Generalization of Foreshortening Through New Reduced Geometrically Nonlinear Structural Formulation

A relevant problem in the development and improvement of numeric analytical methods for the research of structures, buildings and construction is studying the stress–strain state of structures and construction with boundaries that have complex shapes. Deformations and stresses arise in a domain with a geometrically non-linear shape of the boundary (cut-outs and cuts). These stresses and deformations have great values and gradients. Experiments carried out using the photoelasticity method show a change in the deformation order ratios for different subareas of the boundary cut-out area depending on proximity to the apex of the angular cut-out. Areas with minor deformations are observed, and areas where linear deformations and shears are more significant than rotations are also observed. In addition, areas where section rotations are more significant than linear and shear deformations are observed. According to the experimental data, the mathematical model of the SSS in the area of the apex of the cut-out of the domain boundary should take into account non-linear deformations. Hence, it is necessary to formulate the boundary value problem of the theory of elasticity, taking into account the geometrical non-linearity. The research aim of this paper is to formulate the problem of the elasticity theory taking into account the geometrical non-linearity in furtherance of the proposed mathematical model justified by the experimental data obtained using the photoelasticity method. The obtained formulation of the elasticity theory problem allows analyzing the form of the system of equations of the boundary value problem depending on the proximity of the considered area to the irregular point of the boundary, i.e., taking into account the difference in the effect of linear and shear deformations, rotations and forced deformations on the solution to the geometrically non-linear elastic problem dealing with forced deformations in the area of an angular cut-out of the boundary of the plane domain.

This paper concerned the stamping process of corrugated sheet metal for sealing device honeycomb manufacturing. GH536 foil with a thickness of 0.05mm was used to produce corrugated sheet metal with semi-hexagonal profile. Finite element models for the corrugated sheet metal stamping process were built to analyze the forces and material deformation. The prototype molds with a pair of pressure dies were designed and tested. Then according to the curling phenomenon occurred in the experimental result of prototype molds, the optimized final molds was designed with three pairs of pressure dies. The formed corrugated sheet metal by the final molds was much better. KEYWORD: Corrugated metal sheet; Stamping; Honeycomb manufacturing; Numerical simulation International Conference on Industrial Technology and Management Science (ITMS 2015) © 2015. The authors Published by Atlantis Press 1281 The demanded specifications of the corrugated sheet metal is shown in Figure 2, the angles between adjacent sides θ1=θ2 =120°, side length l1 = l2 = l3 = 0.8mm, thickness t=0.05mm. Figure 2. The demanded specifications of the corrugated sheet metal 2 NUMERICAL SIMULATION The principle of stamping is that the part with desired shape and size is obtained by a method of permanent plastic deformation of work piece. This process is achieved by the normal contact force and tangential frictional forces between the mold and the work piece. The stamping process involves three nonlinear problems in mechanics. (1) Geometric nonlinearity (Large displacement, rotation and deformation in the stamping process). (2) Physical nonlinearity (Also known as material nonlinearity which refers to elastic-plastic deformation of the material). (3) Boundary nonlinearity (Refer to non-linear relationship caused by contact friction between the die and work piece). The development of computer technology and finite element method provides a new way for complex engineering calculation. This paper used explicit dynamic software ANSYS\LS-DYNA to simulate the corrugated sheet metal stamping process 2.1 Original Design of Stamping Molds For the corrugated sheet metal stamping, special molds was designed as illustrated in Figure 3 and Figure 4. All the fillets are 0.1 mm. The material of the molds is P20 # steel. Since the GH536 foil is very thin, the deformation of molds is negligible in the forming process, so the molds can be regarded as rigid in simulation. The material properties of the molds are listed in Table 3. Figure 3. Illustration of upper die Figure 4. Illustration of lower die Table 3 The material properties of the molds Desinty (g/cm 3 ) Elastic Modulus (GPa) Possion’s ratio 7.8 206 0.3 The finite element model is illustrated in Figure 5. The molds were modeled using solid elements, and the foil was modeled using shell elements. The foil feed from the left with a preset motion, and the rigid molds moved in vertical direction. The foil and rigid molds were constrained to prevent undesired motion. Two stamping steps were simulated to form two semi-hexagonal profiles. The calculation time was set to 0.3s. The simulated stamping process is shown in Figure 6.

Wind resistant size optimization of geometrically nonlinear lattice structures using a modified optimality criterion method

Object-oriented structural and geometric analysis methods for truss structures are proposed. The methods are based on the iterative relaxation of nodal unbalanced forces by moving the nodes. The entities appeared in truss analysis are modeled as classes, and their class hierarchy is established. The knowledge about a truss node, a truss member, a material, and a sectional shape of the member are extracted and constructed as a knowledge base in an object-oriented manner. The object-oriented structural analysis is found to be valid for material and geometric nonlinearities, and the object-oriented geometric analysis where a kinematic problem is converted into a problem for a force equilibrium at the nodes is also found to be effective for determining the geometry of variable geometry trusses. The whole system is written in an object-oriented language, Smalltalk, and several results on structural and geometric analysis are presented. The speed of the computation is discussed along with the size of the structures and the over-relaxation factor. The extension of the knowledge base is also discussed.

A matrix DEIM technique for model reduction of nonlinear parametrized problems in cardiac mechanics

An energy balancing strategy for solution of combined geometrical and material nonlinearity problems

In theoretical research pertaining to sealing, a contact model must be used to obtain the leakage channel. However, for elastoplastic contact, current numerical methods require a long calculation time. Hyperelastic contact is typically simplified to a linear elastic contact problem, which must be improved in terms of calculation accuracy. Based on the fast Fourier transform, a numerical method suitable for elastoplastic and hyperelastic frictionless contact that can be used for solving two-dimensional and three-dimensional (3D) contact problems is proposed herein. The nonlinear elastic contact problem is converted into a linear elastic contact problem considering residual deformation (or the equivalent residual deformation). Results from numerical simulations for elastic, elastoplastic, and hyperelastic contact between a hemisphere and a rigid plane are compared with those obtained using the finite element method to verify the accuracy of the numerical method. Compared with the existing elastoplastic contact numerical methods, the proposed method achieves a higher calculation efficiency while ensuring a certain calculation accuracy (i.e., the pressure error does not exceed 15%, whereas the calculation time does not exceed 10 min in a 64 × 64 grid). For hyperelastic contact, the proposed method reduces the dependence of the approximation result on the load, as in a linear elastic approximation. Finally, using the sealing application as an example, the contact and leakage rates between complicated 3D rough surfaces are calculated. Despite a certain error, the simplified numerical method yields a better approximation result than the linear elastic contact approximation. Additionally, the result can be used as fast solutions in engineering applications.

Given the vast advancements in computing power in the last several decades, nonlinear dynamic analysis has gained wide acceptance by practicing engineers as a useful way of assessing and improving the seismic performance of structures. Nonlinear structural analysis software packages give engineers the ability to directly model nonlinear component behavior in detail, resulting in improved understanding of how a building will respond under strong earthquake shaking. One key component, in particular, for understanding the behavior of moment-resisting frames is the plastic rotation of the flexural hinges. Performance-based standards typically use plastic rotation as the primary parameter for defining the acceptance criteria in moment-resisting frames. Since plastic rotation is a key parameter in the seismic damage assessment, the concept as well as the method to calculate this quantity must be understood completely. Though engineers rely on the plastic rotation output from seismic structural analysis software packages to determine acceptable performance, the actual calculation methods used in achieving such plastic rotation quantities usually lay within a so-called “black box”. Based on the outputs obtained from most structural analysis software packages, it can be shown that running an algorithm considering material nonlinearity by itself will produce reasonably accurate results. Moreover, separately running an algorithm considering geometric nonlinearity also can produce accurate results. However, when material nonlinearity is combined with geometric nonlinearity in an analysis, obtaining accurate results or even stable solutions is more difficult. The coupling effect between the two nonlinearities can significantly affect the global response and the local plastic rotation obtained from the analysis and therefore needs to be verified through some analytical means. Yet, the verification process is difficult because a robust analytical framework for calculating plastic rotation is currently unavailable and urgently needed. In view of this gap, an improved analytical approach based on small displacement theory is derived to calculate the plastic rotations of plastic hinges at various locations of moment-resisting frames. Both static and dynamic analysis with nonlinear geometric effects will be incorporated in the derivation. Here the element stiffness matrices are first rigorously derived using a member with plastic hinges in compression, and therefore the coupling of geometric and material nonlinearity effects is included from the beginning of the derivation. Additionally, plastic rotation is handled explicitly by considering this rotation as an additional nonlinear degree-of-freedom. Numerical simulation is performed to calculate the nonlinear static and dynamic responses of simple benchmark models subjected to seismic excitations. Results are compared with various software packages to demonstrate the feasibility of the proposed method in light of the output results among software packages in calculating plastic rotations.

Adequate mechanical properties make stainless steel an excellent construction material for structures. The current stainless steel European structural code EN1993‐1‐4 (2015) is largely based on provisions given for carbon steel EN1993‐1‐1 (2005) and does not establish specific design rules for the global analysis of stainless steel structures. Thus, the same classification criterion applies in both codes for sway and non‐sway structures, as well as providing the same predicting expressions for the amplification of bending moments in basic structures, despite the considerable differences between the mechanical behaviour of these two materials. However, recent numerical research on stainless steel frames with H cross‐sections by Walport et al. (2019) showed that the degradation of stiffness due to material nonlinearities considerably affects the response of stainless steel frames, causing greater deformations and increasing second order effects.In order to contribute to the optimal design of stainless steel structures, this paper investigates the influence of the nonlinear material response of stainless steel alloys on frame‐type structures with Rectangular Hollow Sections based on the experimental results conducted by Arrayago et al. (2019), and the interaction of material and geometric nonlinearities.

Adaptive stabilization of discontinuous Galerkin methods for nonlinear elasticity: Analytical estimates

A priori and a posteriori error analyses of augmented twofold saddle point formulations for nonlinear elasticity problems

A parallel spectral element method for dynamic three-dimensional nonlinear elasticity problems

Applied methods of mathematical modeling, on the basis of which adequate mathematical models and effective calculation technologies that investigate peculiarities of nonlinear deformation of nonhomogeneous thin-walled constructions at different types of static and dynamic loading taking into account geometrical and physical nonlinearity are developed, are represented. One-layer and multilayer shell structures made of both traditional and perspective composite materials are considered. When describing deformation processes for thin-walled structures, mathematical models developed on the basis of Kirchhoff-Love and Timoshenko-Reissner hypothesis are used. Geometrical nonlinearity is considered within the framework of mean-bending theory relations and for taking into account physical nonlinearity relations of deformation plasticity theory with Mises yield criterion supposing elastic unloading are used. Thin-walled shells weakened by cutouts of different forms are considered as well. On external and internal boundaries (cut-out contours) of the shell, different variants of boundary conditions are realized. As loads acting on the shell, a system of boundary and surface loads of general and local character is considered. When solving an original integral-differential problem numerically, the method of finite differences is used for discretization in space and time coordinates. Applying variational-difference method for constructing mesh analogs of equilibrium and motion equations results in conservative difference schemes even for multiconnected regions with complex form of the boundaries. For numerical solutions of stationary problems, the quasi-dynamic form of establishment method is used, which develops calculation algorithms of the same type for solving both static and dynamic problems of nonlinear theory of shells. Relations for determination of the optimal values of iteration process parameters are represented, as is the original technology of critical value of time step significantly decreasing computing time when solving nonlinear problems of shells theory on the basis of explicit difference schemes numerically.

Asymptotically correct 3D displacement of the Mooney–Rivlin model using VAM

Geometric nonlinear analyses are used in many structural problems, such as the determination of failure load, as well as the study of buckling mechanism. Nevertheless, due to the complex nature of this type of problems and the absence of a comprehensive analytical solution for them, numerical methods are utilized in practice to approximate the exact response of these systems. In the application of numerical methods, there are also some difficulties such as divergence or finding the correct path of equilibrium, especially in the case of bifurcation points. Hence, the main purpose of this research is to apply the modified energy method (introduced in the dynamics of structures) in quasi-static problems with geometric nonlinearity and bifurcation points so that the efficiency of this method can be compared to others, such as Newtonian numerical techniques and force-displacement-constraint approaches. To achieve the objectives of this research, after briefly reviewing the current force-based computational methods in practice, the energy method is described for such problems, and then the step-by-step process of its computer implementation will be presented. Afterward, by coding in MATLAB software and applying the method to numerical examples employed by other researchers such as truss and frame structures, the numerical results are verified by analytical solution as well as those obtained by other methods, such as Newton-Raphson and Arc Length techniques. Generally, the interpretation of the results obtained from performed simulations has shown that the presented numerical method in analyzing nonlinear geometric problems has better accuracy compared to the Arc Length method; moreover, it can well pass through bifurcation points in the force-displacement curve without divergence in comparison with the Newton-Raphson method.