Formulation Of Stiffness Matrix Research Articles

Exact closed-form dynamic stiffness matrix of damaged frames comprising Timoshenko–Ehrenfest beams

The present work proposes the extension of an exact distributional model for cracked beams to include the local influence on axial deformation on shear deformable beams. Axial, shear and flexural concentrated flexibilities are taken into account to model the damaged sections. Shear deformability is accounted for by adopting the Timoshenko–Ehrenfest model and exact closed-form solution of the mode shapes for multiple cracked beams are derived. On this ground, the final achievement of the work consists in the original formulation of the Dynamic Stiffness Matrix (DSM) with multiple cracked sections in exact closed form. The closed-form expression of the DSM allows the assemblage of the global DSM of a structure to conduct the free vibration analysis of damaged framed structures as well as for the study of frames with semi-rigid joints. This approach has the advantage to avoid any numerical procedure for the construction of the DSM as well as the introduction of further degrees of freedom (dofs) in correspondence of the damaged sections, keeping the size of the problem the same as in the intact configuration. Natural frequencies of the frames are determined by means of the Wittrick–Williams algorithm (W–W), then the mode shapes and modal forces are subsequently obtained in closed form. The proposed approach is validated with results available in literature for two portal frames characterized by the presence of cracks and semi-rigid joints, respectively. Finally, the potentiality of the approach, also in view of practical applications, is demonstrated analysing a multi-storey frame with semi-rigid connections.

Propagation and scattering of Rayleigh wave from an interfacial crack in a viscoelastic waveguide layer bonded to a piezoelectric substrate

In this article, the propagation and scattering of Rayleigh wave from an interfacial crack in a bilayer structure, a viscoelastic waveguide layer attached to a piezoelectric substrate, have been examined. The Principle of Superposition is used to formulate the problem. The Fourier transform method and stiffness matrix formulation have been used to arrive at singular integral equations. These singular integral equations are then reduced to a system of linear algebraic equations by applying the Gauss-Jacobi quadrature technique. A parametric study has been examined for phase velocity, attenuation, and dynamic stress intensity factor of mode-I and mode-II at both crack-tip.

Analysis of contact behaviours and vibrations in a defective deep groove ball bearing

Rolling element bearings are vital parts for rotating machinery. When a defect occurs on the surface of the bearing raceway or rolling element, giving rise to the abnormal vibration of the bearing and even the rotating machine. Thus, it is of great significance to understand the vibration characteristics of the bearing for fault diagnosis and machine condition monitoring. In recent decades, researches on defect-induced bearing vibration in machine condition monitoring have attracted increasing attention. Model-based investigation is a highly useful method for enhancing our understanding of defect-induced vibration and fault excitation mechanism of the bearing. It also serves as a basis for establishing the theory of bearing failure. With this in mind, a novel dynamic model of deep groove ball bearing (DGBB) with a localized outer race defect is proposed in this paper. In this model, the additional displacement excitation which is modelled as a piecewise function based on kinematics is considered, and the angle-based contact force is developed and modelled as a piecewise function taking into account the angular position, collision velocity, and impact force. In addition, an analytical formulation of the angle-varying effective stiffness matrix of the DGBB with a localized raceway defect of varying size is established. The proposed dynamic model is used to study the influences of different defect scenarios (i.e. angular positions and widths) on the time-varying contact stiffness, contact force, spectral characteristics, and bearing vibrations. The comparisons between the simulations and the experiments show that the proposed model is effective.

Second-order improved refined plastic hinge method with implementation of continuous strength method

Stainless steel, aluminium alloy, and cold-formed steel structures have been increasingly used as the main load-bearing components in civil engineering. However, owing to the rounded stress-strain response of the materials with significant strain hardening, the traditional plastic hinge model based on elastic-perfect plastic material response is not applicable, and the strength reduction of slender cross-sections by local buckling is also not considered. This paper proposed an improved refined plastic hinge method that can apply to different materials and fully consider the strain hardening and local buckling effect. A beam-column element was proposed, in which the second-order effect between axial load and bending moment was incorporated in the stiffness matrix formulation by using the approximate sixth-order polynomial function. The continuous strength method was introduced to the refined plastic hinge model to consider the strain hardening effect of non-slender and the local buckling effect of slender cross-sections by controlling the strength and deformability of the sections. Combining the elastic and plastic limit state criteria, the stiffness degradation function for SHS and RHS was obtained. The accuracy and efficiency of the proposed method were verified by the experimental results for stainless steel, aluminium alloy, and cold-formed steel columns of SHS and RHS under axial and eccentric compression. Finally, the proposed method was used for the design of frames for the materials, the results indicated that it was an effective tool that was ready for design practice.

Isogeometric stiffness matrix formulations for buckling analysis of multi-layered shells based on lamination parameters for solid-shell elements

Multi-layered composite shells are extensively used in many industrial fields since they have superior performance in applications. As load-carrying components, buckling failure is one of the most common failure modes of multi-layered shells. Hence, the layup design of multi-layered shells based on optimal buckling factors is widely concerned, where lamination parameters are commonly used in the representation of stiffness. However, the common formulations of laminate stiffness are based on the Classical Laminate Plate Theory (CLT) or the First-order Shear Deformation Plate Theory (FSDT). This paper aims at developing stiffness formulations based on 3D continuum solid theory. The accuracy and efficiency of the stiffness formulations derived in this study are analysed and compared to those obtained by layer-by-layer integration and summation of stiffness calculations.

Second-order improved refined plastic hinge method for high strength steel box-section columns under cyclic loads

Plastic hinge beam-column model is widely used in the direct design method, but it cannot directly simulate the elastic unloading and cumulative damage effect in hysteresis analysis. This paper presents a theoretical study on the cyclic performance of high strength steel box-section columns. Firstly, the second-order effect and the additional axial strain due to the element bending are incorporated in the stiffness matrix formulation of the beam-column element. Then, the plasticity development in box section under axial force and bending moment is analyzed and a practical calculation method for the rotational stiffness of plastic hinges considering the cumulative damage effect is derived. Based on a new hysteresis criterion, an improved refined plastic hinge method is proposed for beam-column element to predict the hysteretic behavior. Existing experimental results of Q460 high strength steel welded box-section columns under cyclic loads are adopted to verify the reasonability and accuracy of the proposed method in which only one element per member is required. The proposed improved refined plastic hinge method is shown to be an efficient tool ready to be implemented into the design practice.

Effective property calculation and its numerical implementation of spatially graded plate structures based on asymptotic homogenization

Spatially graded heterogeneous plate structures that are mapped from periodic plates based on mapping functions are attracting increasing attention due to their excellent mechanical performances and high tailorability, and homogenization approaches are a powerful tool for their efficient numerical analysis. In this work, based on asymptotic homogenization, a unified framework of the effective properties prediction and the corresponding unit cell problems is first theoretically established for spatially-varying plates with arbitrarily-shaped microstructures, which are distinct from those of periodic plates due to the Jacobian of mapping functions. Moreover, the FE formulation for efficient numerical implementation are also proposed for not only solid elements, but also shell and beam elements, where numerical treatment details in stiffness matrix formulation and periodic boundary conditions are elaborated, so that highly efficient effective stiffness computation can be achieved for thin-walled heterogeneous plate structures. At last, unit cells under different Jacobian matrices, which are related to different kinds of geometric deformation patterns, and two spatially graded plates are analyzed to corroborate the correctness and efficiency of the proposed method.

The stiffness characteristics analysis of the duplex angular contact ball bearings based on a comprehensive multi-degree-of-freedom mathematical model

Although the duplex angular contact ball bearings (DACBBs) are widely used in industry, the published mathematical model of DACBBs is sparse. To analyze the stiffness characteristics of DACBBs, this paper proposes a comprehensive multi-degree-of-freedom (multi-DOF) mathematical model for the DACBBs in three traditional configurations. The implicit differential method is adopted to derivate the analytical stiffness matrix formulation of DACBBs. The geometrical constraints and interactions inside the DACBBs are presented based on the vector-and-matrix method. The detailed iterative algorithm with two layers for solving the presented model is given based on the Newton-Raphson method. A systematic study for evaluating the impacts of angular misalignment on the stiffness characteristics of DACBBs has been carried out under three typical load conditions: the pure axial load condition, the pure radial load condition, and the combined-loaded condition. The results indicate that the angular misalignment remarkably affects the stiffness characteristics of DACBBs. Under the pure axial and radial load conditions, the stiffness curves of DACBBs can be divided into two segments according to the value of misalignment angle, and the angular misalignment will be the predominant factor when the angular misalignment is great. Under the combined-loaded condition, the mode (single or combined angular misalignment) and the direction of angular misalignment significantly influence the stiffness characteristics of DACBBs. Also, the angular misalignment effects depend on the radial load. The proposed model can also analyze the mechanical performance of double-row angular contact ball bearings.

Exact solution by dynamic stiffness method for the natural vibration of porous functionally graded plate considering neutral surface

This paper presents the formulation of dynamic stiffness matrix for the natural vibration analysis of porous power-law functionally graded Levy-type plate. In the process of formulating the dynamic stiffness matrix, Kirchhoff-Love plate theory in tandem with the notion of neutral surface has been taken on board. The developed dynamic stiffness matrix, a transcendental function of frequency, has been solved through the Wittrick–Williams algorithm. Hamilton’s principle is used to obtain the equation of motion and associated natural boundary conditions of porous power-law functionally graded plate. The variation across the thickness of the functionally graded plate’s material properties follows the power-law function. During the fabrication process, the microvoids and pores develop in functionally graded material plates. Three types of porosity distributions are considered in this article: even, uneven, and logarithmic. The eigenvalues computed by the dynamic stiffness matrix using Wittrick–Williams algorithm for isotropic, power-law functionally graded, and porous power-law functionally graded plate are juxtaposed with previously referred results, and good agreement is found. The significance of various parameters of plate vis-à-vis aspect ratio ( L/b), boundary conditions, volume fraction index ( p), porosity parameter ( e), and porosity distribution on the eigenvalues of the porous power-law functionally graded plate is examined. The effect of material density ratio and Young’s modulus ratio on the natural vibration of porous power-law functionally graded plate is also explained in this article. The results also prove that the method provided in the present work is highly accurate and computationally efficient and could be confidently used as a reference for further study of porous functionally graded material plate.

Dynamic Stiffness Formulation for Out-of-Plane Natural Vibration of Elastically Supported Functionally Graded Plates

In this paper, the natural vibration characteristics of elastically supported functionally graded material plate are investigated using the dynamic stiffness method (DSM). Power-law functionally graded (P-FG) plate, the material properties of which vary smoothly along the thickness direction following the power-law function, that has been used for the analysis. Classical plate theory and Hamilton’s principle are used for deriving the governing differential equation of motion and associated edge conditions for P-FG plate supported by elastic foundation. During the formulation of dynamic stiffness (DS) matrix, the concepts of rotary inertia and neutral surface are implemented. Wittrick–Williams (W-W) algorithm is used as a solving technique for the DS matrix to compute eigenvalues. The results thus obtained by DSM for the isotropic, P-FG plate, and the P-FG plate with elastic foundation compare well with published results that are based on different analytical and numerical methods. The comparisons indicate that this approach is very accurate. Furthermore, results are provided for elastically supported P-FG plate under four different considerations in order to see the differences in frequencies with the inclusion or exclusion of neutral surface and/or rotary inertia. It is noticed that the inclusion of rotary inertia and neutral surface influences the eigenvalues of P-FG plate, and that cannot be discounted. The study also examines the influence of plate geometry, material gradient index, edge conditions, and elastic foundation modulus on the natural frequency of P-FG plate.

Simple approach for including foundation–soil–foundation interaction in the static stiffnesses of multi-element shallow foundations

In common engineering practice, foundation–soil–foundation interaction of shallow foundations is frequently ignored. This is presumably for cost/benefit reasons, since computationally demanding finite-element and/or boundary element models are required for that purpose, and its effects are usually assumed to be negligible. In this sense, the present paper provides a simple and inexpensive way of incorporating foundation–soil–foundation interaction through a numerically explicit stiffness matrix formulation. The necessary ingredients for homogeneous and non-homogeneous (shear modulus power-law variation with depth) half-spaces are given. The proposed approach is then applied to offshore wind turbines’ multiple suction caisson foundations (tripod and tetrapod), where it is observed that the foundation–soil–foundation interaction is significant. Its range of validity is also established, and valuable ready-to-use closed-form formulas for the correction factors of the stiffnesses of tripod and tetrapod groups are also derived. The methodology is applicable as long as the spacing between foundations is somewhat greater than the foundation depth.

Semi-analytical model development for preliminary study of 3D woven Composite/Metallic flange bolted assemblies

This research deals with the development of a new semi-analytical model for composite-metal eccentric tensile bolted joints of casing structures. The model is based on a direct stiffness method considering structural elements, where flanges are modeled with axisymmetric shell elements, the bolt with a beam element and contact behavior with a specific number of compressive linear springs, identified by means of convergence tests. A hybrid element is added in order to set the link between flange and bolt. Since the model is intended for the preliminary design of bolted joints including composite materials, we propose an extension of the stiffness matrix formulation of isotropic axisymmetric shell elements to orthotropic ones. This formulation is validated with respect to a 2D FE axisymmetric model. Then, a semi-analytical model is established for a casing assembly approximately corresponding to a real casing of an aircraft engine. Design criteria such as normal stress on the bolt, bending moment and axial load at a specific section of the composite flange, given by the semi-analytical model are compared to a 3D FE computation. Results are in very good agreement and the semi-analytical model offers considerable time saving where the computation time was 120 times faster than the 3D FE model. Moreover, relative error does not exceed 2% for normal stress of the bolt and 9% for the flange bending moment which is satisfactory in a preliminary design approach. However, due to the formulation of hybrid element, less accurate results were observed for axial displacement of the composite flange and it would be advisable to consider a new formulation. This will be detailed in future work. Finally, a parametric study showed the ability of the semi-analytical model to predict the physical behavior of joints and proved its usability for running optimization studies and its reliability as a preliminary design tool.

Refined second-order plastic-hinge dynamic analysis for planar steel frames

The paper presents the method of the refined second-order plastic hinge analysis for planar steel frames subjected to dynamic loads. The material model used is the generalized Clough model in consideration of effects of axial and shear force on plastic limit state of cross-sections. Geometric nonlinearities of member and frame are taken into account by the use of stability functions in framed stiffness matrix formulation. Non-linear motional equation of structures was established by finite element method and solved by Newmark method combined with the modified Newton-Raphson method. Based on the obtained algorithms, the author has programmed a computational program and calculated the planar steel frames. From the calculated results, discussions and conclusions are presented.

Effect of angular misalignment on the stiffness of the double-row self-aligning ball bearing

The double-row self-aligning ball bearing commonly runs with angular misalignment between the inner and outer rings, and the angular misalignment significantly affects the bearing performance. However, the effect of angular misalignment on the dynamic characteristics of the double-row self-aligning ball bearing has not been studied thoroughly. This paper investigates the effect of angular misalignment on the stiffness of the double-row self-aligning ball bearing. The quasi-static model for the double-row self-aligning ball bearing is established with five degrees of freedom, namely, three translational displacements along x, y, and z directions and two tilting angles around x-axis and y-axis. The internal clearance between balls and raceways is included in the presented model. The formulation of the three-dimensional stiffness matrix for the double-row self-aligning ball bearing is analytically derived, and is verified by comparing with the available dada in published literature. Finally, the stiffness of the double-row self-aligning ball bearing under various angular misalignment conditions is analyzed systematically. The results show that the tilting angles vary the contact angles and contact forces of the compressed balls in the double-row self-aligning ball bearing, thus affect the bearing stiffness; the bearing stiffness decreases with the internal clearance; the angular misalignment significantly impacts the stiffness of the bearing running at a low speed.

Geometric nonlinear formulation of plate buckling structure

The analytical or exact mathematical formulation of stresses and displacements for plate buckling structure become impossible to develop if the plate geometry is so complicated. Numerical technique is one another approach to solve this problem and it is chosen in this study for plate structure under in-plane and out-plane load. The formulation of elastic stiffness matrix (ke) and geometric nonlinear stiffness matrix (kg) of the plate structure due to buckling is presented and based on virtual displacement principle. The geometric nonlinear stiffness matrix (kg) is found function of internal stresses. The direct iteration technique is applied to find nodal displacements. Under this technique, the Gauss points stresses are initialized as zeros, then the kg matrix is updated, and then a new nodal displacement vector is found for the next approximation of internal stresses. Iterative process is done until convergence of displacement is satisfied. The rectangular plate with one fixed edge supported is used to test the proposed nonlinear formulation and procedure. The compressive in-plane load and moment is considered and applied for the tested plate. The plate is discretized with appropriate number of triangular finite element mesh. It is found that, the convergence of displacement is satisfied by using direct iteration technique. The load - deflection curve shows nonlinear relationship and approach to critical load. This finding shows that the direct iteration method can be accepted for the analyzing of plate buckling by considering geometric nonlinear assumption.

A mixed edge-based smoothed solid-shell finite element method (MES-FEM) for laminated shell structures

We in this study present a mixed solid-shell finite element formulation for laminated shell analysis. With the commonly adopted truncations applied to the strain components and the Jacobian determinant, a generalized stiffness matrix is established. Discrete shear gap (DSG) technique together with assumed natural strain (ANS) method is employed in order to alleviate shear locking and trapezoidal locking. Stiffness matrix formulation is analytically calculated based on the edge-based cell backgrounds, which result in simple and efficient computation. The proposed formulation does not require using cross-diagonal meshes, which can be an obstacle for several existing three-node degenerated shell elements. This aspect makes the present method much more appealing than others through examples tested. Numerical validations show that the present method performs well for both structured and unstructured triangular meshes.

Discontinuous Deformation Analysis Based on Wachspress Interpolation Function for Detailed Stress Distribution

In the original discontinuous deformation analysis (DDA) method, the complete first-order displacement function is used to describe block movement and deformation, which induce constant stress and strain throughout the block. To achieve a more detailed stress distribution, Wachspress interpolation displacement function is employed to express the displacement of blocks in DDA, and the interactions between blocks are still governed under the original DDA. Displacements of the vertexes of all blocks constitute new freedom vectors, and the stiffness and force matrix formulations are derived again. In the new formulation, Wachspress interpolation ensures that the edges of the blocks are straight; therefore, contact detection can be processed based on the original DDA. Several classical examples are analyzed. The results show that the new formulation obtains similar configurations as the original DDA but provides more detailed and continuous stress distributions within block element.

Implementation of an advanced beam model in BHawC

The implementation of a new structural beam model in the in-house aeroelastic tool BHawC is presented. The model is based on a recently published equilibrium based formulation of a beam stiffness matrix as well as a novel stiffness proportional damping formulation. The performance of the new beam implementation is assessed via modelling a commercial wind turbine. Four analyses are presented to illustrates the increased accuracy in recovered section forces and representation of the dynamic behaviour gained from the new structural model.

Damage Tolerant Analysis of Cracked Al 2024-T3 Panels repaired with Single Boron/Epoxy Patch

It is known that damage tolerant analysis has two objectives, namely, remaining life prediction and residual strength evaluation. To achieve the these objectives, determination of accurate and reliable fracture parameter is very important. XFEM methodologies for fatigue and fracture analysis of cracked aluminium panels repaired with different patch shapes made of single boron/epoxy have been developed. Heaviside and asymptotic crack tip enrichment functions are employed to model the crack. XFEM formulations such as displacement field formulation and element stiffness matrix formulation are presented. Domain form of interaction integral is employed to determine Stress Intensity Factor of repaired cracked panels. Computed SIFs are incorporated in Paris crack growth model to predict the remaining fatigue life. The residual strength has been computed by using the remaining life approach, which accounts for both crack growth constants and no. of cycles to failure. From the various studies conducted, it is observed that repaired panels have significant effect on reduction of the SIF at the crack tip and hence residual strength as well as remaining life of the patched cracked panels are improved significantly. The predicted remaining life and residual strength will be useful for design of structures/components under fatigue loading.

Free vibration of fully functionally graded carbon nanotube reinforced graphite/epoxy laminates

This study provides the first-known vibration analysis of fully functionally graded carbon nanotube reinforced hybrid composite (FFG-CNTRHC) laminates. CNTs are non-uniformly distributed to reinforce the graphite/epoxy laminates. Some CNT distribution functions in the plane and thickness directions are proposed to more efficiently increase the stiffening effect. The rule of mixtures is modified by considering the non-homogeneous material properties of FFG-CNTRHC laminates. The formulation of the location dependent stiffness matrix and mass matrix is derived. The effects of CNT volume fraction and distribution on the natural frequencies of FFG-CNTRHC laminates are discussed. The results reveal that the FFG layout may significantly increase the natural frequencies of FFG-CNTRHC laminate.