Torsional vibrations of composite bars of variable cross-section by BEM

In this paper a boundary element method is developed for the nonuniform torsional vibration problem of doubly symmetric composite bars of arbitrary variable cross-section. The composite bar consists of materials in contact, each of which can surround a finite number of inclusions. The materials have different elasticity and shear moduli and are firmly bonded together. The beam is subjected to an arbitrarily distributed dynamic twisting moment, while its edges are restrained by the most general linear torsional boundary conditions. A distributed mass model system is employed which leads to the formulation of three boundary value problems with respect to the variable along the beam angle of twist and to the primary and secondary warping functions. These problems are solved employing a pure BEM approach that is only boundary discretization is used. Both free and forced torsional vibrations are considered and numerical examples are presented to illustrate the method and demonstrate its efficiency and wherever possible its accuracy. The discrepancy in the analysis of a thin-walled cross-section composite beam employing the BEM after calculating the torsion and warping constants adopting the thin tube theory demonstrates the importance of the proposed procedure even in thin-walled beams, since it approximates better the torsion and warping constants and takes also into account the warping of the walls of the cross-section.     

Nonuniform torsion of multi-material composite bars by the boundary element method

In this paper a boundary element method is developed for the nonuniform torsion of arbitrary constant cross-section multi-material composite bars. The materials have different elasticity and shear moduli and are firmly bonded together. The bar is subjected to an arbitrarily concentrated or distributed twisting moment while its edges are restrained by the most general linear torsional boundary conditions. Since warping is prevented, beside the Saint-Venant torsional shear stresses, the warping normal stresses are also computed. Two boundary value problems with respect to the variable along the beam angle of twist and to the warping function with respect to the shear center are formulated and solved employing a BEM approach. Both the warping and the torsion constants are computed by employing an effective Gaussian integration over domains of arbitrary shape. Numerical results are presented to illustrate the method and demonstrate its efficiency and accuracy. The contribution of the normal stresses due to restrained warping is investigated.     

An improved formulation of space stiffeners

This paper presents a displacement based finite element model for predicting the constraint torsion effect of stiffeners. In structural modelling, the plate/shell and the stiffeners are treated as separate elements where the displacement compatibility transformation between these two types of elements takes into account the constraint torsional warping effect in the stiffeners. The development is based on a general beam theory which includes flexural-torsion coupling, constrained torsion warping, and shear-centre location. The virtual work principle includes the second order terms of finite beam rotations. For finite element analysis, cubic Hermitian polynomials are used as shape functions of the straight space frame element with two nodes. Elastic stiffness and geometric stiffness matrices for an arbitrary cross-section are evaluated in a closed form, and load correction stiffness for eccentric stiffener loads are considered. To demonstrate the importance of torsion warping constraints and to illustrate the accuracy of this formulation, finite element solutions are presented and compared with available solutions.     

Dynamic stiffness matrix of non-symmetric thin-walled curved beam on Winkler and Pasternak type foundations

Using the technical computing program Mathematica, the dynamic stiffness matrix for the spatially coupled free vibration analysis of thin-walled curved beam with non-symmetric cross-section on two-types of elastic foundation is newly presented based on the power series method. For this, the elastic strain energy considering the axial/flexural/torsional coupled terms, the kinetic energy including the rotary inertia effect, and the energy due to the elastic foundation are introduced. Then, equations of motion are derived from the energy principle and explicit expressions for displacement parameters are derived based on power series expansions of displacement components. Finally, the exact dynamic stiffness matrix is determined using force–displacement relations. In order to demonstrate the validity and the accuracy of this study, the natural frequencies of thin-walled curved beams with mono-symmetric and non-symmetric cross-sections are evaluated and compared with the analytical solutions and finite element solutions using Hermitian curved beam elements and ABAQUS’s shell elements. In addition, some results by a parametric study are reported.     

Solution of non-uniform torsion of bars by an integral equation method

In this paper, a boundary element method (BEM) is developed for the non-uniform torsion of simply or multiply connected cylindrical bars of arbitrary cross-section. The bar is subjected to an arbitrarily distributed twisting moment while its edges are restrained by the most general linear torsional boundary conditions. Since warping is prevented, besides the Saint-Venant torsional shear stresses, the warping normal stresses are also computed. Two boundary value problems with respect to the variable along the beam angle of twist and to the warping function are formulated and solved employing a BEM approach. Both the warping and the torsion constants are computed by employing an effective Gaussian integration over the domains of arbitrary shape. Numerical results are presented to illustrate the method and demonstrate its efficiency and accuracy. The contribution of the normal stresses due to a restrained warping is investigated, by numerical examples, with great practical interest.     

Exact dynamic stiffness matrix of non-symmetric thin-walled beams on elastic foundation using power series method

Exact dynamic element stiffness matrix for the flexural–torsional free vibration analysis of the shear deformable thin-walled beam with non-symmetric cross-section on two-types of elastic foundation is newly presented using power series method based on the technical computing program Mathematica. For this, the shear deformable beam on elastic foundation theory is developed by introducing Vlasov's assumption and applying Hellinger–Reissner principle. This beam includes the shear deformation effects due to the shear forces and the restrained warping torsion and due to the coupled effects between them, and rotary inertia effects and the flexural–torsional coupling effects due to the non-symmetric cross-sections. And then equations of motion and force–deformation relations are derived from the energy principle and explicit expressions for displacement parameters are derived based on power series expansions of displacement components and the exact dynamic element stiffness matrix is determined using force–deformation relationships. In order to verify the accuracy of this study, the numerical solutions are presented and compared with the analytical solutions and the finite element solutions using the isoparametric beam elements. Particularly the influences of the coupled shear deformation on the vibrational behavior of non-symmetric beam on elastic foundation are investigated.     

Warping shear stresses in nonuniform torsion of composite bars by BEM

In this paper a complete approximation of the nonuniform torsion problem of composite bars of arbitrary constant cross section using the boundary element method is developed. The composite bar consists of a matrix surrounding a finite number of inclusions. The materials have different elasticity and shear moduli and are firmly bonded together. The bar is subjected to an arbitrarily concentrated or distributed twisting moment, while its edges are restrained by the most general linear torsional boundary conditions. Since warping is prevented, beside the Saint-Venant torsional shear stresses, the warping normal and shear stresses are also computed. Three boundary value problems with respect to the variable along the beam angle of twist and to the primary and secondary warping functions are formulated and solved employing a pure BEM approach, that is only boundary discretization is used. Both the warping and the torsion constants together with the torsional shear stresses and the warping normal and shear stresses are computed. Numerical results are presented to illustrate the method and demonstrate its efficiency and accuracy. The magnitude of the warping shear stresses due to restrained warping is investigated by numerical examples with great practical interest.     

Inelastic uniform torsion of steel members

This paper develops a mitre model for the shear strain distribution in steel members under uniform torsion. The mitre model provides reasonable approximations for the shear stress distributions, except near re-entrant corners, and accurate predictions of the fully plastic uniform torque. The elastic torque-twist relationship is predicted with high accuracy, and the inelastic relationship with reasonable accuracy. The mitre model uses a simple approximation to the shear strain distribution caused by uniform torsion. It can be expected to provide a relatively simple method of predicting the inelastic behaviour of steel members when uniform torsion acts in combination with axial force and bending moment.     

Simulation of strength difference for adhesive materials in finite deformation elasto-plasticity

The experimental investigation of certain adhesive materials reveals elastic strains, plastic strains and hardening, respectively. Additionally a pronounced strength difference between tension, torsion or combined loading is observed. The purpose of this work is the simulation of these phenomena in the framework of large strain elasto-plasticity. To this end a yield function dependent on the first and second basic invariants of the Cauchy stress tensor is introduced. Furthermore, a plastic potential with the same mathematical structure is used to formulate the evolution equations under the assumption of small elastic strains. Upon considering thermodynamic consistency of the model equations some restrictions on the material parameters are derived. Furthermore numerical aspects are addressed concerning the integration of the constitutive relations and the finite element equilibrium iteration. In the numerical examples, firstly, we compare simulated and experimental results exhibiting the yield strength difference between tension and torsion for the adhesive material Betamate 1496. A second example investigates the deformation evolution of a compact tension specimen with an adhesive zone.     

Topology optimization of multi-material negative Poisson’s ratio metamaterials using a reconciled level set method

Metamaterials are defined as a family of rationally designed artificial materials which can provide extraordinary effective properties compared with their nature counterparts. This paper proposes a level set based method for topology optimization of both single and multiple-material Negative Poisson’s Ratio (NPR) metamaterials. For multi-material topology optimization, the conventional level set method is advanced with a new approach exploiting the reconciled level set (RLS) method. The proposed method simplifies the multi-material topology optimization by evolving each individual material with a single level set function and reconciling the result level set field with the Merriman–Bence–Osher (MBO) operator. The NPR metamaterial design problem is recast as a variational problem, where the effective elastic properties of the spatially periodic microstructure are formulated as the strain energy functionals under uniform displacement boundary conditions. The adjoint variable method is utilized to derive the shape sensitivities by combining the general linear elastic equation with a weak imposition of Dirichlet boundary conditions. The design velocity field is constructed using the steepest descent method and integrated with the level set method. Both single and multiple-material mechanical metamaterials are achieved in 2D and 3D with different Poisson’s ratios and volumes. Benchmark designs are fabricated with multi-material 3D printing at high resolution. The effective auxetic properties of the achieved designs are verified through finite element simulations and characterized using experimental tests as well.     

A structural dynamics study of a wing-pylon-tiltrotor system

A simple structural model for a three-bladed tiltrotor-pylon-wing assembly is presented, which accounts for chordwise, transverse, and torsional wing deformations, rigid pylon pitching motion with respect to the wing tip cross-section in its deformed position, lead-lag, flap, and torsional deformations of rotor blades. The model considers equivalent viscous damping associated with blade and wing elastic deformations and with rigid pylon pitching motion. It is established that blade-to-wing bending rigidity ratio, pylon pitching frequency, equivalent viscous damping associated with blade elastic deformations, and rotational speed, are the most important design parameters, whose effect on system frequencies and stability boundaries is evaluated.     

Stochastic elastic–plastic finite elements

A computational framework has been developed for simulations of the behavior of solids and structures made of stochastic elastic–plastic materials. Uncertain elastic–plastic material properties are modeled as random fields, which appear as the coefficient term in the governing partial differential equation of mechanics. A spectral stochastic elastic–plastic finite element method with Fokker–Planck–Kolmogorov equation based probabilistic constitutive integrator is proposed for solution of this non-linear (elastic–plastic) partial differential equation with stochastic coefficient. To this end, the random field material properties are discretized, in both spatial and stochastic dimension, into finite numbers of independent basic random variables, using Karhunen–Loève expansion. Those random variables are then propagated through the elastic–plastic constitutive rate equation using Fokker–Planck–Kolmogorov equation approach, to obtain the evolutionary material properties, as the material plastifies. The unknown displacement (solution) random field is then assembled - using polynomial chaos - as a function of known basic random variables and unknown deterministic coefficients, which are obtained by minimizing the error of finite representation, by Galerkin technique.     

The effect of nonlinear elasticity on the large amplitude free vibration behavior of elastic plates at small scale

The effect of nonlinear elasticity on the free vibration behavior of elastic plates has been evaluated by employing continuum mechanics model. The second-order non-linear stress–strain relationship has been considered and the Kirchhoff’s hypothesis has been applied on the elastic plate. The large deformation during vibration has also been considered. By using the Hamilton principle, the governing equations of the free vibration of the plate under different boundary condition have been obtained. In order to get the explicit solutions of the governing equations, the Galerkin’s method and the harmonic balance method have been utilized. The relationship between the vibration frequency and the vibration amplitude has been discussed and the vibration frequencies of different shaped plate have been compared. It is perceived that the nonlinear elasticity has a distinct effect on the free vibration of the plate.

     

材料性质和试样几何形状对应力三维度的影响

为研究应力三维度和塑性应变对金属材料微孔聚合型损伤发展和延性断裂过程的影响,用有限元法计算16Mn钢和两种铝合金材料不同缺口根半径拉伸试样的应力、应变分布. 结果表明:缺口根半径越小,最小横截面上周向和径向应变相差越大且越不均匀,并导致最小横截面上的应力三维度与Bridgman公式预测结果不一样. 材料性质和试样几何形状对应力三维度的影响很大:缺口根半径相同的3种材料试样最小横截面的应力三维度分布形态相似,但应力三维度峰值及其所处位置有所不同;同种材料的试样缺口根半径不同,应力三维度分布形态也不同.     

Finite element elastic-plastic-creep analysis of two-dimensional continuum with temperature dependent material properties

A technique is presented for performing finite element elastic-plastic-creep analysis of two-dimensional continuum composed of material with temperature dependent elastic, plastic, and creep properties. The plastic analysis utilizes the Prandtl-Reuss flow equations assuming isotropic material properties and linear strain-hardening. A power creep flow law formulated by Odquist is used to determine the steady state creep strain rate. The plastic and creep flow laws are employed to derive a ‘softened’ plastic-creep stress-strain matrix. These modified stress-strain relations are then used to formulate the element stiffness matrix in the usual manner. The differences in the elastic, plastic, and creep properties of the material due to the temperature change during the increment result in the formation of pseudo stresses, which in turn lead to load terms that appear on the right hand side of the equilibrium equations. The load terms resulting from these pseudo stresses not only keep the solution on the temperature dependent stress-strain curve of the material, but also correct for the elastic ‘overshoot’ that occurs when an element changes from an elastic to a plastic state. The effect of large displacements is included by the formulation of the geometric stiffness matrix for each element being used in the computer code. With this procedure it becomes economically feasible to perform elastic-plastic-creep stress analysis of two-dimensional continuum subjected to transient thermal and mechanical loadings. Several examples of both elastic-plastic and creep analyses are presented, and the finite element solutions are compared to either other theoretical solutions or experiment.     

Post-buckling behaviour and strength of cold-formed steel lipped channel columns experiencing distortional/global interaction

This paper reports the results of a numerical investigation concerning the elastic and elastic–plastic post-buckling behaviour of cold-formed steel lipped channel columns affected by distortional/global (flexural–torsional) buckling mode interaction. The results presented and discussed were obtained by means of analyses performed using the finite element code Abaqus and adopting column discretisations into fine 4-node isoparametric shell element meshes. The columns analysed (i) are simply supported (locally/globally pinned end sections that may warp freely), (ii) have cross-section dimensions and lengths that ensure equal distortional and global (flexural–torsional) critical buckling loads, thus maximising the distortional/global mode interaction effects, and (iii) contain critical-mode initial geometrical imperfections exhibiting different configurations, all corresponding to linear combinations of the two “competing” critical buckling modes. After briefly addressing the lipped channel column “pure” distortional and global post-buckling behaviours, one presents and discusses in great detail a fair number of numerical results concerning the post-buckling behaviour and strength of similar columns experiencing strong distortional/global mode interaction effects. These results consist of (i) elastic (mostly) and elastic–plastic non-linear equilibrium paths, (ii) curves or figures providing the evolution of the deformed configurations of several columns (expressed as linear combination of their distortional and global components) and, for the elastic–plastic columns, (iii) figures enabling a clear visualisation of (iii₁) the location and growth of the plastic strains and (iii₂) the characteristics of the failure mechanisms more often detected in the course of this research work.     

Elastic–plastic modeling of heat-treated bimorph micro-cantilevers

This paper models the residual stress distributions within micro-fabricated bimorph cantilevers of varying thickness. A contact model is introduced to calculate the influence of contact on the residual stress following a heat treatment process. An analytical modeling approach is adopted to characterize bimorph cantilevers composed of thin Au films deposited on thick poly-silicon or silicon-dioxide beams. A thermal elastic–plastic finite element model (FEM) is utilized to calculate the residual stress distribution across the cantilever cross-section and to determine the beam tip deflection following heat treatment. The influences of the beam material and thickness on the residual stress distribution and tip deflections are thoroughly investigated. The numerical results indicate that a larger beam thickness leads to a greater residual stress difference at the interface between the beam and the film. The residual stress established in the poly-silicon cantilever is greater than that induced in the silicon-dioxide cantilever. The results confirm the ability of the developed thermal elastic–plastic finite element contact model to predict the residual stress distributions within micro-fabricated cantilever structures with high accuracy. As such, the proposed model makes a valuable contribution to the development of micro-cantilevers for sensor and actuator applications.     

Dynamic response simulation for reinforced concrete slabs

Present investigation comprises development of a new finite element numerical formulation for nonlinear transient dynamic analysis of reinforced concrete slab structures. Depending on many experimental data, new material constitutive relationships for concrete material have been formulated. A regression analysis of available experimental data in the SPSS-statistical program has been employed for formulating the proposed material finite element models, and the appropriateness of the models are confirmed through the histograms and measured indices of determination. Concrete slab structures were analyzed using eight-node serendipity degenerated plate elements. The constitutive models of the nonlinear materials are introduced to take into account the nonlinear stress–strain relationships of concrete. For studying the stress profile of the concrete slab through its thickness, a layered approach is adopted. Elastic perfectly plastic and strain hardening plasticity approaches have been employed to model the compressive behavior of concrete. Assumptions for strain rate effect were included in dynamic analysis by supposing the dynamic yield function as a function of the strain rate, in addition to be the total plastic strain. The yield condition is formulated in terms of the first two stress invariants. Geometrical nonlinearity was considered in analysis as a mathematical model based on the total lagrangian approach taking into account Von Karman assumptions. Implicit Newmark with corrector–predictor algorithm was used for time integration solution of the equation of the motion for slab structures. An incremental and iterative procedure is adopted to trace the entire response of the structure; a displacement convergence criterion is adopted in the present study. A computer program coded in FORTRAN has been developed and used for the dynamic analysis of reinforced concrete slabs. The numerical results show good agreement with other published studies’ results which include deflections.     

Numerical analysis of anisotropic ductile continuum damage

The paper deals with the numerical analysis of large elastic–plastic deformation behavior of anisotropically damaged ductile solids based on a generalized macroscopic theory within the framework of nonlinear continuum damage mechanics. Estimates of the stress and strain histories are obtained from a straightforward numerical integration algorithm based on operator split methodology which employs an inelastic (damage–plastic) predictor followed by an elastic corrector step. The finite element method is used to approximate the linearized variational problem. Furthermore, identification of material parameters is discussed. Numerical simulation of the elastic–plastic deformation behavior of damaged tension specimens demonstrate the efficiency of the formulation.