Elastic Stiffness Matrices Research Articles

Application of multiscale elastic homogenization based on nanoindentation for high performance concrete

The paper aims at developing a simple two-step homogenization scheme for prediction of elastic properties of a high performance concrete (HPC) in which microstructural heterogeneities are distinguished with the help of nanoindentation. The main components of the analyzed material include blended cement, fly-ash and fine aggregate. The material heterogeneity appears on several length scales as well as porosity that is accounted for in the model. Grid nanoindentation is applied as a fundamental source of elastic properties of individual microstructural phases in which subsequent statistical evaluation and deconvolution of phase properties are employed. The multilevel porosity is evaluated from combined sources, namely mercury intrusion porosimetry and optical image analyses. Micromechanical data serve as input parameters for analytical (Mori–Tanaka) and numerical FFT-based elastic homogenizations at microscale. Both schemes give similar results and justify the isotropic character of the material. The elastic stiffness matrices are derived from individual phase properties and directly from the grid nanoindentation data with very good agreement. The second material level, which accounts for large air porosity and aggregate, is treated with analytical homogenization to predict the overall composite properties. The results are compared with macroscopic experimental measurements received from static and dynamic tests. Also here, good agreement was achieved within the experimental error, which includes microscale phase interactions in a very dense heterogeneous composite matrix. The methodology applied in this paper gives promising results for the better prediction of HPC elastic properties and for further reduction of expensive experimental works that must be, otherwise, performed on macroscopic level.

Discussion of “Second-Order Analysis of Plane Frames with One Element per Member” by Richard J. Balling and Jesse W. Lyon

The authors presented an analytical model that, as indicated by thetitle, can be used to determine the second-order response of planeframes with one element per member. The authors stated that “Acorotational element is developed directly from the governingsecond-orderdifferentialequationsofbeamtheory.Thecorotationalelement includes not only the effect of axial force on the bendingmoment (P-delta effect) but also the additional axial strain causedby end rotations. Hinged and semirigid end conditions are also in-cludedsothatplastichingescouldbeconsidered.Theresultinglocalelementtangentstiffnessmatrixiscomparedtotraditionallocalele-ment elastic and geometric stiffness matrices. The method is imple-mented and executed on two example problems in which only oneelement per member is used. Results compare favorably to thosefromanonlinearcommercialprograminwhichseveralelementspermemberareused.Athirdexampleproblemincludesbothgeometricandmaterial nonlinearitytodevelopapushovercurve.”Finally,theauthors concluded that “The elastic and first geometric stiffnessmatrices degenerate to traditional elastic and geometric stiffnessmatrices if the initial member length L

Parametric Instability of Laminated Composite Doubly Curved Shell Panels Subjected to Hygrothermal Environment

The dynamic stability behavior of laminated composite shells subjected to hygrothermal loadings are studied in the present investigation. A simple laminated model is developed for the vibration and stability analysis of laminated composite shells subjected to hygrothermal conditions. A computer program based on FEM in MATLAB environment is developed to perform all necessary computations. An eight-node isoparametric element is employed in the present The analysis with five degrees of freedom per node. Element elastic stiffness matrices, mass matrices, geometric stiffness matrix due to mechanical and hygrothermal loads and load vectors are derived using the principle of minimum potential energy. Quantitative results are presented to show the effects of curvature, ply-orientation, degrees of orthotropy and static load factors of laminate on dynamic stability of composite shells for different temperatures and moisture concentrations.

Inelastic Static and Dynamic Stability Analyses of a Column Subjected to a Nonconservative Force

This study performs the inelastic static and dynamic stability analyses of a column subjected to a nonconservative force using the finite element method. Based on the tangent modulus theory, the nonlinear compressive stress–strain and tangent modulus–stress relationships of an inelastic column are derived from the column strength curves given in two design codes. The extended Hamilton principle is employed to obtain the mass, elastic stiffness, and geometric stiffness matrices. Evaluation procedures for the critical values of buckling and flutter of the nonconservative systems are briefly introduced. In numerical examples, the influence of various parameters on the inelastic static and dynamic stability of the nonconservative systems is addressed as follows: (1) the effect of the nonconservativeness parameter on the critical buckling and flutter loads of the column for the inelastic stability analysis from the characteristic curves, (2) the variation of the effective length factor (K-factor) for inelastic columns with respect to the nonconservativeness parameter.

A beam element for coupled torsional-flexural vibration of doubly unsymmetrical thin walled beams axially loaded

A coupled torsional-bending finite element with shear deformations and rotatory inertia for vibration of nonsymmetric thin walled beams axially loaded is developed. The equations of motion are based on Vlasov’s theory of thin-walled beams, which are modified to include an axial load. The formulation is also applicable to solid beams. The Hermite cubic polynomials are adopted as shape functions. Mass, elastic stiffness and geometrical stiffness matrices of unsymmetrical cross-section beams are presented. In order to verify the accuracy of this theory and the corresponding beam element developed, a numerical study is presented and compared with the literature and experimental tests.

Second-Order Analysis of Plane Frames with One Element Per Member

A corotational element is developed directly from the governing second-order differential equations of beam theory. The corotational element includes not only the effect of axial force on the bending moment (P-delta effect) but also the additional axial strain caused by end rotations. Hinged and semirigid end conditions are also included so that plastic hinges could be considered. The resulting local element tangent stiffness matrix is compared to traditional local element elastic and geometric stiffness matrices. The method is implemented and executed on two example problems in which only one element per member is used. Results compare favorably to those from a nonlinear commercial program in which several elements per member are used. A third example problem includes both geometric and material nonlinearity to develop a pushover curve.

붐(Boom)의 탄성을 고려한 해상크레인의 비선형 정적/동적 거동을 위한 수치 해석

해상크레인은 크레인을 탑재한 선박으로서, 조선소에서 대형 블록이나 구조물의 탑재 및 해상 운송 작업에 사용된다. 본 논문에서는 해상크레인과 중량물의 전후 동요(Surge), 상하 동요(Heave), 종 동요(Pitch)에 대한 정적/동적 거동을 분석하였다. 이 때, 유연 다물체계 동역학을 적용하여 해상크레인의 붐(boom)을 탄성으로 고려하였으며, 플로팅 프레임(floating frame)과 노드 좌표(nodal coordinates)를 사용하였다. 질량 행렬, 탄성 강성 행렬, 2 차 속도 벡터, 일반화 좌표 방향으로 작용하는 외력 등을 고려하여 모든 운동이 연성된 비선형 운동 방정식을 구성하였다. 외력으로는 비선형 유체정역학 힘, 선형화된 유체동역학 힘, wire rope 힘, 계류력이 고려되었다. 수치 해석을 위해 Hilber-Hughes-Taylor 방법을 비선형 운동방정식에 적용하였다. 정적 거동 분석을 통한 정적 평형 자세를 고려한 경우와 고려하지 않은 경우에 대해 결과를 비교하였으며, 수치 해석 방법에 대한 정적/동적 거동 분석 결과를 비교하였다.

Energy equations for elastic flexural–torsional buckling analysis of plane structures

Lateral–torsional buckling is a critical mode of failure of metal structures. When the values of the loadings on a member of a structure reach a limiting state, the member will experience out-of-plane bending and twisting. This type of failure occurs suddenly in members with a much greater in-plane bending stiffness than torsional or lateral bending stiffness. Slender members of a structural system may buckle laterally and twist before their in-plane capabilities can be reached. Energy equations are derived by considering the total potential energy of a beam-column element. The second variation of the total potential energy equal to zero indicates the transition from a stable state to an unstable state, which is the critical condition for buckling. Several energy equations are derived analytically by calculating the second variation of the total potential energy of a double symmetric thin wall beam-column element. In this article, in-plane deformations of the beam-column element are disregarded. Then energy equations are derived expressing in dimensional and non-dimensional forms. These energy equations will be implemented in a future article to derive elastic and geometric stiffness matrices for the beam-column element and calculate the lateral–torsional buckling of plane structures. Examples are provided to show the accuracy of the equations and applications.

An improved formulation of space stiffeners

This paper presents a displacement based finite element model for predicting the constraint torsion effect of stiffeners. In structural modelling, the plate/shell and the stiffeners are treated as separate elements where the displacement compatibility transformation between these two types of elements takes into account the constraint torsional warping effect in the stiffeners. The development is based on a general beam theory which includes flexural-torsion coupling, constrained torsion warping, and shear-centre location. The virtual work principle includes the second order terms of finite beam rotations. For finite element analysis, cubic Hermitian polynomials are used as shape functions of the straight space frame element with two nodes. Elastic stiffness and geometric stiffness matrices for an arbitrary cross-section are evaluated in a closed form, and load correction stiffness for eccentric stiffener loads are considered. To demonstrate the importance of torsion warping constraints and to illustrate the accuracy of this formulation, finite element solutions are presented and compared with available solutions.

Stiffness Matrices for Linear and Buckling Analyses of Composite Beams with Partial Shear Connection

AbstractThis paper is concerned with linear and buckling analyses of composite beams with partial shear connection (partial composite beams) using the finite element method. Two elements derived from different types of shape functions are proposed in this study. The first element, referred to as exact, is based on the exact shape functions obtained by solving the differential equations governing the transverse displacement and the slip of the shear connector layer of a partial composite beam. The second element, referred to as approximate, is based on the conventional linear and cubic shape functions used in conventional axial and beam elements. By making use of these two types of shape functions, the elastic and geometric stiffness matrices can be derived explicitly from the strain energy and the load potential, respectively. Both types of elements can be used to carry out linear static and buckling analyses. As expected, the exact element is more accurate than the approximate element if the same discretization is adopted. However, the approximate element has the advantage of easy implementation since the expressions of its elastic and geometric stiffness matrices are very simple. Also, the solutions obtained from the approximate element converge very fast; with reasonable discretization, say 8 elements per member, very accurate solutions can be obtained.

Local Instability and Free Vibration of Supported Plates Using the Basic Plate Triangle

This paper extends the numerical method, originally developed by Onate and his colleagues for the static analysis of plates, to enable modal (vibration) and bifurcative (buckling) analyses of plates to be undertaken. The procedure uses a rotation-free triangular element, and is based on combining the finite element method and the finite volume method so as to produce a triangular element with only three displacement degrees of freedom. Elastic stiffness, stability (geometric stiffness) and mass matrices are developed using the mixed formulation, and these are implemented in routine computer coding in order to demonstrate the validity and efficacy of the solution technique when compared with results reported by independent investigators. Natural frequencies and local buckling coefficients are given for plates with interior line supports, and these are used to illustrate the prowess of the numerical technique for the analysis of a wide class of plate buckling and vibration problems, whose analysis by non-discretisation techniques has found favour recently in deference to general finite element modelling which tends to introduce some inaccuracies. The element developed in this paper, however, is both accurate and has the attractive meshing characteristics of the finite element method for analysing more arbitrary geometric structural topologies.

Effect of Tapering on Natural Frequencies of Rotating Beams

The problem of free vibration of a rotating tapered beam is investigated by developing explicit expressions for the mass, elastic and centrifugal stiffness matrices in terms of the taper ratios. This investigation takes into account the effect of tapering in two planes, the effect of hub radius as well as the stiffening effect of rotation. The equations of motion are derived; the associated generalized eigenvalue problem is defined in conjunction with a suitable Lagrangian form and solved for a wide range of parameter changes. The effect of tapering on the natural frequencies of the beam is examined with all parameter changes present. Results are compared with those available in literature and are found to be in excellent agreement.

Buckling mode decomposition of single-branched open cross-section members via finite strip method: Application and examples

The objective of this paper is to provide implementation details of, and practical examples for, modal decomposition of the cross-section stability modes of thin-walled members by constraining a traditional finite strip method (FSM) solution. The theoretical development of the proposed method is provided in a companion to this paper [Ádány S, Schafer BW. Buckling mode decomposition of single-branched open cross-section members via finite strip method: derivation. Thin-walled Structures, submitted for publication, companion to this paper.] The constraint matrix, which is directly applied to the elastic and geometric stiffness matrices of a traditional FSM solution in order to constrain the deformations, is provided along with all formulae necessary in its construction. In addition, a completely worked out numerical example is provided to aid in implementing the constrained FSM solution. The authors implemented the constrained FSM in the open source program CUFSM. This modified version of CUFSM is then used to provide a series of numerical examples that illustrate (i) the advantages of performing modal decomposition, (ii) the importance of understanding and defining the deformation fields related to a desired mode, and (iii) the behavior of constrained FSM stability solutions compared with classical analytical solutions, GBT, and unconstrained FSM. Decomposition of the cross-section buckling classes related to global and distortional modes is demonstrated. Further, the impact of how to select the deformation fields and perform modal decomposition for cross-section stability modes within a class, e.g., for the traditional three global modes (weak-axis flexure, strong-axis flexure and flexural-torsional buckling), is explored and the impact of the deformation field definitions demonstrated. Comparisons of the constrained FSM solutions with other available solutions demonstrate the importance of properly determining when beam theory and plate theory should apply to the cross-section stability of thin-walled members.

Stability of tapered I-Beams under torsional moments

Previous investigations on the buckling of torsionally loaded bars indicate that the critical loads depend upon the mechanical device that generates the end torque. By representing the torsional moment on an I-section as the combination of St. Venant and warping torsions, this study presents a finite element model for buckling analysis of tapered I-beams subjected to torsional moments. Regarding a tapered I-beam as the formation of three tapered narrow beams (the top flange, web, and the bottom flange), one can directly express the total potential energy equation of the tapered I-beam by summing up all the potential energy equations of these narrow beams based on Yang et al.'s well-developed potential equation of a tapered solid beam. As a result, the linear elastic and geometric stiffness matrices of a torsionally loaded I-beam element with tapered cross-section can be established from the total potential energy derived herein. It is worthy mentioning that the second-order effects of warping moments and non-uniform torsions have been included in the present geometric stiffness matrix. By buckling analysis, the numerical examples indicate that the warping and St. Venant torsions may significantly affect the flexural buckling resistance of tapered I-beams against torsional loadings. Besides, the critical loads of the tapered I-beams with higher warping rigidity would be reduced by the instability effect of warping moments.

Free vibration and stability analysis of stiffened sandwich plates

Description is given of the development of a B-spline finite strip method for predicting the natural frequencies of vibration and the buckling stresses of rectangular sandwich panels. Each panel comprises a flat sandwich main plate which is reinforced by a number of longitudinal compact plate stiffeners attached to the faceplates. The core of the main plate is represented as a three-dimensional solid whilst the faceplates and the stiffening plates may in general be composite laminates which are represented, in turn, as either shear-deformable plates or classically thin plates. Details are given of the development of the elastic stiffness, geometric stiffness and consistent mass matrices of the finite strip models. The nature of the structural equations and the sophisticated procedure for their solution are discussed. Both vibrational and buckling applications of the developed analysis capability are presented and these include situations in which the corresponding mode shapes are of an overall nature and, for some of the buckling problems, of a highly localised nature.

Push-over analysis for performance-based seismic design

The paper presents a simple computer-based push-over analysis technique for performance-based design of building frameworks subject to earthquake loading. The technique is based on the conventional displacement method of elastic analysis. Through the use of a ‘plasticity-factor’ that measures the degree of plastification, the standard elastic and geometric stiffness matrices for frame elements (beams, columns, etc.) are progressively modified to account for nonlinear elastic–plastic behavior under constant gravity loads and incrementally increasing lateral loads. The behavior model accounts for material inelasticity due to both single and combined stress states, and provides the ability to monitor the progressive plastification of frame elements and structural systems under increasing intensity of earthquake ground motion. The proposed analysis technique is illustrated for two building framework examples.

Consistent tangent stiffness for nonlocal damage models

This paper deals with the computational analysis of strain localization problems using nonlocal continuum damage models of the integral type. The general framework for a consistent derivation of the “nonlocal” tangent stiffness is presented. The properties of the tangent stiffness matrix are discussed and the corresponding assembly procedure is described. The quadratic rate of convergence of the Newton–Raphson iteration procedure is demonstrated and the efficiency of the proposed technique is compared to the standard approach based on the secant or elastic stiffness matrices. In this context, performance of direct and iterative solvers for the linearized equilibrium equations is also examined.

Incrementally small-deformation theory for nonlinear analysis of structural frames

An incrementally small-deformation theory that is physically self-explainable is presented for the large-displacement nonlinear analysis of structural frames. Strictly based on the assumption of small strains, small rotations, and small displacements within each incremental step, the elastic and geometric stiffness matrices for the beam element are derived from the force–displacement relations. Due consideration is taken of the 3D rotational behavior of nodal moments. The geometric stiffness matrix derived for the element is asymmetric. However, by enforcing all the joints to remain in equilibrium in the deformed configuration, the antisymmetric parts of the geometric stiffness matrices cancel out, resulting in a symmetric stiffness matrix for the structure. Also described is the procedure for updating the element forces and geometry in an incremental-iterative analysis. The present approach in its entire set is demonstrated to be robust and efficient for solving the nonlinear, postbuckling response of structural frames.

Combined loading of spudcan foundations on clay: numerical modelling

The load–displacement response of a spudcan foundation on clay is described by means of an incremental plasticity model with three degrees of freedom (vertical, rotational, horizontal). The model is termed ‘Model B’ and employs a yield surface and flow rule that are derived from a programme of carefully controlled small-scale laboratory tests. Behaviour inside the yield surface is defined by a set of elastic stiffness factors for embedded conical footings, determined from three-dimensional finite element analysis. The hardening law, which defines the vertical bearing capacity as a function of plastic spudcan penetration, is based on a set of theoretical lower bound bearing capacity factors for embedded conical footings. Care is needed to ensure consistent treatment of partially and fully penetrated spudcans under combined loading. Full details of the elastic and elasto-plastic stiffness matrices needed to predict incremental load–displacement behaviour are provided, and the performance of the model is demonstrated by simulating some of the laboratory calibration tests numerically. While Model B succeeds in capturing many important features of the observed spudcan behaviour, its prediction of sudden rather than gradual yielding is a noticeable weakness, particularly in the simulation of cyclic loading events. Practical applications of the foundation model are discussed with reference to the soil–structure interaction analysis of independent leg jack-up units.

Combined loading of spudcan foundations on clay: numerical modelling

The load–displacement response of a spudcan foundation on clay is described by means of an incremental plasticity model with three degrees of freedom (vertical, rotational, horizontal). The model is termed ‘Model B’ and employs a yield surface and flow rule that are derived from a programme of carefully controlled small-scale laboratory tests. Behaviour inside the yield surface is defined by a set of elastic stiffness factors for embedded conical footings, determined from three-dimensional finite element analysis. The hardening law, which defines the vertical bearing capacity as a function of plastic spudcan penetration, is based on a set of theoretical lower bound bearing capacity factors for embedded conical footings. Care is needed to ensure consistent treatment of partially and fully penetrated spudcans under combined loading. Full details of the elastic and elasto-plastic stiffness matrices needed to predict incremental load–displacement behaviour are provided, and the performance of the model is demonstrated by simulating some of the laboratory calibration tests numerically. While Model B succeeds in capturing many important features of the observed spudcan behaviour, its prediction of sudden rather than gradual yielding is a noticeable weakness, particularly in the simulation of cyclic loading events. Practical applications of the foundation model are discussed with reference to the soil–structure interaction analysis of independent leg jack-up units.