Flexibility Matrix Method Research Articles

Cracking-Induced Mistuning in Bladed Disks

Abstract : A study of the mechanisms of cracking-induced mistuning in bladed disks is presented in this paper. An analytical model for a bladed disk was formulated using lumped-mass-beams, and a cracked blade was represented by a beam with a through-crack at the root. Local stiffness reduction due to cracking was incorporated using a flexibility matrix method. The dynamic characteristics of the bladed disk were analysed for the tuned system and then for the mistuned system with cracks of various depths introduced at the root of a blade. The dynamic characteristics of the mistuned system with a cracked blade were evaluated and the mistuning pattern due to blade cracking was investigated. The mechanisms of cracking-induced mistuning were elucidated by analysing the mode localisations and the reductions in natural frequency for different coupling ratios between the disk and blades. The results show that the occurrence of a cracked blade significantly increases the response amplitude of the other uncracked blades and therefore gives rise to enhanced HCF damage for all coupling ratios studied. In the present study, the relationship between the cracked blade and its vibration response signature is established, hence providing the essential guidance required for blade crack detection procedures and for blade failure investigations.

Flexibility matrix method applied to the finite element elastohydrodynamic analysis of plain journal bearings

The main reasons for using the finite element method (FEM) in the analysis of fluid film bearings are its flexibility in handling complex configurations, and particular boundary conditions, and the ease of its coupling with a structural finite element analysis to produce an elastohydrodynamic (EHD) solution. In this work an approach based on combining separate hydrodynamic and structural analyses is followed by introducing the structural information into a modular code in the form of an externally calculated flexibility matrix. In order to study a wide range of bearing housing and shaft configurations, a semi-automatic numerical procedure has been developed involving the use of commercial and in-house codes making it possible to use different grids for FEM hydrodynamic and elastic analyses, with higher computational efficiency with respect to uniform grids. The final pressure distribution and bearing deformed geometry are then obtained by an iterative calculation. Some results are reported for different bearing configurations.

A coupled BEM‐flexibility force method for bending analysis of internally supported plates

AbstractThis paper presents a new method for the analysis of plates in bending with internal supports. The proposed method can be regarded as an extension of the well‐known force method (the flexibility matrix method) in the matrix analysis of structures. The solution is performed through two phases: the released plate phase, in which the plate is released from all internal supports and solved using the Boundary Element Method (BEM). The effect of internal supports is considered in the second phase, where a series of unit virtual loads is placed instead of the unknown redundant reactions at internal supports. The flexibility matrix is formed and compatibility of deformations at the locations of internal supports is satisfied. Hence, the corresponding system of equations is solved for the unknown redundant forces at internal supports. The final solution of the problem consists of the summation of two phases: the released plate phase and the cases of virtual unit loads phase. An efficient solution algorithm is developed to solve both phases simultaneously. The main advantages of the present formulation are: (1) the present formulation increases the versatility of the BEM as it allows the re‐usability of standard BEM codes for solution of plates in bending to be used in solving problems having internal supports, with even no modifications; and (2) the two solution phases are completely uncoupled; therefore it is easy to trace behaviour of the plate due to failure of one or more of the internal supports without re‐analysis.Several numerical examples are analysed. The results are compared to those of analytical and finite element models to demonstrate the accuracy and the validity of the present formulation. The present formulation is used also to study the differences between the finite element and boundary element modelling for building slabs. Copyright © 2002 John Wiley & Sons, Ltd.

The scaled flexibility matrix method for the efficient solution of boundary value problems in 2D and 3D layered elastic media

We present a method that extends the flexibility matrix method for multilayer elasticity problems to include problems with very thin layers. This method is particularly important for solving problems in which one or a number of very thin layers are juxtaposed with very thick layers. The standard flexibility matrix method suffers from round-off errors and poor scaling of the flexibility equations which occur when one of the layers in the multilayered medium becomes much smaller than the others. The method proposed in this paper makes use of power series expansions of the various components of the flexibility matrix in order to arrive at a system of equations that is appropriately scaled. The effectiveness of the scaled flexibility matrix method is demonstrated on a number of test problems.

Estimation of Overall Settlement of Piled Rafts

Estimation of overall (or average) settlement and differential settlement are the two most critical issues in the design of moderate to large sized raft and piled foundations. Recent studies have shown that a central pile group can largely eliminate differential settlements, but it is still necessary to estimate the average settlement of the combined pile group and raft foundation. This paper presents a simple method of estimating the overall stiffness of piled rafts in a non-homogeneous soil with finite depth, based on the method of Clancy and Randolph (1996). Although the method enables the estimation of the piled raft’s stiffness even when piles are installed beneath the full raft area, particular emphasis is placed on the situation where the pile support is limited to the central region of the raft. The method combines the so called ‘equivalent pier method’ (Poulos and Davis, 1980) and the ‘flexibility matrix method’ (Randolph, 1983). In the presented method, a piled raft is replaced by a ‘capped pier’. Firstly, a simple method of estimating the overall stiffness of stubby piers in a homogeneous or a non-homogeneous soil is presented. The estimated stiffnesses are compared with those calculated by a finite difference approach (FLAC, Itasca Corporation, Version 3.22). The comparison shows that the simple method gives approximate overall stiffness of piers with a wide range of slenderness ratios (length/radius) and pile-soil stiffness ratios. Secondly, the applicability of the equivalent pier method to pile group analysis is examined for homogeneous soil conditions. The calculated stiffnesses are compared with those obtained by FLAC, and the so called ‘hybrid’ method (HyPR: Clancy, 1993) where the existence of individual piles is considered. The applicability of the ‘capped pier’ method to piled raft analysis is then examined. The results show the presented simple method gives satisfactorily accurate stiffness of pile groups and piled rafts without any long computations. Finally, a short example of the application is presented for a centrifuge model piled raft. It was found that the presented method gave close overall stiffness to those obtained by observations, and by alternative rigorous calculations.

A study concerning elastic analysis of disk centrifugal separators

The paper presents a simple method of computerised analysis of empty centrifugal disks separators; the method of linear-elastic analysis is a practical development of the classical flexibility matrix method. In the first step, the separator's bowl is divided in structural elements; each of the structural elements are at first considered separately and then the global generalised forces and bending moments are obtained from the displacement compatibility conditions at each element junction. The idea is exemplified with experimental works and theoretical diagrams concerning total stresses for an empty bowl of disks centrifugal separator. The advantages of the proposed method lies in: (1) low computational cost; (2) reasonable accuracy of the results; (3) expeditive evaluation of the critical junction; and (4) suitable method for preliminary design of such structure.

Influence of microstructure on impact properties of iron-molybdenum alloy steels

The three methods for determining the shell thickness of steel cylindrical liquid storage tanks designed in conformance with API Standard 650, Welded Tanks for Oil Storage (API 650) are: (1) one-foot method (1FM), (2) variable-design-point method (VDM) and (3) linear analysis. We compared the shell designs based on these three methods for different tank properties: diameter, height and allowable stress. For linear analysis, we developed a stiffness–flexibility matrix method based on thin shell theory that gives the theoretical displacements and stresses at each shell course without any approximation or simplification. Results show that shell designs using VDM may produce overstressed shell courses for some of the large steel liquid storage tanks when VDM is permissible to use. Linear analysis would give more accurate shell designs for those cases.