Explicit Stiffness Matrix Research Articles

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.

Mixed Finite Element Matrix for Plate Bending Analysis Using Weighted-Residual Integral and Weak Formulation

The Conventional Finite Element Method for the analysis of plate bending based on a the principle of stationarity of total potential energy which is well known as Stiffness Formulation , using a variety of conformal and no-conformal elements was widely applicable in early of numerical solutions for such problems , later in order to improve the efficiency of such solutions , a more general and flexible formulation called Mixed Formulation based on variational principles which can be regarded as an extension of the Stiffness principle become important alternative. The explicit stiffness matrix for plate bending was given in the most if it`s not in all literatures so the reader can easy follow the solving procedure for such problems and verifying any published results , however such like-matrix (Augmented Matrix) in case of mixed formulation not given , at least for simple elements in the literatures as well as concerned researches . This quietly leads to some difficulties concerning the verification as well as understanding the published results. The main objective of this paper is to introduce this matrix in abbreviated size followed by a applicability through a some detailed examples for some bending models. The derived matrix will be helpful subject of research work in addition to , reveals a very good feeling with understanding and verification the published results in plus to comparing with the analytical solutions of different plate bending problems as the reader can do that.

Mixed Finite Element Matrix for Plate Bending Analysis Using Weighted-Residual Integral and Weak Formulation

The Conventional Finite Element Method for the analysis of plate bending based on a the principle of stationarity of total potential energy which is well known as Stiffness Formulation , using a variety of conformal and no-conformal elements was widely applicable in early of numerical solutions for such problems , later in order to improve the efficiency of such solutions , a more general and flexible formulation called Mixed Formulation based on variational principles which can be regarded as an extension of the Stiffness principle become important alternative. The explicit stiffness matrix for plate bending was given in the most if it`s not in all literatures so the reader can easy follow the solving procedure for such problems and verifying any published results , however such like-matrix (Augmented Matrix) in case of mixed formulation not given , at least for simple elements in the literatures as well as concerned researches . This quietly leads to some difficulties concerning the verification as well as understanding the published results. The main objective of this paper is to introduce this matrix in abbreviated size followed by a applicability through a some detailed examples for some bending models. The derived matrix will be helpful subject of research work in addition to , reveals a very good feeling with understanding and verification the published results in plus to comparing with the analytical solutions of different plate bending problems as the reader can do that.

A high-performance four-node flat shell element with drilling degrees of freedom

This paper presents a new four-node quadrilateral flat shell element, named QFSUQ, for analysis of shell structures. The element is formed by assemblage of a new membrane element and a plate-bending element. The membrane component is an unsymmetric quadrilateral element with drilling degrees of freedom. The trial functions of the membrane element are determined using the element stress fields formulated based on the analytical solutions of the Airy stress function in global Cartesian coordinate system. The corresponding test functions are obtained through the four-node isoparametric-based displacement fields which are enhanced by drilling rotations. The bending component is based on the Hellinger–Reissner variational principle for analysis of Reissner–Mindlin plates. To validate the performance of the proposed element several numerical benchmark problems are employed and the obtained results are compared with other shell elements. The results prove that the QFSUQ element preserves the advantages of the parent element formulation namely explicit stiffness matrix, free of membrane locking, shear locking and singularity problems and is also appropriate in analysis of shell structures with complex geometry, loading and boundary conditions.

Static and dynamic analysis of circular beams using explicit stiffness matrix

Two new elements with six degrees of freedom are proposed by applying the equilibrium conditions and strain-displacement equations. The first element is formulated for the infinite ratio of beam radius to thickness. In the second one, theory of the thick beam is used. Advantage of these elements is that by utilizing only one element, the exact solution will be obtained. Due to incorporating equilibrium conditions in the presented formulations, both proposed elements gave the precise internal forces. By solving some numerical tests, the high performance of the recommended formulations and also, interaction effects of the bending and axial forces will be demonstrated. While the second element has less error than the first one in thick regimes, the first element can be used for all regimes due to simplicity and good convergence. Based on static responses, it can be deduced that the first element is efficient for all the range of structural characteristics. The free vibration analysis will be performed using the first element. The results of static and dynamic tests show no deficiency, such as, shear and membrane locking and excessive stiff structural behavior.

Linear and geometrically nonlinear analysis with 4-node plane quasi-conforming element with internal parameters

A linear 4-node quadrilateral quasi-conforming plane element with internal parameters is proposed. The element preserves advantages of the quasi-conforming technique, including an explicit stiffness matrix, which can be applied to nonlinear problems. The weak patch test guarantees the convergence of the element. Then the linear element is extended to the geometrically nonlinear analysis in the framework of Total Lagrangian (TL) formulation. The numerical tests indicate that the present element is accurate and insensitive to mesh distortion.

The explicit stiffness matrix of the 15-noded cubic strain triangle

This paper presents the explicit stiffness matrix of the 15-noded cubic strain triangle, for a cross-anisotropic medium, which was evolved through the use of area coordinates. This stiffness matrix would be very useful in finite-element analysis of geomechanics problems such as collapse, vibration and consolidation because it will drastically reduce computer run-time through a faster assembly of the global stiffness matrix.

STATIC AND DYNAMIC NONLINEAR ANALYSIS OF SEMI-RIGID STEEL FRAMES WITH NEW BEAM-COLUMN ELEMENT

One of the important issues in the study of steel frames is to find a suitable formulation for semi-rigid connections. In this paper, the explicit stiffness matrix for a two-dimensional beam- column element having end-flexibilities is derived. The effects of the lateral uniformly distributed load on the deflection are considered. Both the tensile and compressive axial loads are also taken into account by one formula. Using the proposed stiffness matrix, some first-order, second-order, buckling, and dynamic analyses for semi-rigid frames are performed. The plastic analysis is also carried out using the plastic hinge approach. Comparing the calculated results with other references shows the accuracy and capabilities of the utilized element. Furthermore, the influences of the semi- rigid connections on the static and dynamic responses are investigated.

Large‐Strain Analysis of Reinforced Membranes

A theoretical formulation is developed for large‐strain analysis of reinforced, multilayer membranes that contain nonlinear, viscoelastic materials and are subjected to unidirectional, in‐plane loading. Regularity conditions, satisfied by the displacement fields of interest, allow the problem to be formulated in terms of a single, in‐plane component of the displacement field, even when strains are large. The theory is implemented in a finite‐element software package called REMA (reinforced membrane analysis), designed to run on a personal computer. An explicit stiffness matrix is used. The software is designed to investigate a wide range of membrane behaviors and to model the sequential stages of reinforcement and matrix subcomponent failure. Here, it is used to predict the load‐deflection behavior and sequence of subcomponent failures produced by typical in situ loading of a reinforced roofing membrane. The predictions are compared with corresponding laboratory tests.

Stress analysis of flat plates with shear using explicit stiffness matrix

AbstractAn explicit expression for the stiffness matrix is worked out for a triangular plate bending element considering the effect of transverse shear deformation. The element has twelve nodes on the sides and four nodes internal to it. The formulation is displacement type and the use of area co‐ordinates makes it possible to obtain the shape functions explicitly. Separate polynomials are assumed for transverse displacement and rotations. To obtain the element stiffness matrix no matrix inversion or numerical integration need be carried out and only a few matrix multiplications of low order are necessary. The element, which is initially of thirty five degrees of freedom, can be reduced to a thirty degrees of freedom one by condensation of the internal nodes. An interesting feature of the element developed is that the values of nodal moments computed at a node point, considering different elements surrounding the node, do not vary significantly. Thus the nodal moments can be obtained directly at node points. Also, the element does not give rise to any inconvenience like locking, even for very thin plates. The straightforward approach in formation of the element stiffness will cut down the storage space considerably and will also call for less CPU time, thus making the use of the element well suited to low capacity computers. A number of plate bending problems have been worked out using the present element for different thickness to side ratios and a comparison has been made with the available results. Good accuracy has been observed in all cases, even for a small number of elements.

Explicit stiffness matrices for linear, quadratic and cubic rectangular prism elements of the serendipity family under three dimensional loading

Analytical integration formulae for the products of shape function local derivatives of linear, quadratic and cubic three dimensional elements of the serendipity family over the square prism are first obtained. By means of a transformation from global to local coordinates, the analytical integration formulae for the products of shape function global derivatives over the rectangular prism are then established. It is shown that these formulae are helpful in the formation of element stiffness matrices of the rectangular prism elements of linear three dimensional elasticity. Numerical integration is totally avoided. Numerical values of all triple integrals for linear and quadratic rectangular prism elements are given in Appendix A. Explicit stiffness matrix coefficients for the 20-node quadratic rectangular prism element of the serendipity family are listed in Appendix B.

Beams, Frames and Shells on Elastic Foundations

Explicit stiffness matrix, present matrix and nodal forces are used as a basis for computer analysis of beams, rings, simple frames, and branched shells of revolution on a three-parameter foundation model. This technique demands neither auxiliary handling nor does it result in a decrease in accuracy when applied to the beam or shell with a general variation of the elastic modulus, thickness, or foundation factors along the beam or shell meridian.

Explicit stiffness matrix of the linearly varying strain triangular element

An explicit stiffness matrix for a linearly varying strain triangular element is derived. It is suggested that this new stiffness matrix is compotationaly efficient to use.