curve and the mode shapes of electric potential, electric displacement, carrier perturbation density and the electric current density.

Fast modification-aimed stress modal analysis of thin plates with holes/notches

Thin plates are widely applied in engineering structures due to their excellent mechanical performance. This paper studies the stress modal analysis (SMA) of thin plates with holes/notches for the purpose of local structural dynamic modification (SDM). Identification of predominant components of stress state at critical locations is theoretically demonstrated. To pursue possible dynamic stress reduction, a fast modification strategy is proposed by using the SMA information, which involves a two-step procedure. First, modal participation factors (MPFs) are validated in SMA and then utilized to determine the predominant modes. Second, components of stress are evaluated for the predominant stress mode shapes (sSMSs11The superscript is to distinguish the stress from the strain in present paper.). Structural dynamic simulations of a representative L-shaped thin plate with hole/notches were implemented. The sSMSs in x-, y-, and xy- directions were compared in details. Together with MPFs, sSMSs can help to identify the dangerous stresses (both the location and the direction), and then to modify the local structure to suppress dynamic stress response. By taking full advantage of numerically-obtained sSMSs along different directions, local structural modification for dynamic stress reduction can be applied in a straightforward way with high efficiency prior to the full dynamic response analysis. This research is aimed to serve as a first step towards the development of SMA-based SDM strategy for real thin-walled structures with more complex geometric details and local changes.

A numerical insight into stress mode shapes of rectangular thin-plate structures

Thin-plate structures are widely exploited in many areas due to their excellent structural properties. This study numerically investigates the mechanics characteristics of the stress mode shapes (SMSs) in thin plates. First, theoretical basis of the SMSs is concisely illustrated for thin-plate structures. Then, finite element models are constructed in numerical simulations. The SMSs of the flat plate are compared with the conventional displacement mode shapes (DMSs). The modal participation factors (MPFs) are used to extract the predominant SMSs. The SMSs in x-, y-, xy- directions and von Mises stress are evaluated in detail. Based on results of the flat plate, local thickness and elastic modulus modifications are performed to examine the changes in SMSs. Modal assurance criterion (MAC) is utilized to evaluate the modal data. The information of SMSs from the modified plates indicates the powerful capability for detecting the local variations of structural properties. Lastly, modal order determination via modal effective mass is numerically verified for accurate dynamic stress calculation.

A Numerical Investigation on Stress Modal Analysis of Composite Laminated Thin Plates

Because of the light weight and high strength, composite laminates have many advantages in aircraft structures; however, they are frequently subjected to severe dynamic loadings during flight. To understand the dynamic properties of composite laminated thin plates at the stress scale, this paper studies the stress modal analysis (SMA) of composite laminated thin plates by finite element method (FEM). Firstly, the basic theory on SMA of composite laminates was given from the classical displacement modal analysis. Secondly, a square laminated thin plate was numerically studied to obtain some distribution laws of the stress mode shapes (SMSs) from the layup and stress component perspectives. Then, based on the characteristics of SMSs in different plies, a modified layup configuration was conducted for possible lower magnitude and more uniform distributions of SMSs. Results indicate that ±45° layups can improve the performance of SMSs of the square plate, without excessively decreasing the modal frequencies. Such fact manifests that ±45° layups are critically vital for the dynamic stress reduction of the square composite laminated plates. Modal participation factor and strain energy were evaluated to assist the determination of critical modes. Lastly, the aspect ratio of the composite plate on layup design was considered. Numerical investigation in this study can serve as a preliminary step of SMSs perspective for the analysis and optimization of dynamic composite laminates.

Local Finite Element Refinement for Accurate Dynamic Stress via Modal Information Only

Accurate stress responses are the basics for failure analysis of aerospace structures, but they still can be challenging in numerical simulations for dynamic systems. This work exploits the stress mode shapes (SMSs) in local finite element (FE) refinement for the purpose of accurate dynamic stress estimation. Toward structural FE modeling, identification of critical locations via predominant SMSs is theoretically demonstrated. To pursue more accurate dynamic stress, a strategy for local FE refinement is proposed by using the information from stress modal analysis. The strategy involves a two-step procedure. First, modal participation factors (MPFs) are used to extract the predominant modes. Second, retained SMSs are evaluated for the local remeshing. Numerical simulations of a representative thin plate with geometric details were conducted. The SMSs were investigated in detail. Together with MPFs, SMSs can help to localize the dangerous stresses and then refine the local meshes to accurately calculate stress responses. By taking full advantage of numerically obtained SMSs, local structural FE modeling can be applied in a straightforward way with high efficiency. Results indicate that SMSs can reveal the underlying mechanism that governs the overall dynamic stress response.

Ponovno razmatranje metoda normalizacije režima delovanja napona

In order to accurately interpret the results of modal analysis for the purpose of stress mode shapes (SMSs), this work revisits the normalization methods in computational structural dynamics. Firstly, the fundamental principle of structural dynamics analysis and mode of normalization were concisely introduced, and the critical parameters affecting SMSs were discussed. Secondly, the displacement and mass normalizations were applied to the case study of a beam structure. Detailed comparisons were considered between the SMSs and the conventional displacement mode shapes (DMSs) with different normalization methods. Lastly, some important characteristics were obtained. Present study can serve as an ongoing research effort aimed at evaluating both the SMSs and DMSs for future dynamic damage/failure analysis.

Hybrid SAFE-GMM approach for predictive modeling of guided wave propagation in layered media

Dispersion curve, and displacement modeshapes of multilayer structures can be obtained using Semi-Analytical Finite Element (SAFE) method. Stress mode shapes calculated from SAFE were discontinuous at the interface, because SAFE method formulation does not apply stress continuity at the interface. A case study of 1 mm aluminum-1 mm steel double-layer plate is presented, showing the SAFE stress predictions at the interface. The results observed were discontinuity of out-of-plane stress modeshapes at the interface. Mesh refinement was performed at the interface to study the convergence of stress mode shapes at the interface, yet the stress discontinuity problem was not solved. A fine mesh at the interface was created by using variable mesh techniques. This also did not provide continuous stress mode shape at the interface. Therefore, in this paper, a novel hybrid SAFE-GMM (Global Matrix Method) approach was used to obtain stress mode shapes accurately and efficiently. GMM develops the displacement and stress equations for individual layers in a multilayered structure and assembles a global matrix by applying the boundary and interface continuity conditions. A case of practical importance, CFRP strengthened concrete structure was analyzed using the SAFE-GMM approach. The drawback of the SAFE technique to this case is also presented, and it shows that the hybrid SAFE-GMM approach gave the interface stresses accurately.

Theoretical and numerical investigation of stress mode shapes in multi-axial random fatigue

Abstract Random fatigue is inherently multi-axial due to the complex stress state in real vibrating structures. This paper investigates the role of Stress Modal Analysis (SMA) in multi-axial random fatigue. Predicting hotspots of multi-axial random fatigue by Stress Mode Shapes (SMSs) was theoretically demonstrated. An improved approach was proposed to exploit SMSs in random fatigue with a multi-axial criterion. First, SMA is conducted to locate the multi-axial fatigue hotspots in a vibrating structure. Second, the frequency-domain approach for random fatigue is performed at the identified hotspots. The capability of SMSs in locating multi-axial fatigue hotspots was verified by numerical investigation. The finite element (FE) model of an L-shaped thin-walled structure containing geometry changes was constructed for case study. Local regions were identified as hotspots by SMSs information. Then, random response and multi-axial fatigue damage of the whole structure were evaluated for verification. Damage map indicates that the critical positions predicted by SMSs have good accuracy. Results show the improved approach with SMA can ensure both accuracy and efficiency for multi-axial random fatigue.

Fast evaluation of stress state spectral moments

In frequency domain evaluation of fatigue damage, (Vibration Fatigue) most of the evaluation criteria are based on spectral moments of the power spectral density function (PSD) of stress signal. In numerical simulation this means to perform a dynamic analysis in frequency domain of the whole model (i.e. Finite Element Model) and, consequently, to evaluate the stress PSD function of each element or node.In this paper the authors introduce a method to fast evaluate the spectral moments of the stress power spectral density functions of a finite element model analyzed by modal approach. The authors theoretically demonstrate and validate that the statistical properties (spectral moments) of the power spectral density functions matrix of the stress tensor of each element of a generic numerical modal model are obtainable only by the evaluation of spectral moments of the power spectral density functions matrix of the model modal coordinates and by a simple linear combination with their modal stress shapes.The proposed method obtains the same results than the standard approach, resulting much faster. Only the evaluation, one time, of the spectral moments of the modal coordinates PSD functions matrix is needed. By a simple linear combination of these with the stress mode shapes is then possible to directly obtain, for each element, stress spectral moments, avoiding a lot of integral calculations, proportional to the elements numerousness.

Utilization of modal stress approach in random-vibration fatigue evaluation

Random-vibration fatigue evaluation can be of considerable importance in the design phase of aerospace structures due to the severe dynamic loads in service. This paper presents the utilization of modal stress approach to the issue of structural random-vibration fatigue evaluation. Prognosis of random fatigue hotspots by using stress mode shapes is theoretically demonstrated. A two-step procedure is proposed for computational efficiency. Firstly, modal stress analysis is conducted to locate the fatigue hotspots in a dynamic structure. Secondly, the frequency domain-based approach for random fatigue evaluation is performed at these hotspots, as opposed to the computation of the entire structure as before. The capability of stress mode shapes to locate fatigue hotspots is verified by numerical investigations. The finite element model of a single-lap plate structure containing various opening holes was constructed for case study. Six elements were identified as hotspots by using modal stress distributions. Then, random responses and fatigue evaluation of the entire structure were carried out for verification. Good agreement was observed between the fatigue damage contour and the modal stress distributions, which can indicate that the critical positions predicted by stress mode shapes have good accuracy. The calculation time and storage space can be significantly reduced by means of the proposed evaluation procedure. Therefore, the accuracy and efficiency of utilization of modal stress approach in random fatigue evaluation can be ensured.

Spectral Formulations for Vortex Induced Vibration Modal Decomposition and Reconstruction

Modal decomposition and reconstruction (MDR) of marine riser vortex induced vibration (VIV) is a technique where vibration is measured using accelerometers and/or angular rate sensors, the modal displacements are solved for and the stress and fatigue damage is reconstructed along the riser. Recent developments have greatly increased the accuracy and reliability of the method. However the computational burden is onerous due to stress time history reconstruction and rainflow cycle counting at every desired location along the riser. In addition, fully synchronous data are required to reconstruct the stress histories. Dirlik’s method for obtaining rainflow damage for Gaussian random stress using only spectral information (four spectral automoments) has proven to be quite accurate with a significant reduction in computational effort. In this paper two spectral formulations of MDR are introduced. The first method is applicable when all the measured data are synchronous. In this method, spectral cross moments of the modal displacements are solved from the spectral cross moments of the measured data using basis vectors consisting of normal mode shapes. The spectral automoments of stress are obtained from the modal displacement cross moments and analytical stress mode shapes. Dirlik’s method is then applied to obtain rainflow damage. The second method is a generalization of the first, where the measured data cross moments are only partially known. This method is applicable when measured data are partially synchronous or asynchronous. A numerical root-finding technique is employed to solve for the modal response cross moments. The method then proceeds in the same manner as the first. The spectral methods are applied to simulated VIV data of a full-scale deepwater riser and to Norwegian Deepwater Program (NDP) scale-model test data on a 38 m long slender riser. Comparisons of reconstructed fatigue damage versus simulated or measured damage indicate that the method is capable of estimating fatigue damage accurately for Gaussian VIV even when data are not fully synchronous. It is also shown that computational cost is greatly reduced.

Overall dynamic constitutive relations of layered elastic composites

A method for the homogenization of a layered elastic composite is presented. It allows direct, consistent, and accurate evaluation of the averaged overall frequency-dependent dynamic material constitutive relations without the need for a point-wise solution of the field equations. When the spatial variation of the field variables is restricted by Bloch-form (Floquet-form) periodicity, then these relations together with the overall conservation and kinematical equations accurately yield the displacement or stress mode-shapes and, necessarily, the dispersion relations. The method can also give the point-wise solution of the elastodynamic field equations (to any desired degree of accuracy), which, however, is not required for the calculation of the average overall properties. The resulting overall dynamic constitutive relations are general and need not be restricted by the Bloch-form periodicity. The formulation is based on micromechanical modeling of a representative unit cell of the composite. For waves in periodic layered composites, the overall effective mass-density and compliance (stiffness) are always real-valued whether or not the corresponding unit cell (representative volume element used as a unit cell) is geometrically and/or materially symmetric. The average strain and linear momentum are coupled and the coupling constitutive parameters are always each others' complex conjugates. We separate the overall constitutive relations, which depend only on the composition and structure of the unit cell, from the overall field equations which hold for any elastic composite; i.e., we use only the local field equations and material properties to deduce the overall constitutive relations. Finally, we present solved numerical examples to further clarify the structure of the averaged constitutive relations and to bring out the correspondence of the current method with recently published results.

Natural Vibrations of Laminated and Sandwich Plates

An exact analytical solution based on the propagator matrix method and a semianalytical solution based on a higher-order mixed approach (displacement and stress interpolation) have been presented in this paper to evaluate the natural frequencies as well as the stress and displacement mode shapes of simply supported, cross-ply laminated and sandwich plates. Continuity of the transverse stresses and displacements has been maintained at the laminae interfaces. Results have been presented for orthotropic plates, symmetric as well as nonsymmetric cross-ply composite and sandwich laminates. Results from the propagator matrix agree well with the published results for frequencies as well as for displacement and stress mode shapes. Furthermore, the frequencies and displacement and stress eigenvectors obtained from the proposed layerwise mixed method are in excellent agreement with those obtained by three-dimensional elasticity theory. Results obtained from the present equivalent single layer theory are in good agreement with those obtained from the displacement based higher order methods. The high accuracy of the present methods is further confirmed by comparing the response of a sandwich plate with significantly different layer properties for which the conventional displacement based formulations yield inaccurate solutions.

Dynamic analysis of laminated composite plates using a layer-wise mixed finite element model

This paper deals with an accurate, three-dimensional, higher order, mixed finite element (FE) modeling for the free vibration analysis of multi-layered thick composite plates. An 18-node, three-dimensional mixed FE model has been developed by using Hamilton’s energy principle. Continuity of the transverse stress and the displacement fields has been enforced through the thickness of laminated composite plate. Further, in addition to the displacement components, the transverse stress components ( σ z , τ xz , and τ yz , where z is the thickness direction) have been invoked as the nodal degrees-of-freedom by applying elastic equations between stress and displacement fields. Thus, the present FE model has a novel feature of maintaining the fundamental elastic relationship throughout an elastic continuum. Natural frequencies of laminated composite plates obtained through the present formulation have been shown to be in good agreement with the available elasticity and closed form solutions. Stress mode shapes have been plotted for the fundamental natural vibration of plates with various lamination schemes. Strain variations through the thickness of laminated plates have also been presented graphically to show the magnitude of discontinuity in the variation of transverse strains at the layer interfaces.

Use of a microstructure theory for modeling steady-state wave propagation in layered elastic materials

Continuum theories with microstructure have been shown to be capable of modeling the dispersion behavior and displacement and stress mode shapes of the lowest propagation modes in a layered elastic material [D.S. Drumbeller and A. Bedford, Int. J. Solids Struct. 10, 61–76 (1974)]. A second-order microstructure theory has been used to solve the problem of a layered half space subjected to steady state displacement and stress boundary conditions by modal superposition using the minimum-square-error and variational solution procedures described by Wu and Plunkett [C.-H. Wu and R. Plunkett, SIAM J. Appl. Math. 15, 107–119 (1967)]. When the solutions were compared to solutions obtained by superpositions of the exact Postma-Rytov modes for the layered material, it was found that, for frequencies below the cutoff frequency of the first optical mode, the microstructure theory predicted the displacement and stress distributions in the material to within a few percent. An important finding is that, due to resonance effects, stresses in the material can substantially exceed the imposed boundary stresses. [Work supported by NSF.]