Effects of shear deformation on large deflection behavior of elastic frames

In this paper, the effects of shear deformation on the large deflection behavior of elastic frames is investigated by the finite element method. A two-node nonlinear beam element with the shear deformation is formulated and employed to analyze some frame structures. The element based on the energy method is developed in the context of the corotational approach. A bracketing procedure u ed the lowest eigenvalue of structural tangent stiffness matrix as indicating parameter is adopted to compute the critical loads. An incremental iterative procedure with the arc-length control technique is employed to trace the equilibrium paths. The numerical results show that the shear deformation plays an important role in the critical load and the large deflection behavior of the frames constructed from the components having low slenderness. A detail investigation is carried out to highlight the influence of slenderness on the behavior of the frames under large deflection.

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Development of Bifurcation Analysis Method for Rate Dependent Materials

Key Engineering Materials ◽

10.4028/www.scientific.net/kem.794.220 ◽

2019 ◽

Vol 794 ◽

pp. 220-225

Author(s):

Daiki Towata ◽

Yuichi Tadano

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Bifurcation Analysis ◽

Stiffness Matrix ◽

Computational Method ◽

Analysis Method ◽

The Finite Element Method ◽

Tangent Stiffness Matrix ◽

Rate Dependent ◽

Element Method

In this study, a novel numerical method to analyze the bifurcation problemof a rate dependent material using the finite element method is proposed. The consistent stiffness matrix, which is required for a bifurcation analysis using the finite element method, for a rate dependent material is generally hard to compute, therefore, a computational method to calculate the tangent stiffness matrix based on a numerical differential is introduced so that exact bifurcation analyses for the rate dependent material can be conducted. A numerical example of the proposed method is demonstrated, and the adequacy of the proposed method is discussed.

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Exact stiffness matrix for a beam element with axial force and shear deformation

Finite Elements in Analysis and Design ◽

10.1016/0168-874x(95)00064-z ◽

1996 ◽

Vol 22 (1) ◽

pp. 1-13 ◽

Cited By ~ 5

Author(s):

Walter D. Pilkey ◽

Weize Kang

Keyword(s):

Shear Deformation ◽

Axial Force ◽

Stiffness Matrix ◽

Beam Element ◽

Exact Stiffness Matrix

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Dynamic Stiffness Analysis of a Beam Based on Trigonometric Shear Deformation Theory

Journal of Vibration and Acoustics ◽

10.1115/1.2775513 ◽

2007 ◽

Vol 130 (1) ◽

Cited By ~ 1

Author(s):

Jun Li ◽

Hongxing Hua ◽

Rongying Shen

Keyword(s):

Shear Deformation ◽

Stiffness Matrix ◽

Deformation Theory ◽

Dynamic Stiffness ◽

Equations Of Motion ◽

Shear Deformation Theory ◽

Beam Element ◽

Mode Shapes ◽

Dynamic Stiffness Matrix ◽

Stiffness Analysis

The dynamic stiffness matrix of a uniform isotropic beam element based on trigonometric shear deformation theory is developed in this paper. The theoretical expressions for the dynamic stiffness matrix elements are found directly, in an exact sense, by solving the governing differential equations of motion that describe the deformations of the beam element according to the trigonometric shear deformation theory, which include the sinusoidal variation of the axial displacement over the cross section of the beam. The application of the dynamic stiffness matrix to calculate the natural frequencies and normal mode shapes of two rectangular beams is discussed. The numerical results obtained are compared to the available solutions wherever possible and validate the accuracy and efficiency of the present approach.

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Beam element for large displacement analysis of elasto-plastic frames

Vietnam Journal of Mechanics ◽

10.15625/0866-7136/26/1/5688 ◽

2004 ◽

Vol 26 (1) ◽

pp. 39-54

Author(s):

Nguyen Dinh Kien ◽

Do Quoc Quang

Keyword(s):

Virtual Work ◽

Internal Force ◽

Beam Element ◽

Large Displacement ◽

Isotropic Hardening ◽

Arc Length ◽

Tangent Stiffness Matrix ◽

Strain Relationship ◽

Internal Force Vector ◽

Finite Element Formulations

The present paper develops a non-linear beam element for analysis of elastoplastic frames under large displacements. The finite element formulations are derived by using the co-rotational approach and expression of the virtual work. The Gauss quadrature is employed for numerically computing the element tangent stiffness matrix and internal force vector. A bilinear stress-strain relationship with isotropic hardening is adopted to update the stress. The arc-length technique based on the Newton-Raphson iterative method is employed to compute the equilibrium paths. A number of numerical examples is employed to assess the performance of the developed element. The effects of plastic action on the large displacement behavior of the structures as well as the expansion of plastic zones in the loading process are discussed.

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An analogy between analytical, approximate and numerical methods in nonlinear buckling of functionally graded columns

Journal of Engineering Design and Technology ◽

10.1108/jedt-03-2021-0158 ◽

2021 ◽

Vol ahead-of-print (ahead-of-print) ◽

Author(s):

Kaveh Salmalian ◽

Ali Alijani ◽

Habib Ramezannejad Azarboni

Keyword(s):

Differential Equation ◽

Finite Element ◽

Finite Difference ◽

Stiffness Matrix ◽

Functionally Graded ◽

The Finite Element Method ◽

Content Type ◽

Tangent Stiffness Matrix ◽

Initial Assumption ◽

Post Buckling

Purpose The purpose of this study is to investigate the post-buckling analysis of functionally graded columns by using three analytical, approximate and numerical methods. A pre-defined function as an initial assumption for the post-buckling path is introduced to solve the differential equation. The finite difference method is used to approximate the lateral deflection of the column based on the differential equation. Moreover, the finite element method is used to derive the tangent stiffness matrix of the column. Design/methodology/approach The non-linear buckling analysis of functionally graded materials is carried out by using three analytical, finite difference and finite element methods. The elastic deformation and Euler-Bernoulli beam theory are considered to establish the constitutive and kinematics relations, respectively. The governing differential equation of the post-buckling problem is derived through the energy method and the calculus variation. Findings An incremental iterative solution and the perturbation of the displacement vector at the critical buckling point are performed to determine the post-buckling path. The convergence of the finite element results and the effects of geometric and material characteristics on the post-buckling path are investigated. Originality/value The key point of the research is to compare three methods and to detect error sources by considering the derivation process of relations. This comparison shows that a non-incremental solution in the analytical and finite difference methods and an initial assumption in the analytical method lead to an error in results. However, the post-buckling path in the finite element method is traced by the updated tangent stiffness matrix in each load step without any initial limitation.

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Tangent Stiffness Matrix of Spatial Beam Element Considering Bend and Torsion Coupling Effect

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.1021.73 ◽

2014 ◽

Vol 1021 ◽

pp. 73-78

Author(s):

Liang Du ◽

Peng Lan ◽

Nian Li Lu

Keyword(s):

Stiffness Matrix ◽

Coupling Effect ◽

Beam Element ◽

High Accuracy ◽

Fem Model ◽

Tangent Stiffness ◽

Tangent Stiffness Matrix ◽

Spatial Beam ◽

Element Pattern ◽

Couple Effect

Based on the spatial beam-column differential equations, the slope deflection equations considering second-order and bend-torsion coupling effect are established. The additional moment caused by torsion and bend deflection are taken into account. Then the finite element pattern of spatial beam-column considering the couple effect of torsion and bend is given. The tangent stiffness matrix and relevant program for nonlinear analysis are further obtained. By the nonlinear calculation and stability analysis of single component which has high accuracy or precise solution, the precision of the FEM model given in this paper is verified by comparing the results with that given in references.

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Numerical equilibrium analysis of a stack of steel post pallets

Technical Sciences ◽

10.31648/ts.3964 ◽

2018 ◽

Vol 4 (21) ◽

pp. 271-280

Author(s):

Józef Pelc

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Quadratic Form ◽

Stiffness Matrix ◽

Equilibrium Analysis ◽

Critical Number ◽

The Finite Element Method ◽

Tangent Stiffness Matrix ◽

Forklift Truck ◽

The Impact

A method for analyzing the equilibrium of a stack of loaded post pallets is presented. The finite element method was used to investigate the behavior of the bottom pallet in the stack during the addition of successive pallets. The stack was regarded as a self-stable multi-storey structure without bracings which is subjected to the weight of loaded pallets, horizontal forces resulting from sway and bow imperfections, and the impact of a forklift truck. The definite quadratic form of the tangent stiffness matrix after every increment in load was determined by nonlinear analysis to indicate the loss of post stability. An analysis of the stacking process of the evaluated pallets did not reveal a buckling trend in the posts of the bottommost pallet and demonstrated that the loss of equilibrium can lead to the collapse of the entire stack when a critical number of pallets is reached.

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Finite element bending analysis of symmetric and non-symmetric functionally graded sandwich beams using a novel parabolic shear deformation theory

Proceedings of the Institution of Mechanical Engineers Part L Journal of Materials Design and Applications ◽

10.1177/14644207211005096 ◽

2021 ◽

pp. 146442072110050

Author(s):

Mohamed-Ouejdi Belarbi ◽

Abdelhak Khechai ◽

Aicha Bessaim ◽

Mohammed-Sid-Ahmed Houari ◽

Aman Garg ◽

...

Keyword(s):

Finite Element ◽

Finite Element Model ◽

Shear Deformation ◽

Transverse Shear ◽

Deformation Theory ◽

Shear Deformation Theory ◽

Element Model ◽

Beam Element ◽

Functionally Graded ◽

Sandwich Beams

In this paper, the bending behavior of functionally graded single-layered, symmetric and non-symmetric sandwich beams is investigated according to a new higher order shear deformation theory. Based on this theory, a novel parabolic shear deformation function is developed and applied to investigate the bending response of sandwich beams with homogeneous hardcore and softcore. The present theory provides an accurate parabolic distribution of transverse shear stress across the thickness and satisfies the zero traction boundary conditions on the top and bottom surfaces of the functionally graded sandwich beam without using any shear correction factors. The governing equations derived herein are solved by employing the finite element method using a two-node beam element, developed for this purpose. The material properties of functionally graded sandwich beams are graded through the thickness according to the power-law distribution. The predictive capability of the proposed finite element model is demonstrated through illustrative examples. Four types of beam support, i.e. simply-simply, clamped-free, clamped–clamped, and clamped-simply, are used to study how the beam deflection and both axial and transverse shear stresses are affected by the variation of volume fraction index and beam length-to-height ratio. Results of the numerical analysis have been reported and compared with those available in the open literature to evaluate the accuracy and robustness of the proposed finite element model. The comparisons with other higher order shear deformation theories verify that the proposed beam element is accurate, presents fast rate of convergence to the reference results and it is also valid for both thin and thick functionally graded sandwich beams. Further, some new results are reported in the current study, which will serve as a benchmark for future research.

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