Hybrid Stress Model Research Articles

Upper and lower bounds for evaluation of nonlinear fracture parameters

Bound theorems for estimating small strain nonlinear fracture parameters are proposed. It is found that the lower bound for the J-integral can be obtained by a compatible displacement finite element method. On the other hand, the upper bound of the I∗-integral, which is the dual counterpart of the J-integral, can be obtained by an equilibrium finite element method. To verify the theorems and avoid the difficulty of designing equilibrium finite element models, the popular Pian–Sumihara hybrid stress model is modified by incorporating a penalty-equilibrium constraint. Moreover, an incremental formulation of the I∗-integral for nonlinear finite element computation is developed. Numerical examples on different crack and loading configurations are presented. All the results indicate the validity of the theorems.

The hybrid stress model for mindlin-reissner plates based on a stress optimization condition

A stress optimization condition for hybrid stress finite elements is applied to develop a Mindlin-Reissner plate element. A modified Hellinger-Reissner energy functional, derived from a stress optimization condition, is employed to relax the patch test constraint of incompatible displacements. An eight-node serendipity thin and moderately thick plate element, which is stable and free from shear locking, is formulated from the optimized stress distribution and the use of incompatible displacements. Numerical results indicate that the resulting element is accurate in both displacements and stresses, and insensitive to mesh distortion.

Improvements on the higher order plate element with partial hybrid stress model

The partial hybrid stress model proposed by Jing and Liao is combined with the higher order plate element to improve the through thickness distributions of deformations and stresses. Through dividing six stress components into flexural ( σ x , σ y , τ xy , σ 2) and transverse shear parts ( τ xz , τ yz ), the combination process can proceed assuming hybrid stress for transverse shear while assuming higher order plate theory for flexural stresses in the Hellinger-Reissner principle. The resulting independent variables are displacements ( u, v, w) and transverse shear ( τ xz , τ yz ) only. From the usual procedures of displacement elements and hybrid stress elements, this partial hybrid-higher order plate element can be established. The improvements of this new element in analyzing thick composite laminates are expressed through comparing with the results from elasticity and general higher order plate elements. The convergence is also discussed.

An efficient global-local model for thick composite laminates

An efficient model based on the recently established partial hybrid stress element is proposed in this paper for global-local analysis of thick composite laminated plates. In this study, for displacement field, the higher order plate theory is used throughout the plate while only the transverse shear stress components are assumed in local regions. Thus, in the region of interest, the flexural stresses ( σ x , σ y , τ xy , σ z ) are still obtained from assumed displacement field while the transverse shear stresses ( τ xz , τ yz ) are calculated by the assumed stress version. The variational principle is also presented in this paper. The proposed functional contains two parts. One is the ordinary potential energy used in the global region; the other is the partial hybrid stress model, which contains displacement and transverse shear as independent variables. The interface equilibrium is guaranteed variationally although no stress parameter is assumed in the global region. Three types of global-local analysis are performed in this paper. These are in-plane analysis, through-thickness analysis, and mixed-type analysis. When compared with elasticity solutions, it is found that the agreement is acceptable, although in the so-called ‘transition region’ the results are less accurate.

A partial hybrid stress model for the refined C1higher‐order plate theory

Abstract The partial hybrid stress model is applied to the refined C 1 higher‐order plate theory in this paper. The displacement model is adopted in the flexural part and the hybrid stress model in the transverse shear part. The plate concept is introduced and the governing equations of plate are derived variationally from the modified Hellinger‐Reissner principle. This new plate element is demonstrated to be more accurate than displacement formulation in the analysis of orthotropic thick laminated plates. Moreover, the through thickness distribution of transverse shear stress is precisely predicted.

On the limitation principles for partial hybrid stress model

Based on the new partial hybrid stress model proposed by the author, the corresponding limitation principles are formulated. The relations between the limitation principles for finite element models, based on the present variational principle and the (two-field) Hellinger-Reissner principle and (single-field) principle of total potential energy, are presented. It is found that, under certain circumstances, the results will be identical to those from the mixed finite element or displacement finite element.

Large deflection analysis of thin elastic shells by a hybrid stress method

Based on the incremental complementary energy principle, a hybrid stress model is developed for the geometrical nonlinear analysis of plates and shells. According to the compatibility of interior and boundary displacements for an element, and the equilibrium of stresses at the interior of the element, an inconsistent model which partially satisfies the stress equilibrium equation is utilized in the present paper, and a three node, 15 degree of freedom, triangular shallow shell element is established. Formulas are derived in the updated Lagrangian coordinate system. Numerical examples demonstrate that the assumed hybrid stress model is very efficient.

A three-dimensional hybrid-stress finite element formulation for free vibrations of laminated composite plates

A three-dimensional eight-node hybrid stress finite element method is developed for the analysis of laminated plates. The hybrid stress model is based on the modified complementary energy principle and takes into account the transverse shear deformation effects. The displacement field is interpolated through shape functions and nodal displacements. All three displacement components are assumed to vary linearly through the thickness of each lamina. All six stresses are included and satisfy the equations of motion. Numerical results for the fundamental frequencies of cross-ply laminates are presented as functions of plate geometry ( a h ) degree of anisotropy ( E L E T ) and number of plies. They are also compared with numerical values obtained from the elasticity solution, the modified shear deformation theory and the classical lamination theory.

A hybrid finite strip analysis of multilayer laminated plates

Abstract A rigorous analysis of the behavior of multilayer laminated plates must take into consideration the heterogeneity through thickness of the laminate. As it is well‐known, negligence of the through‐thickness effect by the classical laminated plate theory may result in unreliable estimation of the stress distribution in the laminates. Herein, a hybrid finite strip method is developed for analyzing the bending problem of multilayer laminated plates of rectangular shape. The method combines the basic ideas of hybrid stress model and the finite strip method. The through‐thickness effect of the laminate can be properly accounted for with relatively few degrees of freedom and less computational effort. Several comparative examples are given to demonstrate the applicability of the method.

A three-dimensional hybrid stress isoparametric element for the analysis of laminated composite plates

In view of the increasing interest in using composite materials for aerospace structures, the analysis of laminated composite plates becomes essential. A three-dimensional eight-node hybrid stress finite element method is developed for the analysis of laminated plates. The hybrid stress model is based on the modified complementary energy principle and takes into account the transverse shear deformation effects. The displacement field is interpolated through shape functions and nodal displacements. All three displacement components are assumed to vary linearly through the thickness of each lamina. The stress field is interpolated through assumed stress polynomials with 55 stress parameters for each lamina. All six stresses are included and satisfy the homogeneous equilibrium equations. The validity of the hybrid stress finite element model is determined by comparing the predicted numerical results with the existing three-dimensional elasticity solutions. Excellent accuracy and fast convergence are observed in the numerical results.

A restricted hybrid stress formulation based on a direct finite element method

AbstractA direct method, which uses stress and displacement modes obtained from the governing equations of a problem, is adopted for finite element formulation. It is shown that this method actually leads to a restricted hybrid stress formulation if the displacement modes are changed to ensure symmetry of the stiffness matrix. Through this direct method, however, the problem of selecting the appropriate number of stress modes in the regular hybrid stress model is bypassed. Only the minimum number of modes that are compatible with the number of nodal degrees‐of‐freedom of an element is needed in the formulation. Using more modes only leads to a combination of stress modes, and will not improve the order of performance of the element. It is shown through numerical examples that the restricted hybrid stress formulation leads to well‐balanced elements.

Relations between incompatible displacement model and hybrid stress model

AbstractAn examination of the variational formulations confirms the similarity between the incompatible displacement model and the assumed stress hybrid model that was pointed out by Irons in 1972. But the basic differences between the two are also identified. For 8‐node solid elements the assumed stress terms obtained through a rational procedure also agree with those deduced by Irons purely from his physical insight.

A computer algorithm for generating analytically integrated hierarchical hybrid stress elements

A simple and accurate computer algorithm is presented for automatic generation of the analytically integrated hybrid stress elements. The elemental matrices for the hybrid stress model are constructed in algorithmic forms, and the analytical integrations are performed symbolically. The resulting algorithms can be evaluated by a computer within the simple DO loops of the related subroutines. The application of this computer algorithm is demonstrated for rectangular hybrid plate elements. Since the algorithms are independent of element geometry, hybrid elements of different geometries can be formed similarly. The versatility of the simple algorithmic operations are exploited by considering a series of stress and displacement field approximations in a hierarchical form. Within this hierarchy, the lower order elements are generated automatically from the higher order approximations by reducing the range of the indices of the related algorithms. The properties and accuracies of the resulting hierarchical hybrid plate elements are assessed and recommendations are made for their possible use in general-purpose computer programs.

合成ばり端部鉄骨フランジのひずみ性状に関する数値解析的研究 その1 : ハイブリッド型応力法に基づく合成ばりの解析法について

In this paper two numerical methods for analyzing a composite beam are proposed, which are based on the hybrid stress model. One assumes changeable effective width of concrete slab. The other assumes two dimensional stress distribution in a concrete slab. Both considers interface slip between top flange of a steel beam and a concrete slab. And these numerical methods are applied to predict the plastic behavior of two composite beams (CB 5-D, CB 5-DV) whose experimental results are reported by Dr. Igarashi et. al. (13)

Elasto-plastic analysis of axisymmetric structures subject to arbitrary loads by hybrid-stress finite elements

A semi-analytic finite element method in conjuction with a hybrid-stress functional based on the initial stress approach is presented for elasto-plastic analysis. A three-dimensional solid element based on hybrid-stress model is also implemented for the elasto-plastic analysis. A procedure to compute the non-linear effects in terms of Fourier series in the hybrid-stress model is described. The accuracy and efficiency of the semi-analytic method is evaluated via numerical examples by comparing the solution with a full 3-D solution. The semi-analytic method is observed to be a viable alternative to 3-D analysis in elasto-plastic analysis of axisymmetric structures subject to arbitrary loads.

Formulation of Quasi-Conforming element and Hu-Washizu principle

A new, convenient and general method called Quasi-Conforming Element Technique is introduced for Finite Element Formulations in which the strain is discretized in terms of nodal values and then the potential energy is used to obtain stiffness matrix. The Quasi-Conforming Element Technique is a multivariate finite element method so that it includes the compatible, incompatible and hybrid stress model as its special cases. The connection between this technique and Hu-Washizu principle is shown.

A direct vector combination procedure for finite element stiffness matrix formulation

AbstractA direct vector combination procedure is developed for the formulation of finite element stiffness matrices. Only the homogeneous solution modes to governing equations of a problem and a uniquely defined interpolation displacement function along the element boundary are used to generate pairs of nodal displacement and nodal force vectors. The stiffness matrix is obtained by a simple multiplication of the nodal displacement and nodal force vectors. The element formed passes all the higher order patch tests for the order of solution modes included in the formulation. However, the stiffness matrix may become unsymmetrical. Restoring symmetry leads to compromising some of the higher order patch test performances. The present procedure is shown to have direct relations to those of the displacement model and the hybrid stress model. The present procedure presents new possibilities for the improvement of element performance. The procedure itself can also be used to study the limitations of element behaviour.

A study of three‐node triangular plate bending elements

AbstractAn assessment of flat triangular plate bending elements with displacement degrees‐of‐freedom at the three corner nodes only is presented, with the purpose of identifying the most effective for thin plate analysis. Based on a review of currently available elements, specific attention is given to the theoretical and numerical evaluation of three triangular 9 degrees‐of‐freedom elements; namely, a discrete Kirchhoff theory (DKT) element, a hybrid stress model (HSM) element and a selective reduced integration (SRI) element. New and efficient formulations of these elements are discussed in detail and the results of several example analyses are given. It is concluded that the most efficient and reliable three‐node plate bending elements are the DKT and HSM elements.

Comparison of hybrid-stress element through-thickness distributions corresponding to a high-order plate theory

A study using the hybrid-stress model is presented to assess various through-thickness element stress distributions to be used in conjunction with a high-order plate theory. The plane-strain example problem of a semi-infinite plate under transverse sinusoidal loading is chosen so that a computationally efficient comparison of the stress assumptions can be made using 2-D plane-strain elements. The displacement assumption for all elements allows for inplane and transverse displacements which are cubic and quadratic, respectively. On the basis of results obtained in this pilot study, recommendations are made for extensions to single- or multi-layer thick plate elements.

An analysis of the crack tip stress field in DCB adhesive fracture specimens

The problem of a cracked adhesive bonded DCB-type fracture specimen has been analyzed using a hybrid stress model finite element analysis which incorporates an advanced crack tip element. Stresses in the near and far fields have been studied as a function of adherend/adhesive modulus ratio and adhesive thickness. The results are compared to monolithic systems with regard to the stress intensity factor and the localization of the singular stress domain associated with the crack tip.