J.L. Meek

Geometrically nonlinear analysis of space frames by an incremental iterative technique

The paper explores the analysis of the geometrically nonlinear behaviour of space structures, using the modified arc length method of Riks and Crisfield. Several problems not previously documented in the literature are encountered in the solution procedure, and means evolved for circumventing the same. The resulting algorithm is robust and able to handle problems that exhibit several negative eigenvalues simultaneously. Several examples are given of the trace of load-deflection paths of space frames.

Formulation of Mindlin-Engesser model for stiffened plate vibration

In this paper, a Mindlin-Engesser model is developed for the vibration analysis of moderately thick plates with arbitrarily oriented stiffeners. The theoretical derivation incorporates the Mindlin theory to account for the effects of transverse shear deformation and rotary inertia of plates, and the Engesser theory to account for the shear deformation of stiffeners with the inclusion of torsion effect. In the method of solution, the resulting energy functionals are minimized using the Ritz procedure with a set of admissible two-dimensional functions expressed in the form of simple polynomials. The key kinematic feature of these shape functions is that they are boundary oriented and no boundary losses are introduced as in discretization methods. With the aim of demonstrating the applicability and versatility of the method, numerical examples including plates of various shapes with arbitrarily oriented stiffeners are presented. Several findings and conclusions regarding the method have been highlighted and discussed.

A study on the instability problem for 2D-frames

This paper presents large deflection, post-buckling analysis of the two-dimensional elastic frame from a dynamic point of view. A co-rotational formulation combined with small deflection beam theory with the inclusion of the effect of axial force is adopted. An increment iterative method based on the modified Newton-Raphson method and extrapolation techniques for improving the convergence behaviour combined with a constant arc length control method is employed for the solution of the non-linear equilibrium equations before the limit point. A non-linear dynamic analysis based on the average acceleration of the Newmark algorithm with a slow rate of load incrementation is carried out to trace the load-deflection path beyond the limit point. As a result, the snap through problem is overcome without down loading. The efficiency of the method proposed in this study is demonstrated by typical two-dimensional frames which show snap-through behaviour in static analysis with any increment of load after the limit point.

Nonlinear static and dynamic analysis of shell structures with finite rotation

A simple, effective finite element incremental formulation and procedure for geometrically nonlinear static and dynamic analysis of a shell structure with finite rotations are presented, based on both the total Lagrangian and updated Lagrangian description of motion. The co-rotation formulation is used for finite rotation. The element employed here to implement the present method is a faceted shell element with Loof nodes (DKL + LST). An incremental iterative method based on the constant arc length method in conjunction with Newton-Raphson method is implemented in the present study for static and Newmark’s integration scheme for dynamic analysis. To demonstrate the accuracy and efficiency of the formulations and to compare the difference between the total and updated Lagrangian formulation, numerical studies are presented. 0 1998 Elsevier Science S.A. All rights reserved.

Dynamic response and instability of frame structures

Abstract This paper presents a geometrically non-linear dynamic instability analysis for both two- and three-dimensional frames, which may be subjected to finite rotations. The finite element displacement method based on the beam–column approach is employed to derive the non-linear equations governing the behaviour of plane and spatial frames. A co-rotational formulation combined with small deflection beam theory with the inclusion of the effect of axial force is adopted. The governing dynamic equilibrium equations are obtained from the static equations by adding the inertia and damping terms. The implicit Newmark time integration with the Newton–Raphson (NR) iteration method is employed. Dynamic critical loads are defined by the Budiansky–Roth criterion. Several numerical examples are illustrated to demonstrate the effectiveness of the present method.

Large displacement analysis of thin plates and shells using a flat facet finite element formulation

The large displacement analysis of flat faceted shell element, previously developed by Meek and Tan [15] is further developed. This element, has been already tested in the nonlinear range via the use of the total Lagrangian formulation and a somewhat cumbersome procedure for the calculation of the joint orientation matrices at each step of the calculations [25]. However, this approach is quite complex and by taking advantage of the positions of the rotational degrees of freedom, being located at the Loof nodes along the edges and normal to the edges, a simple and elegant formulation is obtained. The theory is developed in incremental form and the updated Lagrangian approach with the co-rotational formulation is employed. Numerical examples are presented and compared with other published results to verify the proposed formulation.

Theoretical and Experimental Investigation of a Shallow Geodesic Dome

This paper presents the nonlinear analytical and experimental study of a shallow geodesic dome comprising thin walled circular hollow sections where the large displacement theory is applicable. The numerical formulation is based on the updated Lagrangian method incorporating geometric and material nonlinearities. The strain unloading is included in the present numerical analysis. Warping effects are ignored. A 156-member shallow geodesic dome that has a rise to span ratio of 1:10 (i.e., a rise of 600 mm to span of 6000 mm) was constructed and tested experimentally to validate the proposed numerical method. The analytical nonlinear analysis of this dome compares well with the experimental results. Both results show a considerable difference with the large displacement analysis in which the material nonlinearity is not considered.

Computer Shape Finding of Form Structures

This paper presents a non-linear finite element procedure for shape finding of form structures and studies an approach wherein the engineer recognises formable curved surfaces focusing on the engineers own imagination and creativity before turning to numerical analysis to assist in the design. With the investigation of a simple numerical method of locating geodesic lines, a convenient shape finding approach is presented. Another purpose of this paper is to present a method by which compression shell shapes can be obtained by inverting the loads on the tension shape obtained by loading a cable net system. This work parallels the physical models used by H. Isler for his work on thin concrete shells. The paper gives examples of the mesh shapes generated and the computer visualisation of several examples. It demonstrates that the computer modelling is a suitable alternative to the physical modelling process. The numerical results given show that the computer software is reliable, and provides the designer with a useful tool for membrane design.

A stiffness matrix extrapolation strategy for nonlinear analysis

The use of the direct stiffness method of analysis of geometrically nonlinear structural problems was pioneered by the work of Turner, Dill, Martin and Melosh [l]. Basically, the solution of a nonlinear problem reduces to that of tracing a nonlinear load-displacement path by solving a system of nonlinear equations. This problem is however normally very involved, with multi-degrees of freedom and its solution occurs in the multi-dimensional space. A host of solution strategies for tackling nonlinear structural analysis has since evolved and several reviews on the subject can be found in the literature [2-6]. Bergan et al. [7] found it convenient to classify the solution techniques according to their mathematical formulation; from which 3 different categories could be identified. For the first category, the problem is represented by an energy potential. Use is then made of the algorithms for unconstrained minimization to search for the stationary value of the energy functional. Techniques such as the random search method [8] and Powell’s method of conjugate directions [9] make use of the object function only; the method of steepest descent [8] and the conjugate gradient method [lo], in addition make use of the gradient (first derivative); while the quasi-Newton [ll-131, secant-Newton [14] and Newton-Raphson [15] methods require the evaluation of the 2nd derivatives as well. The second category of solution techniques is based on the equilibrium conditions, which are usually formulated by setting to zero the variation of the total potential energy of the system. The method of functional iteration 1161 belongs to this group, but it is characterised by poor convergence properties. The most powerful of the solution techniques here is the Newton-Raphson algorithm [17]. In the third category, use is made of the incremental form of the equilibrium equations. The easiest technique is the Euler-Cauchy method (or incremental stiffness procedure) [l]. Use of more accurate integration schemes such as the Runge-Kutta [18] or the predictor-corrector [18] methods will reduce the drifting effect which is associated with the Euler integration. The most popular techn,ique is to combine both the incremental and iterative methods; a simple Euler incrementation is first ‘applied and then either the full or modified Newton-Raphson method is evoked to iterate to equilibrium [7]. The above techniques as they stand, are unsatisfactory for tracing the post limit response of the structure. Strategies which allow one to follow the post limit behaviour are the method of artificial springs [19], the ‘current stiffness parameter’ method of Bergan [7], the displacement

Computer Cutting Pattern Generation of Membrane Structures

This paper sets out a simple design procedure for generating the cutting patterns of a membrane surface which can then be used for a variety of purposes, including fabric of tension surface. It does this by pre-defining the requirements of the cutting pattern which is set out by the designer on an initially flat surface. A cable net is used for the approximating surface and the cables approximate the warp and weft directions of the fabric (see our previous paper on shape generation). The present work designs the procedure and gives examples of results of the shape finding of a surface with a number of different cutting patterns. An example of a compression membrane surface is also given.