On the eigensolution of elastically connected columns

On the eigensolution of elastically connected column

Exact natural frequencies of multi-level elastically connected taut strings and related problems

The dynamics of a family of simple, but extremely useful structural elements is governed by a second order Sturm-Liouville equation. This equation allows for the uniform distribution of mass and stiffness and enables the motion of strings and shear beams, together with the axial and torsional motion of bars to be described exactly. As a result, each member type in this family has been treated exhaustively when considered as a single member or when joined contiguously to others. However, when such members are linked in parallel by uniformly distributed elastic interfaces, their complexity becomes significantly more intractable and it is this class of structures that has led to renewed interest and which forms the basis of the work that follows. Initially, differential equations governing the coupled motion of the system are developed from first principles. These are organised into the form of a generalised linear symmetric eigenvalue problem, from which a family of uncoupled differential operators can be established. These operators define a series of exact substitute systems that together describe the complete motion of the original structure. These equations can then be used in either of two ways. In their most powerful form they can be developed into exact dynamic stiffness matrices that enable all the powerful features of the finite element method to be utilised. This subsequently enables sets of members carrying point masses and subject to point spring supports to be analysed easily. Alternatively, the equations are able to yield an exact relational model that links any uncoupled frequency of an original member to the corresponding set of coupled system frequencies. This approach enables ‘back of the envelope calculations’ to be undertaken quickly and efficiently. The exact mode shapes of the original structure can be recovered in either case. Due to space limitations, only aspects of the first technique are described briefly herein, but both are covered exhaustively elsewhere [1]

On the provenance of hinged-hinged frequencies in Timoshenko beam theory

An exact differential equation governing the motion of an axially loaded Timoshenko beam supported on a two parameter, distributed foundation is presented. Attention is initially focused on establishing the provenance of those Timoshenko frequencies generated from the hinged-hinged case, both with and without the foundation being present. The latter option then enables an exact, neo-classical assessment of the ‘so called’ two frequency spectra, together with their corresponding modal vectors, to be undertaken when zero, tensile or compressive static axial loads are present in the member. An alternative, ‘precise’ approach, that models Timoshenko theory efficiently, but eliminates the possibility of a second spectrum, is then described and used to confirm the original eigenvalues. This leads to a definitive conclusion regarding the structure of the Timoshenko spectrum. The ‘precise’ technique is subsequently extended to allow, either the full foundation to be incorporated, or either of its component parts individually. An illustrative example from the literature is solved to confirm the accuracy of the approach, the nature of the Timoshenko spectrum and a wider indication of the effects that a distributed foundation can have

Homogeneous trees of second order Sturm-Liouville equations: a general theory and program

Quantum graph problems occur in many disciplines of science and engineering and they can be solved by viewing the problem as a structural engineering one. The Sturm–Liouville operator acting on a tree is an example of a quantum graph and the structural engineering analogy is the axial vibration of an assembly of bars connected together with a tree topology. Using the dynamic stiffness matrix method the natural frequencies of the system can be determined which are analogous to the eigenvalues of the quantum graph. Theory is presented that yields exact solutions to the Sturm–Liouville problem on homogeneous trees. This is accompanied by an extremely efficient and compact computer program that implements the theory. An understanding of the former is enhanced by recourse to a structural mechanics analogy, while the latter program is fully annotated and explained for those who might wish to extend its capability. In addition, the use of the program as a ‘black box’ is fully described and a small parametric study is undertaken to confirm the accuracy of the approach and indicate its range of application including the computation of negative eigenvalues

Eigenvalues and eigenvectors of a system of Bernoulli Euler beams connected together in a tree topology

Eigenvalues and eigenvectors of a system of Bernoulli Euler beams connected together in a tree topolog

Exact eigensolution of a class of multi-level elastically connected members

Attention is given to determining the exact natural frequencies and modes of vibration of a class of structures comprising any number of related parallel members that are connected to each other, and possibly also to foundations, by uniformly distributed elastic interfaces of unequal stiffness. The members themselves are considered to have a uniform distribution of mass and stiffness and account can be taken of additional point masses and spring supports. The formulation is general and applies to any structure in which the motion of the component members is governed by a second order Sturm-Liouville equation. Closed form solution of the governing differential equations leads either, to a series of exact substitute systems that are easy to solve through a stiffness approach and which together yield the complete spectrum of natural frequencies and corresponding mode shapes of the original structure, or to simple exact relationships between the natural frequencies corresponding to coupled and uncoupled motion that enable hand solution of the more standard problems to be achieved. An appropriate form of the Wittrick-Williams algorithm is presented for converging on the required natural frequencies to any desired accuracy with the certain knowledge that none have been missed. Examples are given to confirm the accuracy of the approach and to indicate its range of application

Eliminating end effects for theoretical panel buckling with FEM

The theoretical buckling performance of thin walled panels under compression can be predicted using classical plate theory (CPT) and using tools such as the VICONOPT, which uses the finite strip method, and Abaqus, a finite element modelling program (FEM). VICONOPT is much more computationally efficient than FEM, and is able to optimise panel design for buckling and pseudo-postbuckling performance. This work forms part of a larger project to use VICONOPT to optimise panel designs that are allowed to buckle in a stable manner below the maximum allowed loading. Because VICONOPT can only make a first approximation of postbuckling performance, a method is required to validate the optimisation results against a more accurate method

Finite element buckling analysis of stiffened plates with filleted junctions

Modern aircraft wings are thin-walled structures composed of ribs, spars and stiffened panels, where the top skin is subject to compressive forces in flight that can cause buckling instability. If these panels are machined from a single billet of metal then the initial buckling performance can be significantly improved by increasing the fillet radius along the line junction between the stiffener webs and skin. Typically thin-walled structures are usually modelled with two dimensional elements. To model the stiffened panel with fillets three dimensional elements are required. For the stiffened panel selected for the analysis the paper shows that the three dimensional model shows a substantial increase in skin initiated buckling if the fillet is taken account of. A 5 mm radius leads to an increase of 34% increase in local buckling load performance for a skin portion of breath to thickness ratio of 100. The associated overall buckling load increases by 1.8%. The mass penalty for a 5 mm radius is 5.1%. To avoid local and overall buckling interaction an accurate measure of both buckling loads is very important and may have impact for designers. The three dimensional models with no fillets show very good agreement with the two dimensional models

A pinned-pinned beam with and without a distributed foundation: A simple exact relationship between their eigenvalues

The body of this paper considers a pinned-pinned Bernoulli-Euler beam, from which the core natural frequencies and critical buckling loads corresponding to in-plane flexure, can be determined easily. The theory is then developed to yield an exact relationship between the static axial load in the beam and the frequency of vibration. This enables the core eigenvalues to be related exactly to their counterparts when the beam is additionally supported on a two parameter elastic foundation.The relationship is simple, exact and obviates the complex problems involved in solving the foundation problem using more traditional techniques. A number of illustrative problems are solved to confirm the accuracy and efficacy of the approach

Structural optimisation using analytical equations

Optimisation of structures requires the minimisation of an objective function subject to a set of constraints. Typically the objective function is mass for mass sensitive structures or cost for heavy engineering projects. However environmental sensitivity can be incorporated without any difficulty. For example in an energy scarce environment energy inputs can be considered as the objective function which can take into account all energy inputs such as material manufacture and structural element forming. Mass and energy are in fact more important considerations than cost. The barrier to this has been shaped by the political world where cost considerations are considered to be of paramount importance