John Argyris

An excursion into large rotations

The present discourse develops an enlarged exploration of the matrix formulation of finite rotations in space initiated in [1]. It is shown how a consistent but subtle matrix calculus inevitably leads to a number of elegant expressions for the transformation or rotation matrix T appertaining to a rotation about an arbitrary axis. Also analysed is the case of multiple rotations about fixed or follower axes. Particular attention is paid to an explicit derivation of a single compound rotation vector equivalent to two consecutive arbitrary rotations. This theme is discussed in some detail for a number of cases. Semitangential rotations—for which commutativity holds—first proposed in [2, 3]are also considered. Furthermore, an elementary geometrical analysis of large rotations is also given. Finally, we deduce in an appendix, using a judicious reformulation of quarternions, the compound pseudovector representing the combined effect of n rotations. In the authors opinion the present approach appears preferable to a pure vectorial scheme—and even more so to an indicial formulation— and is computationally more convenient.

On large displacement-small strain analysis of structures with rotational degrees of freedom

Abstract The matrix displacement analysis of geometrically nonlinear structures becomes an intricate task as soon as finite elements in space with rotational degrees of freedom are considered. The fundamental reason for these difficulties lies in the non-commutativity of successive finite rotations about fixed axes with different directions. In order to circumvent this difficulty, a new definition of rotations — the so-called semitangential rotations — is introduced in this paper. Our new definition leads to a reformulation of the theory of [1,2]which in itself is clearly consistent and correct. In contrast to rotations about fixed axes these semitangential rotations which correspond to the semitangential torques of Ziegler [3]possess the most important property of being commutative. In this manner, all complexities involved in the standard definition of rotations are avoided ab initio . A specific aspect of this paper is a careful exposition of semitangential torques and rotations, as well as the consequences of the semitangential definitions for the geometrical stiffness of finite elements. In fact, these new definitions permit a very simple and consistent derivation of the geometrical stiffness matrices. Moreover, the semitangential definition automatically leads to a symmetric geometrical stiffness which clearly expresses that the nonlinear strain-displacement relations must satisfy the condition of conservativity of the structure itself — independently of any loading. The general theory of geometrical stiffness matrices as evolved in this paper is applied to beams in space. The consistency of the theory is demonstrated by a large number of numerical examples not only of straight beams but also of the lateral and torsional buckling and post-buckling behaviour of stiff-joined frames. Most of the former developments appear to be inadequate.

Recent developments in the finite element analysis of prestressed concrete reactor vessels

Abstract This paper describes recent developments in the nonlinear deformation and ultimate load analysis of prestressed concrete reactor vessels using finite elements. First, a number of finite element models are called into attention for the idealization of composite structures such as reinforced and prestressed concrete components. Then different inelastic constitutive models are proposed for the behaviour of concrete in the pre- and post-failure regime. Subsequently various numerical techniques are examined for the solution of nonlinear problems, especially with regard to their distortion of the constitutive model. In conclusion these modelling techniques are applied to the analysis of four typical examples, the nonlinear deformation analysis of a concrete specimen subjected to biaxial compression, the crack analysis of a thick-walled concrete cylinder, the overload analysis of the THTR 1 : 5 scale model, and the ultimate load analysis of a concrete top closure model.

Finite elements in time and space

Abstract The idea of finite element discretisation is applied to time dependent phenomena. Hamiltons principle is used as a suitable variational statement which means that the time is discretised into a set of finite elements which are taken to be the same for all structural elements. The method is introduced for the unidimensional case and thereafter generalised for multidegree of freedom systems. The use of arbitrary time elements is outlined. The method appears particularly suitable for the investigation of time dependent dynamic phenomena without prior deduction of the natural modes and frequencies. Linear as well as nonlinear phenomena may be analysed.

Computerized integrated approach for design and stress analysis of spiral bevel gears

Abstract An integrated computerized approach for synthesis, analysis and stress analysis of enhanced spiral bevel gear drives is proposed. The approach is accomplished by application of computerized methods of local synthesis and simulation of meshing and contact of gear tooth surfaces. The developed methods enable to provide: (i) a localized bearing contact less sensitive to errors of alignment, and (ii) reduced vibration and noise. Application of finite element analysis allows to investigate the formation of bearing contact during the cycle of meshing and perform the stress analysis. The design of finite element models and the settings of boundary conditions are automatized. The developed approach is illustrated with numerical examples.

On the geometrical stiffness of a beam in space—a consistent V.W. approach

The application of the standard virtual work expressions to the large displacement-small strain domain merely requires the replacement of the standard linear strain-displacement relations by the quadratic ones. In fact, the geometrical stiffness matrix of an arbitrary finite element can be derived immediately from the virtual work of the second order terms in the strains. A correct geometrical stiffness is, however, only obtained if the prerequisites of the energy theorems are strictly observed. This proves to be a rather difficult task as soon as the elements contain rotational freedoms. In this case, the solution demands a comprehensive understanding of the true nature of the strains, stresses and nodal displacements, rotations as well as forces, moments. After an extensive study of the relevant entities the above principle is successfully applied to the derivation of the geometrical stiffness of a beam element in space. The consistency of the present approach is demonstrated by the full agreement with the prior results of the authors based on the natural mode technique [2,3]. Some numerical examples demonstrate the practical importance of the present development for the geometrically nonlinear analysis of three-dimensional frame structures.

Computational aspects of welding stress analysis

Abstract The residual stresses and deformations due to arc-welding are determined in a rectangular steel plate. The thermomechanical response behaviour is computed in two steps (i), the heat flow analysis of the entire plate due to the moving electrode, and (ii) the thermoelastic-viscoplastic response analysis of the midsection for the transient temperature history at this section. The staggered solution strategy leads to a fully integrated approach of the thermal and mechanical solution steps within each time increment providing for weak thermomechanical coupling. The main difficulties arise due to the transient character of the heating-cooling cycle which spans the entire temperature regime from room temperature up to melting and vice versa. Clearly, highly localized material phase changes and associated stress redistributions complicate the time history analysis in addition to volume changes and latent heat effects during recristallization. The incremental solution of the thermoelastic-viscoplastic process is illustrated with the arc-welding of a rectangular steel plate for which limited experimental data are also available.

A simple triangular facet shell element with applications to linear and non-linear equilibrium and elastic stability problems

Abstract This paper has two main objectives. First to describe a very simple facet triangular plate and shell finite element called TRUMP which includes, if required, transverse shear deformation and is based on physical lumping ideas with a simple mechanical interpretation [ 1,2,4,5 ]. Second to give an account of some non-trivial numerical examples of large deflection and post-buckling of shells. There are two types of non-linear structural problems which give rise to particularly delicate numerical experimentation. They are those involving deflections of the order of the structural dimensions, such as three-dimensional elastica, and the instability phenomena of the type leading to dynamic snapthrough, e.g. in cylindrical panels. To tackle such problems using a highly sophisticated shell element such as SHEBA is neither easy nor inexpensive. It is shown that the TRUMP element with only 18 displacement and rotation degrees of freedom is relatively economical to use and yet capable of engineering accuracy. The paper makes use of the theory of simplified geometrical stiffness based on the natural mode method which has been described fully in previous publications [1,2].

Nonlinear finite element analysis of elastic systems under nonconservative loading—natural formulation part II. Dynamic problems

Abstract The paper presents a nonlinear finite element analysis of elastic structures subject to nonconservative forces. Attention is focused on the stability behaviour of such systems. This leads mathematically to non-self-adjoint boundary-value problems which are of great theoretical and practical interest, in particular in connection with alternative modes of instability like divergence of flutter. Only quasistatic effects are however considered in the present part. The methodology of our theory is general, but the specific thrust of the present research is directed towards the analysis of structures acted upon by displacement-dependent nonconservative (follower) forces. In a finite element formulation the analysis of geometrically nonlinear elastic systems subject to such forces gives, in general, rise to a contributory nonsymmetric stiffness matrix known as the load correction matrix. As a result, the total tangent stiffness matrix becomes unsymmetric - an indication of the non-self-adjoint character of the problem. Our theory is based on the natural mode technique [1, 2, 3]and permits i.a. a simple but elegant derivation of the load correction matrix. The application of the general theory as evolved in this paper is demonstrated on the beam element in space. A number of numerical examples are considered including divergence and flutter types of instability, for which exact analytic solutions are known. The problems demonstrate the efficiency of the present finite element formulation. The paper furnishes also a novel and concise formulation of finite rotations in space which may be considered as a conceptual generalization of the theory presented in [2, 3].

TRIC : a simple but sophisticated 3-node triangular element based on 6 rigid-body and 12 straining modes for fast computational simulations of arbitrary isotropic and laminated composite shells

TRIC is a simple but sophisticated 3-node shear-deformable isotropic and composite flat shell element suitable for large-scale linear and nonlinear engineering computations of thin and thick anisotropic plate and complex shell structures. Its stiffness matrix is based on 12 straining modes but essentially requires the computation of a sparse 9 by 9 matrix. The element formulation departs from conventional Cartesian mechanics as well as previously adopted physical lumping procedures and contains a completely new implementation of the transverse shear deformation; it naturally circumvents all previously imposed constraints. The methodology is based on physical inspirations of the Natural-Mode finite element method (NM-FEM) formalized through appropriate geometrical, trigonometrical and engineering mathematical relations and it involves only exact integrations; its stiffness, mass and geometrical matrices are all explicitly derived. The kinematics of the element are hierarchically decomposed into 6 rigid-body and 12 straining modes of deformation. A simple congruent matrix operation transforms the elemental natural stiffness matrix to the local and global Cartesian coordinates. The modes show explicitly how the element deforms in axial straining, symmetrical and antisymmetrical bending as well as in transverse shearing; the latter has only become clear in the formulation presented here and has brought about a completion of the understanding of natural modes as they apply to the triangular shell element. A wide range of numerical examples substantiate the conception and purpose of the element TRIC; fast convergence is observed in many examples.