Terms Of Nodal Displacements Research Articles

On the Free Vibration and the Buckling Analysis of Laminated Composite Beams Subjected to Axial Force and End Moment: A Dynamic Finite Element Analysis

This work presents the bending–torsion coupled free vibration analysis of prestressed, layered composite beams subjected to axial force and end moment using the traditional finite element method (FEM) and dynamic finite element (DFE) techniques. Current trends in the literature, in terms of different types of modeling techniques and constraints, were briefly examined. The Galerkin-type weighted residual method was applied to convert the coupled differential equations of motion into a discrete problem using a polynomial interpolation function in the finite element method. In the dynamic finite element method, trigonometric shape functions were implemented to describe the equations in terms of nodal displacements. The eigenvalue problem resulting from the discretization along the length of the beam was solved in order to determine the system’s natural frequencies and modes. The results, showing the effects of axial load, end moment, and combined loading on natural frequencies, are discussed and are followed by some concluding remarks.

Optimal design of thin laminate plates for efficient airflow in ventilation via buckling

This work studies an optimal design problem to maximize the airflow opening of a thin-walled structure for efficient ventilation. The airflow opening is defined in terms of nodal displacements as the area swept by a deformed free edge, or the volume formed between the deformed and undeformed configurations or their combination. In this study, a novel formulation for the airflow opening in a plate or shell with large deflections is derived by using the principle of virtual work and the constructed generalized virtual force. In this formulation, the airflow opening can be evaluated via only two finite element analyses for one real and one virtual load cases. By using the present formulation, the maximum airflow problem is formulated as a topology optimization problem for a laminated thin-walled structure, and is subsequently solved by using a moving iso-surface threshold method. Numerical examples are presented for the optimum laminated flat plates of three edges simply supported and one edge free with designable and non-designable layers and under in-plane compression along the two opposite simply-supported edges. The numerical results show that the airflow opening can be maximized when the optimized laminated plate deforms in a shape similar to its critical buckling mode and the opening is much larger than that of the flat plate with the same amount of material and constant thickness.

A Review on Dynamic Analysis and Design of R.C.C Solids Storage Structure

Aim of research paper is to compare and briefly describe about the advantage and limitations of solid storage structure by using Staad Pro Structural software. Solid storage structures are considered as special structures as its design is based on the properties of materials stored. The pressure exerted by the stored material on the side of a bin varies with the processes and arrangements of filling and emptying operations. Due to this variation, it is extremely difficult to analyze the pressure exerted on the walls of the bins. In our research work, we are designing the RCC solid storage structure located in all seismic zones with the help of structural software Staad Pro. The design concept include, all dimensions of structural component based on trial and error method, using Equivalent lateral force method in term of Comparison of different models of concrete solid storage structure for earthquake such as nodal displacement, stress and vertical or horizontal pressure on walls etc. for volume of 180 m3. All the designs have been based on the recommendations of I.S 4995 -1974 (part 1&2) and I.S 456 – 2000 codes, Based on these designs, that dimension of solid storage structures shows least amount of concrete and steel. Main objective of our research work is to compare of different models of concrete solid storage structure for earthquake in terms of nodal displacement, stress and vertical or horizontal pressure on walls etc.

Fundamental-solution-based hybrid finite element with singularity control for two-dimensional mixed-mode crack problems

In this work, a fundamental-solution-based hybrid finite element method is presented for modeling mixed-mode cracks in two-dimensional (2D) isotropic elastic media. In the method, a double-variable hybrid functional for each element is formulated to derive element stiffness equation in terms of nodal displacements, which includes line integrals along the element boundary only. The element interior displacement field is approximated using a linear combination of displacement fundamental solutions at different source locations, while the independent element frame displacement field is approximated by one-dimensional shape function interpolation. To correctly model the behavior of crack-tip displacement, the discontinuous quarter-point crack-tip singular hybrid element formulation is developed, so that crack tip stress intensity factors can be easily evaluated by the near-tip displacement method. Three numerical examples of internal cracks in 2D elastic domains are presented to show the efficiency of the proposed method.

A unified approach for analysis of cable and tensegrity structures using memoryless quasi-newton minimization of total strain energy

A unifying approach is presented for the nonlinear static analysis of cable structures and for the form-finding of tensegrity structures. The novelty lies in the possibility of static analyses of structures where the stiffness matrix is singular throughout the path to equilibrium. The unification of static analyzes and form-finding procedures allows the understanding and treatment of tensegrity and cable structures as a single type of structure. A total potential energy function is derived in terms of nodal displacements which are the unknowns of a nonlinear programming problem. The proposed approach uses a Quasi-Newton method overcoming a limitation of the Newton Raphson Method employed by widel-used commercial Finite Element software packages. Example analyses are presented and compared with experimental results reported in the literature to demonstrate the feasibility of the proposed approach which is particularly useful for under-constrained structures that contain pre-tensioned elements.

A Strut Finite Element for Exact Incompressible Isotropic Hyperelastic Analysis

Abstract This text describes a mathematical model of a strut finite element for isotropic incompressible hyperelastic materials. The invariants of the Right Cauchy-Green deformation tensor are written in terms of nodal displacements. The equilibrium problem is formulated as an unconstrained nonlinear programming problem, where the objective function is the total potential energy of the structure and the nodal displacements are the unknowns. The constraint for incompressibility is satisfied exactly, thereby eliminating the need for a penalty function. The results of the examples calculated by the proposed mathematical model show five significant digits in agreement when compared with commercial finite element analysis software.

Static analysis of shear flexible beams and frames in adhesive contact with an isotropic elastic half-plane using a coupled FE–BIE model

Shear deformable beams and frames in perfect adhesion with an isotropic elastic half-space are analysed in plane strain or plain stress states. By means of a mixed variational formulation, the beam elements are described in terms of nodal displacements and rotations using locking-free shape functions, whereas the soil substrate is represented in terms of surface tractions through a boundary integral equation that incorporates a suitable Green’s function. The formulation ensures full continuity between structure and substrate in terms of displacements and rotations.A variety of numerical examples is presented to show the effectiveness of the proposed model. For very stiff beams subjected to a point force or moment, comparisons are made with available closed-form solutions to the contact problem of a rigid indenter. For foundation beams bonded to the substrate and loaded by a vertical point force the proposed model exhibits a superior convergence rate in comparison with other standard numerical models. Moreover, the shear deformations are shown to play a crucial role on both beam displacements and soil surface tractions. Finally, the soil–structure interaction analysis of a two-bay plane frame is presented.

A meshless local Petrov–Galerkin method for nonlinear dynamic analyses of hyper-elastic FG thick hollow cylinder with Rayleigh damping

This paper is devoted to the geometrically nonlinear analysis of a functionally graded (FG) thick hollow cylinder with Rayleigh damping. The hollow cylinder is subjected to axisymmetric mechanical shock loading on its bounding surfaces. First, the meshless local Petrov–Galerkin (MLPG) method is developed for geometrically nonlinear problems based on total Lagrangian approach. During this process, the local integral equations are obtained using the weak formulation on local sub-domains for the set of governing equations by employing a Heaviside test function. The radial point interpolation method is used to approximate the field variables in terms of nodal displacements. The iterative Newmark/Newton–Raphson method is employed to solve the system of resulting nonlinear equations in suitable time steps. Because of large deformations compared to linear elastic materials, the hyper-elastic neo-Hookean model is considered for the problem. The hollow cylinder is supposed to be in plane strain condition. The properties of the FG cylinder are varied in the thickness direction using the volume fraction that is an exponential function of radius. At the end, to prove the robustness of the proposed method, several numerical tests are performed and effects of relative parameters on the dynamic behavior of the cylinder for various kinds of FGMs are discussed in detail. Findings demonstrate the effectiveness of the presented MLPG method for large deformation problems because of vanishing of the mesh distortion. This paper furnishes a ground to develop the MLPG method for dynamic large deformation problems.

Large displacement geometrically nonlinear finite element analysis of 3D Timoshenko fiber beam element

Based on continuum mechanics and the principle of virtual displacements, incremental total Lagrangian formulation (T.L.) and incremental updated Lagrangian formulation (U.L.) were presented. Both T.L. and U.L. considered the large displacement stiffness matrix, which was modified to be symmetrical matrix. According to the incremental updated Lagrangian formulation, small strain, large displacement, finite rotation of three dimensional Timoshenko fiber beam element tangent stiffness matrix was developed. Considering large displacement and finite rotation, a new type of tangent stiffness matrix of the beam element was developed. According to the basic assumption of plane section, the displacement field of an arbitrary fiber was presented in terms of nodal displacement of centroid of cross-area. In addition, shear deformation effect was taken account. Furthermore, a nonlinear finite element method program has been developed and several examples were tested to demonstrate the accuracy and generality of the three dimensional beam element.

Interpolation–free discrete modeling with gradient matrix: Case study of edge dislocation in linearly elastic crystal

A meshless and interpolation-free (MIF) method of numerical modeling in solid mechanics has been formulated with a gradient matrix extending the gradient operation to discrete data. Nodal strains and, consequently, stresses in this method are expressed immediately in terms of nodal displacements and the stress divergence in terms of nodal stresses that makes it possible to get the stress balance equation in a truly discrete form. A trial MIF model where nodal points correspond to atom positions is employed for a rectilinear edge dislocation in a linearly elastic crystal. Both the resulting stress level at the dislocation core, close to the theoretical crystal strength, and respective core dimensions prove to be realistic physically whereas calculated long-range stresses asymptotically approach the related continuous fields known in an analytical form for the dislocation in linearly elastic isotropic continuum.

Fundamental-solution-based hybrid FEM for plane elasticity with special elements

The present paper develops a new type of hybrid finite element model with regular and special fundamental solutions (also known as Green's functions) as internal interpolation functions for analyzing plane elastic problems in structures weakened by circular holes. A variational functional used in the proposed model is first constructed, and then, the assumed intra-element displacement fields satisfying a priori the governing partial differential equations of the problem under consideration is constructed using a linear combination of fundamental solutions at a number of source points outside the element domain, as was done in the method of fundamental solutions. To ensure continuity of fields over inter-element boundaries, conventional shape functions are employed to construct the independent element frame displacement fields defined over the element boundary. The linkage of these two independent fields and the element stiffness equations in terms of nodal displacements are enforced by the minimization of the proposed variational functional. Special-purpose Green's functions associated with circular holes are used to construct a special circular hole element to effectively handle stress concentration problems without complicated local mesh refinement or mesh regeneration around the hole. The practical efficiency of the proposed element model is assessed via several numerical examples.

A numerical framework for material characterisation of inhomogeneous hyperelastic membranes by inverse analysis

An inverse method for estimating the distributions of the elastic properties of hyperelastic, inhomogeneous membranes is proposed. The material description of the membrane is based on a versatile constitutive model, in which two stiffness parameters govern the nonlinear elastic behaviour of the material. The estimation procedure includes a finite element framework. The two stiffness parameters in the constitutive law are assumed to vary continuously over the inhomogeneous membrane, and in the finite element framework the distributions of the two parameters are approximated using standard linear shape functions. Experimental results are assumed to exist in terms of nodal displacements from a test with known geometry and boundary conditions. The experimental membrane is modelled in the finite element framework, and the deformation of it is predicted. An error function, quantifying the discrepancy between the experimentally obtained deformation pattern and the numerically predicted pattern, is then minimised with respect to the nodal values of the two interpolated parameters. In a number of numerical examples, the proposed procedure is assessed by attempting to reproduce the given random reference distributions of material properties. The proposed estimation method is fully able to reproduce the reference distributions of the two material parameters with excellent accuracy.

A MLPG(LBIE) approach in combination with BEM

The local boundary integral equation (LBIE) approach is a promising meshless method, recently proposed as an effective alternative to the boundary element method (BEM), for solving non-homogeneous, anisotropic and non-linear problems. Since the LBIE method utilizes in its weak form fundamental solutions as test functions, it can be considered as one of the six meshless local Petrov–Galerkin (MLPG) methods proposed by Atluri and coworkers. This explains the use of the initials MLPG(LBIE) in the title of the present paper. This work addresses a coupling of a new MLPG(LBIE) method, recently proposed by the authors for elastodynamic problems, and the BEM. Because both methods conclude to a final system of linear equations expressed in terms of nodal displacement and tractions, their combination is accomplished directly with no further transformations as it happens in other MLPG/BEM formulations as well as in typical hybrid finite element method/BEM schemes. The coupling approach is demonstrated for static and frequency domain elastodynamic problems. Three representative examples are provided in order to illustrate the achieved accuracy of the proposed here MLPG(LBIE)/BE methodology.

A planar flexible robotic manipulator

Presents the dynamic modelling of a flexible robotic manipulator with two flexible links and two revolute joints, which rotates in the horizontal plane. The dynamic equations are derived using the Newton‐Euler formulation and the finite element method, based on elementary beam theory, which is used to discretize the displacements such that the small motion is represented in terms of nodal displacements. Computer simulation results are presented to illustrate this study. The dynamic model becomes necessary for use in future design and control applications.

An analytical method for mixed-mode propagation of pressurized fractures in remotely compressed rocks

An analytical method for mixed-mode (mode I and mode II) propagation of pressurized fractures in remotely compressed rocks is presented in this paper. Stress intensity factors for such fractured rocks subjected to two-dimensional stress system are formulated approximately. A sequential crack tip propagation algorithm is developed in conjunction with the maximum tensile stress criterion for crack extension. For updating stress intensity factors during crack tip propagation, a dynamic fictitious fracture plane is used. Based on the displacement correlation technique, which is usually used in boundary element/finite element analyses, for computing stress intensity factors in terms of nodal displacements, further simplification in the estimation of crack opening and sliding displacements is suggested. The proposed method is verified comparing results (stress intensity factors, propagation paths and crack opening and sliding displacements) with that obtained from a boundary element based program and available in literatures. Results are found in good agreements for all the verification examples, while the proposed method requires a trivial computing time.

A NEW SYMMETRIC AND POSITIVE DEFINITE BOUNDARY ELEMENT FORMULATION FOR LATERAL VIBRATIONS OF PLATES

A new symmetric and positive definite boundary element method in the time domain is presented for the dynamic analysis of thin elastic plates. The governing equations of the problem are obtained from a variational principle in which a hybrid modified functional is employed. The functional is expressed in terms of the domain and boundary basic variables in plate bending, assumed to be independent of each other. In the discretized model the boundary variables are expressed by nodal values, whereas the internal displacement field is modelled by a superposition of static fundamental solutions. The equations of motion are deduced from the functional stationarity conditions and they constitute a linear system of ordinary differential equations expressed in terms of nodal displacements. The structural operators are frequency independent and preserve the symmetry and definiteness properties of the continuum. Moreover these operators are calculated by performing boundary integrations of regular kernels only. Some applications to free vibrations and self-excited vibrations under aerodynamic loads have been worked out to test the accuracy of the proposed method. The numerical results have been found in very good agreement with those obtained by using other solution techniques by which the accuracy of the present boundary element model is demonstrated.

Nonlinear Dynamic Analysis of Fluid‐Structure Systems

A numerical procedure for computing the dynamic response of coupled fluid‐structure systems is developed and used to evaluate the effects of nonlinear behavior of the structure and fluid. The procedure includes cavitation of the fluid, which is modeled as a bilinear compressible material, in addition to nonlinear models of the structure. A mixed displacement‐pressure formulation for the fluid is used, in which the pressure is approximated independently of the fluid displacements, allowing satisfaction of the bilinear equation of state in the weak sense and not requiring reduced integration for the tangent stiffness matrices. A standard displacement finite‐element formulation is used for the structure. The symmetric coupled equations of motion are in terms of nodal displacements and are solved by a fully implicit time integration method. Response results from analysis of a simple fluid‐structure system show that cavitation and nonlinear structural behavior can interact to affect the response of the coupled...

An efficient finite element scheme for elastic porous media

A semi-analytical finite element scheme for the analysis of diffusion process in linear elastic porous media is presented. Variational principle based on Biot's Theory serves to establish discretized equilibrium and flow equations in terms of nodal displacements and fluid pore pressures. Time dependency of the system is removed by the Laplace transformation.The Laplace transforms of nodal pore pressures can be obtained by solving a standard eigenvalue system after releasing all nodal displacements through condensation. In the case of step loads commonly found in consolidation problems, die Laplace transforms can be inverted analytically. For general loading however, a series of step loads is introduced as its approximation, and the final solution is obtained by superposiiton.Three types of quadrilateral plane-strain finite elements are tested for their performance in solving 2-D consolidation problems by the present scheme. The quadrilateral element with eight displacement and four pore pressure nodes shows the best overall performance. Results from numerical examples indicate that the present scheme is extremely efficient. Solution of both the pore pressure and the displacement fields at any specified time can be determined explicitly without intermediate solutions.

The use of assumed displacement fields in finite-element analysis to calculate the gravity effect of crustal deformation

Abstract An expression is derived which relates a displacement field u defined on a continuum to the change in gravity (ΔgD) brought about by deformation: x[ (z − R 3 − U 3 α 3 (R 1 , R 2 , R 3 - (z − R 3 − U 3 α 3 [T D (R 1 , R 2 , R 3 ] ] dR 1 dR 2 dR 3 where GG is the gravitational constant; ρR is the pre-deformation density function; VR is the volume occupied by the continuum before deformation; TD is a transformation which represents the one-to-one mapping of VR onto the volume VA occupied by the continuum after deformation: A1 = R1 + U1(R1, R2, R3)TD: A2 = R2 + U2 (R1, R2, R3)A3 = R3 + U3 ( R1, R2, R3) where A1, a2, A3, are the Cartesian coordinates of a point in VA; R1, R2, R3 are the Cartesian coordinates of a point in VR; U1, U2, U3, are the components of u ; and α is the distance from the field point (x, y, z) to a general point in VA or VR. The derivation is based on the principle of conservation of mass as stated in continuum mechanics. The formulation of the finite-element procedure for problems in linear elasticity typically involves the selection of an assumed function for u within each element. Unknown constants which characterize the function are determined in terms of nodal displacements. The integral in the first equation above may therefore be evaluated for each element, and the results summed to obtain ΔgD. Convergence to the exact solution for ΔgD depends on the convergence of the finite-element solution for u . The procedure is demonstrated for axisymmetric, torsion-free deformation of a solid of revolution in which the elements are tori of triangular cross-section. These results may be useful in the analysis of gravity change associated with tectonic deformation of the earth's crust.

Finite element analysis of rubber‐like materials by a mixed model

AbstractA Reissner type variational principle is utilized to formulate a mixed finite element model for a finite‐strain analysis of Mooney‐Rivlin rubber‐like materials. An incremental and stationary Lagrangian formulation is adopted. The functional consists of incremental displacements and incremental hydrostatic and distortional stresses as variables. In the finite element formulation the displacements are interpolated in terms of nodal displacements while the two different strss components are approximated independently. The stress parameters for the distortional stresses are eliminated at the element level and the resulting matrix equations for each incremental solution involve the incremental nodal displacements and the average hydrostatic pressure in each element as unknowns. Four‐node quadrilateral plane stain elements were used to analyze the inflation of an infinitely long thick‐walled cylinder subjected to internal pressure. Both resulting displacements and stresses are found to converge to exact values as the magnitude of the loading increments is decreasing.