Stress oscillations in plane stress modelling of flexure—a field-consistency interpretation

Exactly integrated isoparametric plane stress elements behave poorly in flexure. The 4-noded element ‘locks’, with errors that progress indefinitely as element aspect ratio increases. Reduced integration of the shear strain energy eliminates this locking entirely. The 8-noded element does not lock, but improves in performance with reduced integration of shear strain energy. Both elements, with their original shape functions, show severe shear stress oscillations in flexure. In this paper we attribute these oscillations to the lack of ‘consistency’ of shear strain fields derived directly from independent field-variable interpolations. We derive error models for specific tractable examples which can confirm the accuracy of this conceptual scheme through digital computation using the finite element models. A field-consistent redistribution strategy for the shear strain field is offered as an elegant procedure to free the elements of spurious oscillations and give a ‘lock’-free performance.     

A fast stiffness formulation for finite element analysis of two-dimensional solids

This paper describes and efficient method for computing two-dimensional element stiffness matrices. It is a generalization of the algorithm given by Taylor. ¹Theory is developed for elements which are loaded under plane stress, plane strain or axisymmetric conditions, and two FORTRAN IV subroutines are included. These subroutines, which are presented in a modular form, are applicable to any type of serendipity, Lagrangian or triangular element, and may be incorporated into a user's library. Alternatively, if desired, the relevant code may be embedded directly into existing stiffness generation routines.     

Machine locking of degenerated thin shell elements

The degenerated shell element is one of the most efficient elements for analysing shell structures. However, it is known to result in rather stiff models when used in thin element applications. The phenomena associated with this behaviour are known as locking phenomena. This paper analyses the machine locking mechanism developed in thin to very thin Lagrangian and serendipity elements. The machine related locking phenomenon is distinguished from the shear and membrane locking phenomena. A remedy for the pure machine locking problem is developed for the two elements. The proposed remedy is based on the technique of the modified transverse shear modulus. It is also extended to control shear locking. The proposed technique is shown to completely eliminate machine locking. Also, it is shown to effectively alleviate stiffening effects due to the presence of spurious shear strain.     

Reduced minimization of Mindlin plate

The use of inconsistent displacement fields for Mindlin plate elements causes unmatched coefficients to appear in shear strain interpolations. The role of the numerical integration order in relation to these unmatched coefficients, and the existence of optimal stress points are explained through the use of the reduced minimization concept. This concept makes it possible to test whether assumed displacement fields or shear strain fields result in inconsistent strain fields which contain spurious constraints. We have included applications of reduced minimization to the conventional C⁰-continuous elements, which employ reduced integration, and to quadrilateral Mindlin plate elements of Hinton and Huang to demonstrate how these elements alleviate shear locking.     

A study on the linear elastic stability of Mindlin plates

The debate on the performance of Lagrangian and Serendipity elements in Mindlin's plate theory has been going on for quite some time. Limited published results for static and vibration analysis based on exact integration demonstrated a drastic deterioration in accuracy as the thickness of the plate decreases, and reduced/selective integration schemes have been suggested to improve their performance. Appreciable improvement for Lagrangian elements has been recorded, but it is only marginal for the Serendipity elements. On the strength of such observations one would then be tempted to rule out the exact integration schemes and Serendipity elements. In this paper, the above problem is reviewed for stability analysis of plates. Two elements are chosen from each family, one representing the higher order and the other the lower order element. Contrary to published results, all elements can attain very accurate solutions independent of the integration schemes for a sufficiently restrained plate, although in general the Serendipity elements will require a more refined mesh than the Lagrangian ones. However, for loosely restrained plates, the solution failed when integration is performed by reduced/selective schemes. The failure marks the limitation of the reduced/selective schemes which have somehow introduced spurious modes into the system, but on the other hand it is ironical that these spurious modes in fact contribute to the improvement of performance of the restrained cases. Therefore, one can equally improve the Serendipity elements by a selective scheme which can introduce additional zero modes. A scheme based on (5 × 5) integration points for flexural stiffness and (3 × 3) integration points for shear stiffness improved the 17SE (Serendipity elements) remarkably. However, because of the lack of bounds in most cases, the use of reduced/selected schemes is still not recommended. Finally, this paper also proposed an approximate formulation of the geometric stiffness matrices to replace the full formulation. Such an approximate formulation reduces the number of variables considerably in the eigenvalue search and can still give reasonably accurate results for thin plates with a/t > 15.     

Quadratic NMS Mindlin‐plate‐bending element

The development of a robust and efficient quadratic Mindlin‐plate‐bending elements mainly by the use of non‐conforming displacement modes is presented in this paper. A brief review on the previous efforts to develop efficient non‐conforming Mindlin plate bending elements is also given. The behaviour of the newly proposed plate element is further improved by the combined use of nonconforming displacement modes, the selectively reduced integration scheme, and the assumed shear strain fields. Thus, the newly developed element has been designated as ‘NMS‐8P’. The improvement achieved may be attributable to the fact that the merits of these improvement techniques are merged in the formation of the new element in a complementary manner. The proposed 8‐node element passes the patch tests, does not show spurious mechanism, and does not produce shear locking phenomena even with distorted meshes. It is also shown that the element produces reliable solutions through numerical tests for standard benchmark problems. Copypright © 1999 John Wiley & Sons, Ltd.     

DESIGN SENSITIVITY ANALYSIS AND OPTIMAL DESIGN OF COMPOSITE STRUCTURES USING HIGHER ORDER DISCRETE MODELS

This paper deals with the structural optimization of multilaminated composite plate structures of arbitrary geometry and layup, using single layer higher order shear deformation theory discrete models. The structural and sensitivity analysis formulation is developed for a family of C° Lagrangian elements. The design sensitivities of static response for objective and/or constraint functions, such as maximum displacements, stress failure criterion and elastic strain energy, with respect to ply angles and ply thickness are presented. The objectives of the design are the minimization of the structural elastic strain energy, minimization of maximum deflection and/or the minimization of the structure volume. The accuracy and relative performance of the proposed discrete models are compared and discussed among developed elements and alternative models. Several test designs are optimized to show the applicability of the proposed refined discrete models.     

Reduced modified quadratures for quadratic membrane finite elements

Reduced integration is frequently used in evaluating the element stiffness matrix of quadratically interpolated finite elements. Typical examples are the serendipity (Q8) and Lagrangian (Q9) membrane finite elements, for which a reduced 2 × 2 Gauss–Legendre integration rule is frequently used, as opposed to full 3 × 3 Gauss–Legendre integration. This ‘softens’ these element, thereby increasing accuracy, albeit at the introduction of spurious zero energy modes on the element level. This is in general not considered problematic for the ‘hourglass’ mode common to Q8 and Q9 elements, since this spurious mode is non‐communicable. The remaining two zero energy modes occurring in the Q9 element are indeed communicable. However, in topology optimization for instance, conditions may arise where the non‐communicable spurious mode associated with the elements becomes activated. To effectively suppress these modes altogether in elements employing quadratic interpolation fields, two modified quadratures are employed herein. For the Q8 and Q9 membrane elements, the respective rules are a five and an eight point rule. As compared to fully integrated elements, the new rules enhance element accuracy due to the introduction of soft, higher‐order deformation modes. A number of standard test problems reveal that element accuracy remains comparable to that of the under‐integrated counterparts. Copyright © 2004 John Wiley & Sons, Ltd.     

Refined 9‐Dof triangular Mindlin plate elements

Based on the Mindlin/Reissner plate theory, two refined triangular thin/thick plate elements, the conforming displacement element DKTM with one point quadrature for the part of shear strain and the element RDKTM with the re‐constitution of the shear strain, are proposed. In the formulations the exact displacement function of the Timoshenko's beam is used to derive the element displacements of the refined elements. Numerical examples are presented to show that the present models indeed possess properties of high accuracy for thin and thick plates, is capable of passing the patch test required for Kirchhoff thin plate elements, and does not exhibits extra zero energy modes. The element RDKTM is free of locking for very thin plate analysis and its convergence can be ensured theoretically. However, the element DKTM is not free of shear locking when the thickness/span ratios less than 10^?2. Copyright © 2001 John Wiley & Sons, Ltd.     

Field- and edge-consistency synthesis of A 4-noded quadrilateral plate bending element

In this paper, we demonstrate the use of two conceptual principles, the field-consistency requirement and the edge-consistency requirement, as the basis for deriving a 4-noded quadrilateral plate bending element based on Mindlin plate theory using Jacobean transformations only. The derivation is now free of the use of such devices as strain-interpolation points and Hrennik off strain reference lines, etc., which have been the basis for many recent formulations of this element. The shear strain constraints are now consistently defined within the element domain, and ‘tangential’ shear strains are consistently matched at element boundaries so that there is no locking even under extreme distortion—e.g. even when two nodes are collapsed so that the quadrilateral becomes a triangle. Numerical experiments show that this synthesis produces an element that should be identical to other recent formulations of this element based on tensorial transformations or on shear constraint condensation on the edges, but now given a more complete and formal logical basis.     

On using different finite elements with an automatic adaptive refinement procedure for the solution of 2-D stress analysis problems

A series of numerical tests is carried out employing some commonly used finite elements for the solution of 2-D elastostatic stress analysis problems with an automatic adaptive refinement procedure. Different kinds of elements including Lagrangian quadrilateral and triangular elements, serendipity quadrilaterals, incompatible elements and hybrid elements have been tested. It is found that for a general problem involving compressible material and when a moderate accuracy of the final solution is sought, the nine-node Lagrangian (L9) element will be the most effective element, while when an extremely accurate solution is needed, higher order Lagrangian quadrilaterals or triangles will be a suitable choice. However, if only linear elements are available, the well known 5βI linear hybrid element is the best choice. © 1997 John Wiley & Sons, Ltd.     

A new 3-D finite element model to evaluate added mass for analysis of fluid-structure interaction problems

The paper presents the results of investigations conducted to evaluate the added mass to represent fluid-structure interaction effects in vibration/dynamic analysis of floating bodies such as ship hulls. While the structural plating is idealized by 9-noded plate/shell finite elements, the fluid domain is modelled by 20-noded/21-noded 3-D finite elements in the investigations conducted. A new 8-noded element has been developed to model the interface between the structure and the fluid. An efficient computational methodology has been used for computation of added mass. The finite element models are validated by comparing the results with those given by analytical solution for a submerged sphere. The efficacy of the finite element model is demonstrated through convergence of the results obtained for a floating barge problem. A better convergence rate and distribution of added mass in three orthogonal directions have been obtained.     

MODIFIED CRACK CLOSURE INTEGRAL (MCCI) FOR 3-d PROBLEMS USING 20-NODED BRICK ELEMENTS

Abstract— The Modified Crack Closure Integral (MCCI) technique based on Irwin's crack closure integral concept is very effective for estimation of strain energy release rates G in individual as well as mixed-mode configurations in linear elastic fracture mechanics problems. In a finite element approach, MCCI can be evaluated in the post-processing stage in terms of nodal forces and displacements near the crack tip. The MCCI expressions are however, element dependent and require a systematic derivation using stress and displacement distributions in the crack tip elements.
Earlier a general procedure was proposed by the present authors for the derivation of MCCI expressions for 3-dimensional (3-d) crack problems modelled with 8-noded brick elements. A concept of sub-area integration was proposed to estimate strain energy release rates at a large number of points along the crack front. In the present paper a similar procedure is adopted for the derivation of MCCI expressions for 3-d cracks modelled with 20-noded brick elements. Numerical results are presented for center crack tension and edge crack shear specimens in thick slabs, showing a comparison between present results and those available in the literature.     

Adding transverse normal stresses to layered shell finite elements for the analysis of composite structures

This work presents a formulation developed to add capabilities for representing the through thickness distribution of the transverse normal stresses, σ_z, in first and higher order shear deformable shell elements within a finite element (FE) scheme. The formulation is developed within a displacement based shear deformation shell theory. Using the differential equilibrium equations for two-dimensional elasticity and the interlayer stress and strain continuity requirements, special treatment is developed for the transverse normal stresses, which are thus represented by a continuous piecewise cubic function. The implementation of this formulation requires only C⁰ continuity of the displacement functions regardless of whether it is added to a first or a higher order shell element. This makes the transverse normal stress treatment applicable to the most popular bilinear isoparametric 4-noded quadrilateral shell elements.

To assess the performance of the present approach it is included in the formulation of a recently developed third order shear deformable shell finite element. The element is added to the element library of the general nonlinear explicit dynamic FE code DYNA3D. Some illustrative problems are solved and results are presented and compared to other theoretical and numerical results.     

A study of quadrilateral plate bending elements with ‘reduced’ integration

The accuracy of certain thick plate elements when used in the context of thin plate problems can be improved by the use of reduced integration of the stiffness matrices. A series of numerical experiments on five different quadrilateral thick plate elements demonstrates the use of reduced integration and indicates the main reason for its success. This is the relaxation of a constraint on the shear strains. It is shown that the performance of a nine-noded Lagrangian element is near optimal.     

Reduced minimization in Lagrangian Mindlin plate element with arbitrary orientation under uniform isoparametric mapping

For the displacement-based Lagrangian Mindlin plate elements oriented arbitrarily under uniform isoparametric mapping without internal distortion, a theoretical interpretation on the conventional shear-reduced integration is presented by introducing the concept of reduced minimization. It reveals that the conventional shear-reduced integration such as URI and SRI prevents the unmatched coefficients of shear strains from becoming spurious constraints. Thus, the shear-reduced integration eliminates spurious constraints and prevents locking.     

QUADRATIC TRIANGULAR C0 PLATE BENDING ELEMENT

In this paper, a six-node triangular C⁰ plate bending element is developed by the assumed strain formulation. The sampled transverse shear strains in the element are chosen such that the latter has a favourable constraint index of shear locking and the strains are optimized with respect to a linear pure bending displacement/rotation field. It happens that the optimal strains are the mean strains along the element edges and medians. Numerical examples reveal that the element is free from shear locking and passes all the patch tests for plate bending elements. Moreover, the element accuracy is close to that of a state-of-the-art seven-node assumed strain element. © 1997 by John Wiley & Sons, Ltd.     

Calculation of strain-energy release rates with higher order and singular finite elements

A general finite element procedure for obtaining strain-energy release rates for crack growth in isotropic materials is presented. The procedure is applicable to two-dimensional finite element analyses and uses the virtual crack-closure method. The procedure was applied to non-singular 4-noded (linear), 8-noded (parabolic), and 12-noded (cubic) elements and to quarter-point and cubic singularity elements. Simple formulae for strain-energy release rates were obtained with this procedure for both non-singular and singularity elements. The formulae were evaluated by applying them to two mode I and two mixed mode problems. Comparisons with results from the literature for these problems showed that the formulae give accurate strain-energy release rates.     

On hybrid stress,hybrid strain and enhanced strain finite element formulations for a geometrically exact shell theory with drilling degrees of freedom

The paper is concerned with the finite element formulation of a recently proposed geometrically exact shell theory with natural inclusion of drilling degrees of freedom. Stress hybrid finite elements are contrasted by strain hybrid elements as well as enhanced strain elements. Numerical investigations and comparison is carried out for a four-node element as well as a nine-node one. As far as the four-node element is concerned it is shown that the stress hybrid element and the enhanced strain one are equivalent. The hybrid strain formulation corresponds to the hybrid stress formulation only in shear dominated problems, that is the case of the plate. © 1998 John Wiley & Sons, Ltd.     

Geometrically nonlinear analysis of shells by quadrilateral flat shell element with drill,shear, and warping

In this paper, a four‐node quadrilateral flat shell element is proposed for geometrically nonlinear analysis based on updated Lagrangian formulation with the co‐rotational kinematics concept. The flat shell element combines the membrane element with drilling degrees of freedom and the plate element with shear deformation. By means of these linearized elements, a simplified nonlinear analysis procedure allowing for warping of the flat shell element and large rotation is proposed. The tangent stiffness matrix and the internal force recovery are formulated in this paper. Several classic benchmark examples are presented to validate the accuracy and efficiency of the proposed new and more proficient element for practical engineering analysis of shell structures. Copyright © 2016 John Wiley & Sons, Ltd.