Updated Lagrangian Method Research Articles

Nonlinear finite element analysis of a laminated cylindrical shell with transverse matrix cracks

A clamped laminated cylindrical shell is presented to investigate nonlinear structural behavior involving geometrically nonlinear deformation. In the investigation, transverse matrix cracks are considered in the stiffness of the laminated cylindrical shell. Stiffness degradation is examined for several laminated angles and transverse crack density. Micro-mechanics theory on the composite material was used to derive the degraded stiffness of the laminated cylindrical shell due to the crack density. Iterative numerical scheme was developed to calculate the degraded composite stiffness which is a complicated relation with the crack density. A nonlinear finite element program was developed using 3-D degenerated shell element and the fist order shear deformation theory to consider the large deformation of the clamped laminated cylindrical shell. The updated Lagrangian method is used for nonlinear finite element analysis. Nonlinear structural responses of the laminated cylindrical shell were examined for various stacking sequences and crack density under transversely loaded pressure. Also, the effect of crack opening/closed was considered in the examination. Through this study, it is realized that the transverse matrix crack causes moderate stiffness reduction and affects the responses of the composite shell.

Numerical simulation of ground flow caused by seismic liquefaction

Flow failure of sandy subsoil induced by seismic liquefaction is known to cause significant damage to structures. It is induced not only by the dynamic forces exerted by seismic acceleration but also by the static gravity force in consequence of the topography of the ground. The ground flow may sometimes continue after the end of the seismic loading and finally the ground is significantly deformed to cause a failure. This paper numerically predicts the magnitude of flow that could occur when soil liquefaction continues for a sufficiently long period. It is considered that liquefied soil behaves like a viscous liquid, and hence, ground flow is governed by the principle of minimum potential energy. In the calculation, liquefied sand is assumed to be a viscous liquid that deforms in undrained conditions with its volume remaining constant. To consider the non-linearity due to large displacement, the updated Lagrangian method is used to solve the equation of motion. The Newmark β method is employed to calculate the time history of the ground motion. Finally, a simulation using this calculation method shows that the proposed method gives reasonable results for the conditions indicated.

Non-linear finite element analysis of composite panels

In order to promote the efficient use of composite materials in civil engineering infrastructure, effort is being directed at the development of design criteria for composite structures. Insofar as design with regard to buckling of composite shells is concerned, it is well known that a key step is to investigate the influence of initial geometric imperfection on the non-linear behaviour of the composite shells. One possible approach is to use the validated numerical model based on the non-linear finite element analysis. Thus, the objective of this article is to present the formulation used in developing a composite shell element and to validate the element from the composite panels . The finite element used in the current study is an eight-noded shell element with six degrees of freedom per node. The non-linear formulation of the shell element is based on the updated Lagrangian method . The shell element is capable of small strain and large displacement analysis with finite rotations. In order to remove the rigid body rotation , a co-rotational method is used. The transverse shear deformation effects using the Reissner–Mindlin theory are included in formulating the linear and geometric stiffness matrix . Thus, the present composite shell element allows modeling of relatively thick composite plates and shells for both the linear and non-linear analyses. The validation of the composite shell element shows that the present results have very good agreement with existing references. Subsequently the postbuckling analyses for the modeling of the curved panel with initial imperfections are performed in order to investigate the effect of the initial geometric imperfection shape and amplitude. The results are used to estimate imperfection sensitivity for such panels.

Updated Lagrangian polar decomposition method in non‐linear elasticity and its shell element implementation

In this paper, the relation between the incremental right stretch tensor by the total Lagrangian (T.L.) description and that by the updated Lagrangian (U.L.) description is obtained. The incremental polar decomposition method without drilling degrees of freedom is developed. The non-linear shell elements using the Biot strain tensor as the strain measure are established. The non-linear finite element formulation based on the polar composition theorem is developed with respect to the U.L. description. As a by-product the material time derivative of the right stretch tensor is given by a new form suitable for numerical evaluations. Two examples demonstrate the feasibility of the proposed incremental polar decomposition method. Copyright © 1999 John Wiley & Sons, Ltd.

Updated Lagrangian polar decomposition method in non-linear elasticity and its shell element implementation

In this paper, the relation between the incremental right stretch tensor by the total Lagrangian (T.L.) description and that by the updated Lagrangian (U.L.) description is obtained. The incremental polar decomposition method without drilling degrees of freedom is developed. The non-linear shell elements using the Biot strain tensor as the strain measure are established. The non-linear finite element formulation based on the polar composition theorem is developed with respect to the U.L. description. As a by-product the material time derivative of the right stretch tensor is given by a new form suitable for numerical evaluations. Two examples demonstrate the feasibility of the proposed incremental polar decomposition method.

A study of strain localization in the fine-blanking process using the large deformation finite element method

Abstract In this paper, an updated Lagrangian large deformation finite-element method was developed for solving metal forming problems with an incrementally objective mid-interval integration algorithm. In order to avoid oscillatory behaviour, the Green–Naghdi stress rate was adopted in the finite-element programme, this stress rate being determined implicitly by means of the rotation tensor. A consistent tangent operator and the Newton–Raphson iterative method were employed to solve the incremental equilibrium equations. The hypoelastic constitutive equations have been integrated numerically by the projection algorithm. The target material is assumed to be a low-carbon steel having a stress–strain relationship which exhibits strain hardening and damage softening. With this model, a numerical computation of strain localization in the fine-blanking process was conducted. The computed deformed mesh agrees with experimental observation. The calculated equivalent plastic strain was high at the clearance zone and the plastic flow was localized in this area. Thus, the localization of plastic flow may cause the initiation of shear bands. Under the high hydrostatic stress of the fine-blanking process, further increase in the strains within the shear band will not contribute significantly to the overall deformation of the material. Hence, shearing fracture will occur at the cutting edges and a good cutting surface can be achieved.

Postbuckling analysis of composite panels with imperfection damage

In order to promote the efficient use of composite materials in civil engineering infrastructure, effort is being directed at the development of design criteria for composite structures. Insofar as design with regard to buckling of composite shells is concerned, it is well known that a key step is to investigate the influence of initial geometric imperfection. At present, imperfection sensitivity study of composite shells has not been explored in detail. Thus, the objective of this paper is to present the formulation used in developing a composite shell element and to validate the element from the composite curved panel. The non-linear formulation of the shell element is based on the updated Lagrangian method. The shell element is capable of small strain and large displacement analysis with finite rotations. In order to remove the rigid body rotation, a co-rotational method is used. Subsequently the postbuckling analyses from the modeling of the curved panel with initial imperfection damage are performed to investigate the effect of initial geometric imperfection shape and amplitude. The results are used to estimate imperfection sensitivity for such panels.

Large deformation finite element analysis of strain localization in fine-blanking process

In this study, an updated Lagrangian large deformation finite element method is developed for solving the metal forming problems with an incrementally objective mid-interval integration algorithm. A numerical simulation of fine-blanking process is hence performed to predicate the strain localization and to investigate the failure mechanism of fine-blanking process. Using the numerical result, The stress and strain fields developed in the fine-blanking process as well as the large deformation elasto-plastic finite element implementation for solving axisymmetric strain localization problems are discussed.

In-Plane Elastic-Plastic Deformation of a Circular Arch under Central Impact by a Flyer.

Dynamic elastic-plastic deformation of pin-ended circular arches has been numerically analyzed with a discrete model composed of rigid bars connected with nonlinear springs. Equations of motion of the discretized arch are solved by the updated Lagrange method. The calculation has been performed for arches impacted by a flyer at the crown and the results are compared with the experimental ones. The temporal change in the deflection at the crown, the deformed shape, strain histories are well estimated by the calculation.

Buckling and postbuckling behavior of composite laminates with a delamination

A finite-element model is presented for studying the buckling and postbuckling behavior of composite laminates with an embedded delamination. Degenerated-shell elements and rigid-beam elements are utilized for the finite-element model. In the non-linear finite-element formulation, the updated Lagrangian description method based on the second Piola-Kirchhoff stress tensor and the Green strain tensor has been used. The buckling and postbuckling behavior are investigated for various delamination sizes and boundary conditions. Three different possible modes of instability are identified at the critical load, i.e. global, mixed, and local buckling modes. It is shown that the buckling load and postbuckling behavior depend on the buckling mode, which is determined by the delamination size and boundary conditions. It is also found that bifurcation buckling occurs for composite laminates with an embedded delamination.

Primal and mixed variational principles for dynamics of spatial beams

A general formulation is presented for static and dynamic analysis of spatial elastic beams capable of undergoing finite rotations and small strains. The tangent maps associated to the finite rotation vector are used to compute the tangent iteration matrices used to integrate implicitly the equations of motion in descriptor form. A total Lagrangian primal corotational method and an updated Lagrangian mixed variational method are proposed to compute the tangent stiffness matrix. The tangent inertia matrices, including the gyroscopic and centrifugal terms, are also obtained by using the tangent maps of rotation. The numerical examples analyzed in this paper include static (pre- and postbuckling) and dynamic analysis of flexible beams structures. The new finite elements show a very good performance, in terms of fewer number of elements used and accuracy during the simulation, both for static and dynamic problems.

Buckling behaviour of composite panels using the finite element method

Buckling analyses of the laminated plates and shells under axial compression are carried out using the finite element method. The formulation of a geometrically non-linear composite shell element based on the updated Lagrangian method is presented to study the buckling behaviour. The element is capable of small strain and large displacement analysis with finite rotations. It has eight nodes with six degrees of freedom per node. Transverse shear deformation effects are included using a firstorder (Mindlin-Reissner) theory which allows modelling of relatively thick plates and shells. Results of buckling behaviour show that the present formulation has very good agreement with existing references.

A sensitivity analysis for the optimal design of metal-forming processes

A major objective in metal-forming processing is the optimum selection of process conditions in the design stage of processes given the material state and the geometry of the final product, conditions on the initial workpiece and possibly some restrictions on the processes. Since the required process conditions are input to the direct forming analysis, the design of forming processes usually consists of the repeated trial and error use of direct modeling techniques. Here, process conditions refer to die surfaces, die lubrication conditions, preform selection and the selection of the required ram forces and velocities. In this work, a sensitivity analysis for large deformation hyperelastic viscoplastic solids is presented that is consistent with the kinematic analysis and the constitutive integration scheme used in the updated Lagrangian method for solving the direct deformation problem. The method is developed with the problem of die design in metal-forming processes in mind. As such, special attention is given to the modeling of the complex boundary conditions that result in the die-workpiece interface. The effectiveness of the method is demonstrated by solving an extrusion axially symmetric die design problem. In particular, an extrusion die is designed such that the material state in the final product has the least possible standard deviation.

Influence of the die arc on formability in cylindrical cup-drawing

The incremental updated Lagrangian elasto-plastic finite-element method was applied in this study towards analyzing cylindrical cup-drawing under axisymmetric conditions. The extended r-minimum technique was employed such that each incremental step size is determined by not only the yielding of an element gaussian point but also by the change of the boundary conditions along the tool-metal interface. The Coulomb friction law was also adopted in calculating the influence of friction coefficients on the drawing process. A low-carbon steel (BA-CQ 2) was used as the test material in the experiments. Also, five various die-arc radii were designed, of value four-times to twelve-times the blank thickness, each die being used to draw the blank in the experiments. Results obtained for the punch-load in drawing, the material thickness variation after drawing and the limit drawing ratio (LDR) were compared both with the results of experiment and with the results of FEM analysis: the experimental and FEM results both confirmed that a greater LDR and a generally greater thickness would be obtained as a larger die-arc radius is used in the die-arc portion. These results have not only verified the reliability and accuracy of the computer simulation, but also could be used as references in the designing and manufacturing of dies for industrial applications.

A Study of the Effect of Pre-Straining on Angular Distortion in One-Pass Fillet Welding Incorporating Large Deformation Theory

By using the finite element method based on the large deformation theory, the effect of pre-straining on angular distortion of one-pass fillet weldments was analysed for the free-end and constrained-end condition. Pre-straining by an additional concentrated load caused a large thermal deformation during heating, and reduced the reverse distortion during cooling and consequently the residual distortion. Geometric change of the fillet weldment caused by pre-straining during welding induced an initial stress effect on the stiffness to the further deformation, which is due to a developed in-plane stress. This initial stress effect could be considered in the large deformation theory to be based on the updated Lagrangian method, while it could not in the small deformation theory. As the pre-straining load increased in the free-end condition, the initial stress effect also increased. In the constrained-end condition, the initial stress effect played an important role in the deformation of the fillet weldment, even without pre-straining. By comparing the results predicted by the large deformation theory with those by the small deformation theory and experiments, it can be shown that the large deformation theory is able to describe the distortion phenomenon in fillet welding reasonably well.

Rolling process analysis of aluminum strip by a coupled thermo-elastic-plastic model

The phenomenon of longitudinal curvature on an aluminum strip caused by different heat conduction boundary conditions on the top and bottom surfaces of the strip while it undergoes a rolling process is studied. This research adopts large deformation-large strain theory to develop rolling process analysis of an aluminum strip by a coupled thentto-elastic-plastic model using the updated Lagrangian formulation (ULF) and incremental method. The flow stress of the materials is considered as a function of strain, strain rate and temperature, and the finite difference method is used simultaneously to solve equations of transient heat transfer. Finally, the numerical analysis method developed from this study is used to determine the temperature change and deformation of an aluminum strip when it undergoes hot rolling. At the same time, the simulation results on normal stress and contact angle are compared with the results of experiments and other published references; the comparison results verify that the present model is reasonable. In addition, the average rolling force during hot rolling is simulated and comparison is also made with the results of experiments provided by the China Steel Corporation. The simulated results in this article are generally considered to be reasonable.

Integrated FEM Formulation for Total/Updated-Lagrangian Method in Geometrically Nonlinear Problems

This paper presents a new integrated FEM formulation for geometrically non-linear analyses. There have been two methods to solve the so-called large displacement problem, i.e., the total-Lagrangian method and the updated-Lagrangian method, and two types of FEM programming have been conventionally written. It is shown in the present paper that these two types of programming can be combined by the use of the covariant components of the incremental Green-Lagrange strain tensor and the contravariant components of the second Piola-Kirchhoff stress tensor in the convected coordinate system. The difference is only seen in the transformation of the constitutive tensorial components. The stiffness matrices for the solid and shell elements by this formulation are illustrated in detail, and advantages and disadvantages of the proposed method are also discussed.

An elasto-plastic finite element analysis of sheet metal U-bending process

The application of updated Lagrangian elasto-plastic finite element method towards analyzing the sheet metal U-bending process under the plane strain condition is discussed in this paper. Different boundary conditions would generally through a U-bending process, including the contact-separation boundary, the slide-sticking friction boundary, and the change of elastic state to plastic state. An extended r-minimum algorithm was developed for satisfying the above boundary conditions. The entire loading process history, the whole deformation history of the forming process and the nodal velocity were obtained by numerical calculation. The effects of process variables, e.g. friction coefficient, tooling geometry, and clearance between punch and die, on the final shape after spring-back or spring-go are discussed in detail.

Nonlinear dynamic analysis of lattice structures

A computational procedure for predicting the geometric and material nonlinear dynamic response of space trusses is presented. An updated Lagrangian method, based on the incremental formulation of the equation of motion, is employed. The constitutive law is cast into the Ramberg-Osgood polynomial model. Numerical studies of two- and three-dimensional truss structures under various dynamic loading are made. The structure dynamic response including different types of nonlinearity is compared with the linear response and with the results obtained from static analysis.

Nonlinear analysis of thin-walled members under biaxial bending

Abstract A numerical method for the nonlinear and instability analysis of thin-walled members (NISAT) subjected to biaxial bending has been developed which includes the warping of degree freedom at each mode. The analysis includes the effects of geometric nonlinearity, out-of-plane bending and the consequent torsion, secondary effects of load applied away from the shear centre, large displacements and rotations in the three-dimensional space and initial imperfections. The effect of large displacements is included in the nonlinear analysis following the updated Lagrangian method where the nodal coordinates are updated before every load increment. The analysis does not consider distortion of the cross-section. The nonlinear biaxial bending behaviour of thin-walled Z- and sigma sections subjected to transverse load is studied analytically using the NISAT program and the results are compared with experimental values. The failure load, midspan displacements and the normal stresses at critical points of the midspan section are also compared.