Geometric Stiffness Matrices Research Articles

Prestressed Vibration of Stiffened Variable-Angle Tow Laminated Plates

Compared to straight fiber path laminates, variable-angle tow (VAT) laminates are known to redistribute the in-plane stress resultants for improving structural buckling response. The VAT laminates can also be used to improve the free vibration response by tailoring the structural elastic stiffness. This paper studies the prestressed vibration response of a stiffened, VAT laminated plate under a uniform end shortening. Because there are both spatially dependent stiffener and fiber path orientations for a stiffened, VAT laminated plate, a separate modeling of the plate and stiffeners is considered. This method avoids placing common nodes at stiffener–plate and stiffener–stiffener interfaces. This allows one to study mesh convergence for the plate and the stiffeners, separately. It also improves the efficiency and robustness of the finite element model. The finite element code based on the present method is verified extensively using examples that are either available in literature or are analyzed using commercial software. Parametric studies show that, depending on the in-plane load and boundary condition, the VAT laminates with linearly varying (LV) fiber paths can increase the prestressed vibration fundamental frequency when compared to the straight fiber path laminates. Optimization studies found that using nonlinearly varying fiber paths, when compared to the LV fiber paths, causes a significant increase in the buckling load but only a slight increase in the case of free or prestressed vibration fundamental frequencies. The VAT laminates appear to mainly tailor the panel’s elastic stiffness for improving the prestressed vibration response for the stiffened plates with clamped edges. However, for the simply supported stiffened plate, the VAT laminates are mainly used to change both the elastic and geometric stiffness matrices for improving the prestressed vibration response.

Dynamic instability of channel-section beams under periodic loading

This paper presents an analytical investigation on the free vibration, static buckling and dynamic instability of channel-section beams when subjected to periodic loading. The analysis is carried out by using Bolotin’s method. By assuming the instability modes, the kinetic energy and strain energy of the beam and the loss of the potential of the applied load are evaluated, from which the mass, stiffness and geometric stiffness matrices of the system are derived. These matrices are then used to carry out the analyses of free vibration, static buckling and dynamic instability of the beams. Theoretical formulae are derived for the free vibration frequency, critical buckling moment, and excitation frequency of the beam. The effects of the lateral restraint applied to the flange, the section size of the beam and the static part of the applied load on the variation of dynamic instability zones are also discussed.

On the nonlinear extension-twist coupling of beams

Abstract In many applications, beams carry large axial forces, such as centrifugal forces, for instance, and the resulting axial stresses affect their torsional behavior. To capture this important coupling effect within the framework of beam models, a nonlinear theory must be developed that takes into account higher-order strain deformations. The proposed development starts with a three-dimensional elasticity model of beam-like structures. While strain components are assumed to remain small, the strain energy expression is expanded to retain strain components up to cubic order. This expansion of the three-dimensional model leads to material and geometric stiffness matrices. For linear problems, the three-dimensional warping fields induced by each of the six unit sectional strain components can be obtained. These warping fields are used to reduce the three-dimensional elasticity model to one-dimensional beam equations. Byproducts of this reduction process include the sectional stiffness matrix and six nonlinear stiffness matrices representing the geometric stiffness induced by unit sectional strains. It is shown that the nonlinear extension-twist coupling is captured by these geometric stiffness matrices. Comparison of the predictions of the proposed approach with analytical and experimental results demonstrate their effectiveness and accuracy.

Buckling of functionally graded plates under thermal, axial, and shear in-plane loading using complex finite strip formulation

ABSTRACTA complex finite strip method was used to study the buckling of functionally graded plates (FGPs) under thermal and mechanical (longitudinal, transverse, and shear in-plane) loading. The mechanical characteristics of FGPs were assumed to vary through the thickness, according to power law distribution. The nonlinear temperature distribution in the direction of the plate thickness was assumed according to thermal conduction steady state conditions. In complex finite strip method, the polynomial Hermitian functions were assumed in the transverse direction and the complex exponential functions were used in the longitudinal direction to evaluate the standard and geometric stiffness matrices that have the ability of calculating the critical shear stress in contrast to trigonometric shape functions. The solution was obtained by the minimization of the total potential energy and solving the corresponding eigenvalue problem. In addition, numerical results for FGPs with different boundary conditions were presented and compared with those available in the literature and the interaction curves of mechanical and thermal buckling capacity of FGPs were obtained.

Dynamic instability of laterally-restrained zed-purlin beams under uplift loading

This paper presents an analytical investigation on the free vibration, static buckling and dynamic instability of laterally-restrained zed-section purlin beams when subjected to uplift wind loading. The analysis is carried out by using the classical principle of minimum potential energy. By assuming the instability modes, the kinetic energy and strain energy of the beam and the loss of the potential energy of the applied load are evaluated, from which the mass, stiffness and geometric stiffness matrices of the system are derived. These matrices are then used to carry out the analyses of free vibration, static buckling and dynamic instability of the beams. Theoretical formulae are derived for the free vibration frequency, critical buckling moment, and excitation frequency of the beam. The effects of the section size of the beam and the static part of the applied load on the change of dynamic instability zone are also discussed.

Local Elastic and Geometric Stiffness Matrices for the Shell Element Applied in cFEM

In this paper local elastic and geometric stiffness matrices of ashell finite element are presented and discussed. The shell finiteelement is a rectangular plane element, specifically designedfor the so-called constrained finite element method. One of themost notable features of the proposed shell finite element isthat two perpendicular (in-plane) directions are distinguished,which is resulted in an unusual combination of otherwise classicshape functions. An important speciality of the derived stiffnessmatrices is that various options are considered, whichallows the user to decide how to consider the through-thicknessstress-strain distributions, as well as which second-order strainterms to consider from the Green-Lagrange strain matrix. Thederivations of the stiffness matrices are briefly summarizedthen numerical examples are provided. The numerical examplesillustrate the effect of the various options, as well as theyare used to prove the correctness of the proposed shell elementand of the completed derivations.

Buckling analysis of functionally graded truncated conical shells under external displacement-dependent pressure

This paper is presented to solve the buckling problem of functionally graded truncated conical shells subjected to displacement-dependent pressure which remains normal to the shell middle surface throughout the deformation process by the semi-analytical finite strip method. Material properties are assumed to be temperature dependent, and varied continuously in the thickness direction according to a simple power law distribution in terms of the volume fraction of a ceramic and metal. The governing equations are derived based on first-order shear deformation theory which accounts for through thickness shear flexibility with Sanders-type of kinematic nonlinearity. The element linear and geometric stiffness matrices are obtained using virtual work expression for functionally graded materials. The load stiffness also called pressure stiffness matrix which accounts for variation of load direction is derived for each strip and after assembling, global load stiffness matrix of the shell which may be un-symmetric is formed. The un-symmetric parts which are due to load non-uniformity and unconstrained boundaries have been separated. A detailed parametric study is carried out to quantify the effects of power-law index of functional graded material and shell geometry variations on the difference between follower and non-follower lateral buckling pressures. The results indicate that considering pressure stiffness which arises from follower action of pressure causes considerable reduction in estimating buckling pressure.

Buckling and free vibration of shear-flexible sandwich beams using a couple-stress-based finite element

In this work, we create a framework for linear buckling and free vibration analyses of sandwich beams using a microstructure-dependent Timoshenko beam model founded on the modified couple-stress theory. The stiffness parameters of a structural web-core sandwich panel are determined by unit cell analysis. An extension to homogeneous cores is also carried out. By employing the exact general solution to the governing equations of the beam, an accurate approximate finite element stiffness matrix is formulated. Furthermore, the static shape functions are used to derive consistent linear geometric stiffness and mass matrices. A convergence study shows that the approximate finite element has good accuracy although the hyperbolic terms of the exact general solution have been expanded into only relatively low-order polynomial series. Results from examples show that the microstructure-dependent beam can predict critical buckling loads and natural frequencies with very good accuracy when compared to more sophisticated finite element models. Unlike the classical Timoshenko beam model, the microstructure-dependent model yields accurate results also when the sandwich assembly is transversely flexible and the bending stiffness of the faces non-negligible.

A discussion on the stiffness matrices used in tensegrity structures

The expression stiffness matrices used in the study of tensegrity structures has inherited a formulation initially conceived for the shape finding problems of tension structures. Nevertheless stiffness and geometrical stiffness matrices are well known concepts of structural analysis and all the formulations which are used need to be congruent. In this paper, we present a formal discussion about the formulations used in several publications showing that they are not equivalent. A classic example is presented, which has been solved using several expressions available in the relevant literature, the results have been compared with a classical finite element software, showing the discrepancies.

Non-linear finite element analysis of an elastic structure loaded by hydrostatic follower forces

This study deals with the non-linear finite element computation of thin flexible structures loaded by hydrostatic forces due to the presence of an internal liquid at rest. In aerospace application, the dynamic behaviour of structures containing an inviscid incompressible fluid (i.e. launcher with liquid propellant, tank of satellite, etc.) is generally computed considering an equilibrium state resulting from a linear fluid-structure interaction problem. It is important to note that in this case, the gravity plays an important role in vibrations and the so called elastogravity operator should be taken into account [1,2]. In the present work, we consider the large deformation behavior of a structure loaded by hydrostatic follower forces in order to obtain an accurate static equilibrium state. The solution is computed using a Newton-Raphson algorithm considering the geometrical and material tangent stiffness matrices as well as the contribution of the follower forces [3]. Those linearized operators of the iterative algorithm are used to compute the dependancy between the incremental horizontal level of the free surface, under the fluid volume conservation constraint [4]. Special attention is paid to the finite element discretization of the wetted surface, considering a level-set method [6,7]. Some numerical examples are analyzed to show the efficiency of the proposed approach.

Shear buckling of FG-CNT reinforced composite plates using Chebyshev-Ritz method

Abstract Shear buckling response of carbon nanotube reinforced composite (CNTRC) rectangular plates in thermal environment is investigated in this research. Distribution of CNTs across the plate thickness may be uniform or graded via a mid-plane symmetric pattern. Properties of the CNTRC plate are obtained using the modified rule of mixtures approach by introduction of efficiency parameters. First order plate theory is adopted to construct the basic equations of the plate. A two-dimensional Ritz formulation with Chebyshev basis polynomials is implemented to obtain the elastic and geometric stiffness matrices of the plate. The proposed solution method may be used for arbitrary in-plane and out-of-plane edge supports. After performing convergence and comparison studies to show the effectiveness and accuracy of the proposed method, parametric studies are conducted to examine the influences of boundary conditions, volume fraction of CNTs, graded pattern of CNTs, thermal environment and plate geometry. It is shown that, shear buckling capacity of the plate may be enhanced through functionally graded distribution of CNTs. Besides, enrichment of matrix with CNTs enhances the shear buckling loads of FG-CNTRC plates.

Non-conservative Instability of Shear Flexible Steel Columns Considering Inelastic Effect

The aim of this paper was to investigate the non-conservative instability behavior of the shear flexible steel columns considering the inelastic effect. The tangent modulus and the effective length factor are newly evaluated using the iterative approach based on the AISC-LRFD and CRC strength curves. The equation of motion of shear flexible inelastic column is derived from the extended Hamilton’s principle. The finite element procedure using the Hermite cubic interpolation functions taking into account the shear effect is employed to obtain the mass, inelastic stiffness, geometric stiffness, and load correction matrices. The evaluation procedure for the critical values of divergence and flutter loads are briefly described, and the obtained results are compared with available results through numerical examples. In particular, the effects of various structural parameters such as the material inelasticity, the shear deformation, the rotary inertia, the effective length factor, and the non-conservativeness factor on the non-conservative instability behavior of steel columns are parametrically investigated.

Dynamic stability analysis for Beck’s type of inelastic Timoshenko steel columns

In this study, the dynamic stability analysis of a Beck’s type of steel column subjected to a sub-tangential follower force is presented considering the effects of gradual yielding along the length of column, the shear deformation, and the rotary inertia. The nonlinear tangent modulus-stress, tangent modulus-slenderness ratio, and compressive stress-strain relationships of an inelastic column are derived from the column strength curve given in Korea Bridge Design Code (KBDC) based on the tangent modulus theory. The equations of motion of the Timoshenko inelastic column are derived from the Hamilton’s principle. The finite element method using the Hermite cubic interpolation functions is employed to obtain the mass, inelastic stiffness, geometric stiffness, and load correction matrices, and the evaluation procedure for critical values of divergence and flutter of the non-conservative system is briefly described. The obtained results are compared with available results of other numerical or analytical methods. Finally, the effects of various structural parameters such as the material inelasticity, the shear deformation, the rotary inertia, the effective length factor, and the sub-tangentiality factor on the divergence and flutter behavior of non-conservative columns are parametrically investigated.

Modeling the Elastic and Damping Properties of the Multilayered Torsion Bar-Blade Structure of Rotors of Light Helicopters of the New Generation. 1. Finite-Element Approximation of the Torsion Bar

A prismatic semiquadratic element with a nonclassical approximation of its displacements is suggested for modeling the composite and soft layers of a torsion bar and multilayered plate-rod structures. The stiffness, weight, damping, and geometric stiffness matrices of the above-mentioned element are obtained. Expressions for computing stresses in the finite element under the action of static loads and vibrations in the resonance zone are presented. Test examples confirming the validity of the element suggested are given. An example of finite element determination of the dynamic response of a multilayered torsion bar in the resonant mode is considered.

Buckling and Vibration Analysis of Cold-Formed Steel CHS Members and Frames Using Generalized Beam Theory

This paper reports the results of an investigation on the use of Generalized Beam Theory (GBT) to assess the buckling and vibration behaviors of thin-walled members and frames built from cold-formed steel circular hollow section (CHS) profiles. Initially, the concepts and procedures involved in performing GBT buckling and vibration analyses are presented, paying particular attention to the derivation of the mass tensors that account for the influence of the inertia forces. Then, the formulation, numerical implementation and validation of a GBT-based beam finite element for isolated members are described. Next, the determination of the frame linear stiffness, geometric stiffness and mass matrices, which incorporate the influence of the frame joints, is addressed. Finally, in order to illustrate the application and capabilities of the proposed GBT finite element formulation, numerical results are presented and discussed — they concern the buckling and vibration behaviors of an "L-shaped" frame. For validation purposes, most GBT-based results are compared with values yielded by shell finite element analyses carried out in the code ANSYS.

A Circumferentially Enhanced Hermite Reproducing Kernel Meshfree Method for Buckling Analysis of Kirchhoff–Love Cylindrical Shells

A circumferentially enhanced Hermite reproducing kernel (HRK) meshfree method is developed for the buckling analysis of Kirchhoff–Love cylindrical shells. In this method, in order to accurately represent the circumferential periodicity of cylindrical shells, the shell mid-surface is first discretized by a set of physical nodes in the two-dimensional parametric space, thereafter another set of dummy nodes are added by a straightforward periodic translation of the physical nodes. Subsequently the meshfree shape functions are constructed using both the physical nodes and the dummy nodes through a periodically linked relationship. The resulting meshfree shape functions exhibit the desired circumferential periodicity. The meshfree shape functions are formulated in the HRK framework which can be degenerated to the standard reproducing kernel (RK) shape functions just by removing the rotational terms. Meanwhile, the cylindrical shell buckling equations are rationally derived from the consistent linearization of the internal virtual work. During the meshfree discretization, the in-plane shell displacements are represented by the conventional RK shape functions, while the out-of-plane shell deflection is approximated by the Hermite meshfree shape functions with both directional and rotational degrees of freedom. The numerical integration of the material as well as the geometric stiffness matrices are carried out by the strain smoothing sub-domain stabilized conforming integration (SSCI) method. Numerical examples show that the proposed approach yields very favorable results for the buckling analysis of cylindrical shells.

An anisotropic large displacement cohesive zone model for fibrillar and crazing interfaces

A new cohesive zone model to describe fracture of interfaces with a microstructurs made of fibrils with statistically distributed in-plane and out-of-plane orientations is proposed. The elementary force–displacement relation of each fibril is considered to obey the peeling theory of a tape, although other refined constitutive relations could be invoked for the adhesive constitutive response without any lack of generality. The proposed consistent 2D and 3D interface finite element formulations for large displacements account for both the mechanical and the geometrical tangent stiffness matrices, required for implicit solution schemes. After a preliminary discussion on model parameters identification, it is shown that by tailoring the spatial density of fibrils at different orientations can be a way to realize innovative interfaces enhancing adhesion or decohesion, depending on the need. For instance, it can be possible to realize microstructured adhesives to facilitate debonding of the glass cover in photovoltaic modules to simplify recycling purposes. Moreover, the use of probability distribution functions describing the density of fibrils at different orientations is a very effective approach for modeling the anisotropy in the mechanical bonding between paper tissues and for simulating the complex process of crazing in amorphous polymers.

Large deflection analysis of plates and cylindrical shells by an efficient four-node flat element with mesh distortions

This study reports an improved finite element computational model using a flat four-node element for nonlinear bending analysis of plates and cylindrical shells with element distortions. The von Karman’s large deflection theory and the total Lagrangian approach are employed in the formulation to describe small strain geometric nonlinearity with large deformations using the first-order shear deformation theory. The most important feature of the developed element is the evaluation of linear membrane bending and nonlinear geometric stiffness matrices based on integration along the boundary of smoothing elements. This technique can give more accurate numerical integrations even with badly shaped elements or coarse meshes when compared to other flat elements using domain integration techniques. The accuracy and predictive capability of the present model is demonstrated by several numerical investigations and comparative studies with analytical/experimental and other numerical solutions available in the literature.

Accurate Element of Compressive Bar considering the Effect of Displacement

By constructing the compressive bar element and developing the stiffness matrix, most issues about the compressive bar can be solved. In this paper, based on second derivative to the equilibrium differential governing equations, the displacement shape functions are got. And then the finite element formula of compressive bar element is developed by using the potential energy principle and analytical shape function. Based on the total potential energy variation principle, the static and geometrical stiffness matrices are proposed, in which the large deformation of compressive bar is considered. To verify the accurate and validity of the analytical trial function element proposed in this paper, a number of the numerical examples are presented. Comparisons show that the proposed element has high calculation efficiency and rapid speed of convergence.

Group-theoretic method for efficient buckling analysis of prestressed space structures

An efficient group-theoretic method is proposed for the buckling analysis of symmetric prestressed space structures. Tangent stiffness matrices and geometric stiffness matrices of trusses, cables, and frames are given, and thus, the proposed method is applicable not only to pin-jointed structures but also to more general structures with frame and cable elements. Important contribution of initial prestresses is considered in formulating the tangent stiffness matrices for the prestressed structures. By adopting irreducible representations of a symmetry group and projection operator theory, the associated stiffness matrices are converted to symmetry-adapted forms. Subsequently, the original buckling problem is decomposed into a series of independent subproblems with smaller dimensions, which obtain precise solutions but lead to significant reduction in computational cost. To describe the typical process for the group-theoretic method, efficient buckling analyses on four types of symmetric prestressed space structures composed of cables, trusses, and/or frames are carried out. The computational efficiency of the proposed method is dramatically improved in comparison with those of FEM and the conventional numerical method. Comparing the results with the corresponding ones from ABAQUS and published data, we verify that the proposed method is elegant and reliable.