• Structural Engineering and Mechanics
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• v.35 no.6
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• pp.677-697
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• 2010

This paper focuses on geometrically non-linear static analysis of a simply supported beam made of hyperelastic material subjected to a non-follower transversal uniformly distributed load. As it is known, the line of action of follower forces is affected by the deformation of the elastic system on which they act and therefore such forces are non-conservative. The material of the beam is assumed as isotropic and hyperelastic. Two types of simply supported beams are considered which have the following boundary conditions: 1) There is a pin at left end and a roller at right end of the beam (pinned-rolled beam). 2) Both ends of the beam are supported by pins (pinned-pinned beam). In this study, finite element model of the beam is constructed by using total Lagrangian finite element model of two dimensional continuum for a twelve-node quadratic element. The considered highly non-linear problem is solved by using incremental displacement-based finite element method in conjunction with Newton-Raphson iteration method. In order to use the solution procedures of Newton-Raphson type, there is need to linearized equilibrium equations, which can be achieved through the linearization of the principle of virtual work in its continuum form. In the study, the effect of the large deflections and rotations on the displacements and the normal stress and the shear stress distributions through the thickness of the beam is investigated in detail. It is known that in the failure analysis, the most important quantities are the principal normal stresses and the maximum shear stress. Therefore these stresses are investigated in detail. The convergence studies are performed for various numbers of finite elements. The effects of the geometric non-linearity and pinned-pinned and pinned-rolled support conditions on the displacements and on the stresses are investigated. By using a twelve-node quadratic element, the free boundary conditions are satisfied and very good stress diagrams are obtained. Also, some of the results of the total Lagrangian finite element model of two dimensional continuum for a twelve-node quadratic element are compared with the results of SAP2000 packet program. Numerical results show that geometrical nonlinearity plays very important role in the static responses of the beam.

• Journal of Theoretical and Applied Mechanics
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• v.1 no.1
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• pp.97-109
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• 1995

In this paper, a finite viscoelastic continuum model for rubber and its finite element analysis are presented. This finite viscoelatic model based on continuum mechanics is an extended model of Johnson and Wuigley's 1-D model. In this extended model, continuum based kinematic measures are rigorously defied and by using this kinematic measures, elastic stage law and flow rule are introduced. In kinematics, three configuration are introduced. In kinematics, three configuration are introduced. They are reference, current and virtual visco configurations. In elastic state law, it is assumed that at a certain time, there exists an elastic potential which describes the recoverable elastic energy. From this elastic potential, elastic state law is derived. The proposed flow rule is based on phenomenological observation. The flow rule gives precise relaxation response. In finite element approximation, mixed Lagrangian description is used, where total and similar method of updated Lagrangian descriptions are used together. This approach reduces the numerical job and gives simple nonlinear syatem of equations. To satisfy the incompressible condition, penalty-type modified Mooney-Rivlin energy function is adopted. By this method nearly incompressible condition is obtain the virtual visco configuration. For verification, uniaxial stretch tests are simulated for various stretch rates. The simulated results show good agreement with experiments. As a practical experiments. As a preactical example, pressurized rubber plate is simulated. The result shows finite viscoelastic effects clearly.

• Structural Engineering and Mechanics
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• v.40 no.3
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• pp.347-371
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• 2011

This paper focuses on post-buckling analysis of Timoshenko beams with various boundary conditions subjected to a non-uniform thermal loading by using the total Lagrangian Timoshenko beam element approximation. Six types of support conditions for the beams are considered. The considered highly non-linear problem is solved by using incremental displacement-based finite element method in conjunction with Newton-Raphson iteration method. As far as the authors know, there is no study on the post-buckling analysis of Timoshenko beams under uniform and non-uniform thermal loading considering full geometric non-linearity investigated by using finite element method. The convergence studies are made and the obtained results are compared with the published results. In the study, the relationships between deflections, end rotational angles, end constraint forces, thermal buckling configuration, stress distributions through the thickness of the beams and temperature rising are illustrated in detail in post-buckling case.

• Structural Engineering and Mechanics
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• v.41 no.6
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• pp.775-789
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• 2012

This paper focuses on post-buckling analysis of functionally graded Timoshenko beam subjected to thermal loading by using the total Lagrangian Timoshenko beam element approximation. Material properties of the beam change in the thickness direction according to a power-law function. The beam is clamped at both ends. The considered highly non-linear problem is solved by using incremental displacement-based finite element method in conjunction with Newton-Raphson iteration method. As far as the authors know, there is no study on the post-buckling analysis of functionally graded Timoshenko beams under thermal loading considering full geometric non-linearity investigated by using finite element method. The convergence studies are made and the obtained results are compared with the published results. In the study, with the effects of material gradient property and thermal load, the relationships between deflections, end constraint forces, thermal buckling configuration and stress distributions through the thickness of the beams are illustrated in detail in post-buckling case.

• Coupled systems mechanics
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• v.6 no.4
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• pp.399-415
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• 2017

This paper deals with the nonlinear static deflections of functionally graded (FG) porous under thermal effect. Material properties vary in both position-dependent and temperature-dependent. The considered nonlinear problem is solved by using Total Lagrangian finite element method within two-dimensional (2-D) continuum model in the Newton-Raphson iteration method. In numerical examples, the effects of material distribution, porosity parameters, temperature rising on the nonlinear large deflections of FG beams are presented and discussed with porosity effects. Also, the effects of the different porosity models on the FG beams are investigated in temperature rising.

• Structural Engineering and Mechanics
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• v.51 no.6
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• pp.955-971
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• 2014

Large post-buckling behavior of Timoshenko beams subjected to non-follower axial compression loads are studied in this paper by using the total Lagrangian Timoshenko beam element approximation. Two types of support conditions for the beams are considered. In the case of beams subjected to compression loads, load rise causes compressible forces end therefore buckling and post-buckling phenomena occurs. It is known that post-buckling problems are geometrically nonlinear problems. The considered highly non-linear problem is solved considering full geometric non-linearity by using incremental displacement-based finite element method in conjunction with Newton-Raphson iteration method. There is no restriction on the magnitudes of deflections and rotations in contradistinction to von-Karman strain displacement relations of the beam. The beams considered in numerical examples are made of lower-Carbon Steel. In the study, the relationships between deflections, rotational angles, critical buckling loads, post-buckling configuration, Cauchy stress of the beams and load rising are illustrated in detail in post-buckling case.

• Structural Engineering and Mechanics
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• v.47 no.1
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• pp.27-44
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• 2013

A finite element computational procedure for the accurate analysis of quasistatic thermorheological complex structures response is developed. The geometrical nonlinearity, arising from large displacements and rotations (but small strains), is accounted for by the total Lagrangian description of motion. The Schapery's nonlinear single-integral viscoelastic constitutive model is modified for a time-stress-temperature-dependent behavior. The nonlinear thermo-viscoelastic constitutive equations are incrementalized leading to a recursive relationship and thereby the resulting finite element equations necessitate data storage from the previous time step only, and not the entire deformation history. The Newton-Raphson iterative scheme is employed to obtain a converged solution for the non-linear finite element equations. The developed numerical model is verified with the previously published works and a good agreement with them is found. The applicability of the developed model is demonstrated by analyzing two examples with different thermal/mechanical loading histories.

• Transactions of Materials Processing
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• v.4 no.2
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• pp.131-140
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• 1995

In this paper damage propagation of a material during forming is investigated with the concept of continuum damage mechanics. An isotropic damage model based on the theory of materials of type N is adopted to describe the damage process of a ductile material with large elasto-viscoplastic deformation. The stiffness degradation of the loaded material is chosen as a damage measure. The highly nonlinear equilibrium equations are reduced to the incremental weak form and approximated by the total Lagrangian finite element method. To simulate contact condition, extended interior penalty method with modified coulomb friction law is adopted. The displacement control method along with the modified Riks' continuation technique is used to solve the incremental iterative equations. As numerical examples, upsetting problem and backward extrusion problem are simulated and the results of damage propagation and $J_2$ stress contours with and without friction are presented.

• Steel and Composite Structures
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• v.30 no.4
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• pp.327-336
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• 2019

This paper presents geometrically nonlinear behavior of cracked fiber reinforced composite beams by using finite element method with and the first shear beam theory. Total Lagrangian approach is used in the nonlinear kinematic relations. The crack model is considered as the rotational spring which separate into two parts of beams. In the nonlinear solution, the Newton-Raphson is used with incremental displacement. The effects of fibre orientation angles, the volume fraction, the crack depth and locations of the cracks on the geometrically nonlinear deflections of fiber reinforced composite are examined and discussed in numerical results. Also, the difference between geometrically linear and nonlinear solutions for the cracked fiber reinforced composite beams.

• Structural Engineering and Mechanics
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• v.66 no.1
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• pp.27-36
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• 2018

The objective of this work is to analyze geometrically nonlinear static analysis a simply supported laminated composite beam subjected to a non-follower transversal point load at the midpoint of the beam. In the nonlinear model of the laminated beam, total Lagrangian finite element model of is used in conjunction with the Timoshenko beam theory. The considered non-linear problem is solved considering full geometric non-linearity by using incremental displacement-based finite element method in conjunction with Newton-Raphson iteration method. There is no restriction on the magnitudes of deflections and rotations in contradistinction to von-Karman strain displacement relations of the beam. In the numerical results, the effects of the fiber orientation angles and the stacking sequence of laminates on the nonlinear deflections and stresses of the composite laminated beam are examined and discussed. Convergence study is performed. Also, the difference between the geometrically linear and nonlinear analysis of laminated beam is investigated in detail.