Symmetries of the free Schrödinger equation in the non-commutative plane

We study all the symmetries of the free Schrödinger equation in the non-commu- tative plane. These symmetry transformations form an infinite-dimensional Weyl algebra that appears naturally from a two-dimensional Heisenberg algebra generated by Galilean boosts and momenta. These infinite high symmetries could be useful for constructing non-relativistic interacting higher spin theories. A finite-dimensional subalgebra is given by the Schröodinger algebra which, besides the Galilei generators, contains also the dilatation and the expansion. We consider the quantization of the symmetry generators in both the reduced and extended phase spaces, and discuss the relation between both approaches.

Fatigue Analysis of Welded Cruciform Joints with Cold Laps

Työssä on tutkittu vetojännityskuormituksen alaisena olevien hitsattujen kuormaa kantamattomien X-liitosten hitsin paikallisen geometrian variaation vaikutusta väsymislujuuteen. Muuttujina olivat reunan pyöristyssäde, kylmäjuoksun suuruus ja kylkikulma. Geometristen muuttujien parametrinen riippuvuussuhde on analysoitu usealla elementtimallilla. Väsymistarkastelu on suoritettu käyttämällä lineaaris-elastista murtumismekaniikkaa (LEFM) tasovenymätilassa ja materiaalina terästä. Särönkasvun suunnan ennustamisessaon käytetty maksimipääjännityskriteeriä sekä jännitysintensiteettikertoimet on määritetty J-integraalilla. Särön ydintymisvaihetta ei ole otettu huomioon. Rakenteen on oletettu olevan hitsatussa tilassa ja jännitysheilahdus on kokonaan tehollinen. Särön kasvunopeuden ennustamiseen on käytetty Paris'n lakia. Väsymislujuustulokset on esitetty karakteristisina väsymisluokkina (FAT) ja sovitettu parametriseksi yhtälöksi. Lopuksi väsymisanalyysin ennustamia tuloksia on verrattu saatavilla oleviin väsytystestituloksiin.

The Mont Collon mafic complex (Austroalpine Dent Blanche nappe) : permian evolution of the Western European mantle

Résumé: Le complexe du Mont Collon (nappe de la Dent Blanche, Austroalpin) est l'un des exemples les mieux préservés du magmatisme mafique permien des Alpes occidentales. Il est composé d'affleurements discontinus et d'une stratification magmatique en son centre (Dents de Bertol) et est composé à 95% de roches mafiques cumulatives (gabbros à olivine et/ou cpx, anorthositiques, troctolites, wehrlites et wehrlites à plagioclase) et localement de quelques gabbros pegmatitiques. Ces faciès sont recoupés par de nombreux filons acides (aphtes, pegmatites quartziques, microgranodiorites et filons anorthositiques) et mafiques tardifs (dikes mélanocrates riches en Fe et Ti). Les calculs thermométriques (équilibre olivine-augite) montrent des températures de 1070-1120 ± 6°C, tandis que le thermomètre amphibole-plagioclase indique une température de 740 ± 40°C à 0.5 GPa pour les amphiboles magmatiques tardives. La geobarométrie sur pyroxène donne des pressions moyennes de 0.3-0.6 GPa, indiquant un emplacement dans la croûte moyenne. De plus, les températures obtenues sur des amphiboles coronitiques indiquent des températures de l'ordre de 700 ± 40°C confirmant que les réactions coronitiques apparaissent dans des conditions subsolidus. Les âges concordants U/Pb sur zircons de 284.2 ± 0.6 et 282.9 ± 0.6 Ma obtenus sur un gabbro pegmatitique et une pegmatitique quartzique, sont interprétés comme des âges de cristallisation. Les datations 40Ar/39Ar sur amphiboles des filons mélanocrates donnent un âge plateau de 260.2 ± 0.7 Ma, qui est probablement très proche de l'âge de cristallisation. Ainsi, cet age 40Ar/39Ar indique un second évènement magmatique au sein du complexe. Les compositions des roches totales en éléments majeurs et traces montrent peu de variations, ainsi que le Mg# (75-80). Les éléments traces enregistrent le caractère cumulatif des roches (anomalie positive en Eu) et révèlent des anomalies négatives systématiques en Nb, Ta, Zr, Hf et Ti dans les faciès basiques. Le manque de corrélation entre éléments majeurs et traces est caractéristique d'un processus de cristallisation in situ impliquant une quantité variable de liquide interstitiel (L) entre les phases cumulus. Les distributions des éléments traces dans les minéraux sont homogènes, indiquant une rééquilibration .subsolidus entre cristaux et liquide interstitiel. Un modèle quantitatif basé sur les équations de cristallisation in situ de Langmuir reproduisent correctement les concentrations en terres rares légères des minéraux cumulatifs montrant la présence de 0 à 35% de liquide interstitiel L pour des degrés de différenciation F de 0 à 45%, par rapport au faciès les moins évolués du complexe. En outre, les valeurs de L sont bien corrélées avec les proportions modales d'amphibole interstitielle et les concentrations en éléments incompatibles des roches (Zr, Nb). Le liquide parental calculé des cumulats du Mont Collon est caractérisé par un enrichissement relatif en terres rares légères et Th, un appauvrissement en terres rares lourdes typique d'une affinité transitionnelle (T-MORB) et une forte anomalie négative en Nb-Ta. Les roches cumulatives montrent des compositions isotopiques en Nd-Sr proches de la terre globale silicatée (BSE), soit 0.6<εNdi<+3.2, 0.7045<87Sr/86Sri<0.7056. Les rapports initiaux en Pb indiquent une source dans le manteau enrichi subcontinental lithosphérique, préalablement contaminé par des sédiments océaniques. Les dikes mélanocrates Fe-Ti sont représentatifs de liquides et ont des spectres de terres rares enrichis, une anomalie positive en Nb-Ta et des εNdi de +7, des 87Sr/86Sri de 0.703 et des rapports initiaux en Pb, similaires à ceux des basaltes d'île océanique, indiquant une source asthénosphérique modérément appauvrie. Ainsi, la fusion partielle du manteau lithosphérique subcontinental est induite par l'amincissement post-orogénique et la remontée de l'asthénosphère. Les filons mélanocrates proviennent, après délamination du manteau lithosphérique, de la fusion de l'asthénosphère. Abstract The early Permian Mont Collon mafic complex (Dent Blanche nappe, Austroalpine nappe system) is one of the best preserved examples of the Permian mafic magmatism in the Western Alps. It is composed of discontinuous exposures and a well-preserved magmatic layering (the Dents de Bertol cliff) crops out in the center part of the complex. It mainly consists of cumulative mafic rocks, which represent 95 vol-% of the mafic complex (ol- and cpx-bearing gabbros and rare anorthositic layers, troctolites, wehrlites and plagioclase-wehrlites) and locally pegmatitic gabbros. All these facies are crosscut by widespread acidic (aplites, quartz-rich pegmatites, microgranodiorites) and late mafic Fe-Ti melanocratic dikes. Olivine-augite thermometric calculations yield a range of 1070-1120 ± 6°C, while amphibole-plagioclase thermometer yields a temperature of 740 ± 40°C at 0.5 GPa. Pyroxene geobarometry points to a pressure of 0.3-0.6 GPa, indicating a middle crustal level of emplacement. Moreover, temperature calculations on the Mont Conon coronitic amphiboles indicate temperatures of 700 ± 40°C, close to those calculated for magmatic amphiboles. These temperatures confirm that coronitic reactions occurred at subsolidus conditions. ID-TIMS U/Pb zircon ages of 284.2 ± 0.6 and 282.9 ± 0.6 Ma obtained on a pegmatitic gabbro and a quartz-pegmatitic dike, respectively, were interpreted as the crystallization ages of these rocks. 40Ar/39Ar dating on amphiboles from Fe-Ti melanocratic dikes yields a plateau age of 260.2 ± 0.7 Ma, which is probably very close to the crystallization age. Consequently, this 40Ar/P39Ar age indicates a second magmatic event. Whole-rock major- and trace-element compositions show little variation across the whole intrusion and Mg-number stays within a narrow range (75-80). Trace-element concentrations record the cumulative nature of the rocks (e.g. positive Eu anomaly) and reveal systematic Nb, Ta, Zr, Hf and Ti negative anomalies for all basic facies. The lack of correlation between major and trace elements is characteristic of an in situ crystallization process involving variable amounts of interstitial liquid (L) trapped between the cumulus mineral phases. LA-ICPMS measurements show that trace-element distributions in minerals are homogeneous, pointing to subsolidus re-equilibration between crystals and interstitial melts. A quantitative modeling based on Langmuir's in situ crystallization equation successfully reproduced the Rare Earth Element (REE) concentrations in cumulitic minerals. The calculated amounts of interstitial liquid L vary between 0 and 35% for degrees of differentiation F of 0 to 45%, relative to the least evolved facies of the intrusion. Furthermore, L values are well correlated with the modal proportions of interstitial amphibole and whole-rock incompatible trace-element concentrations (e.g. Zr, Nb) of the tested samples. The calculated parental melt of the Mont Collon cumulates is characterized by a relative enrichment in Light REE and Th, a depletion in Heavy REE, typical of a transitional affinity (T-MORB), and strong negative Nb-Ta anomaly. Cumulative rocks display Nd-Sr isotopic compositions close to the BSE (-0.6 < εNdi < +3.2, 0.7045 < 87Sr/86Sri < 0.7056). Initial Pb ratios point to an origin from the melting of an enriched subcontinental lithospheric mantle source, previously contaminated at the source by oceanic sediments. The contrasted alkaline Fe-Ti melanocratic dikes are representative of liquids. They display enriched fractionated REE patterns, a positive Nb-Ta anomaly and εNdi of +7, 87Sr/86Sri of 0.703 and initial Pb ratios, all reminiscent of Ocean Island Basalt-type rocks, pointing to a moderately

Increased risk of cardiovascular disease (CVD) with age in HIV-positive men : a comparison of the D:A:D CVD risk equation and general population CVD risk equations

OBJECTIVES: The aim of the study was to statistically model the relative increased risk of cardiovascular disease (CVD) per year older in Data collection on Adverse events of anti-HIV Drugs (D:A:D) and to compare this with the relative increased risk of CVD per year older in general population risk equations. METHODS: We analysed three endpoints: myocardial infarction (MI), coronary heart disease (CHD: MI or invasive coronary procedure) and CVD (CHD or stroke). We fitted a number of parametric age effects, adjusting for known risk factors and antiretroviral therapy (ART) use. The best-fitting age effect was determined using the Akaike information criterion. We compared the ageing effect from D:A:D with that from the general population risk equations: the Framingham Heart Study, CUORE and ASSIGN risk scores. RESULTS: A total of 24 323 men were included in analyses. Crude MI, CHD and CVD event rates per 1000 person-years increased from 2.29, 3.11 and 3.65 in those aged 40-45 years to 6.53, 11.91 and 15.89 in those aged 60-65 years, respectively. The best-fitting models included inverse age for MI and age + age(2) for CHD and CVD. In D:A:D there was a slowly accelerating increased risk of CHD and CVD per year older, which appeared to be only modest yet was consistently raised compared with the risk in the general population. The relative risk of MI with age was not different between D:A:D and the general population. CONCLUSIONS: We found only limited evidence of accelerating increased risk of CVD with age in D:A:D compared with the general population. The absolute risk of CVD associated with HIV infection remains uncertain.

A simple model to describe the thixotropic behavior of paints.

We propose a simple rheological model to describe the thixotropic behavior of paints, since the classical hysteresis area, which is usually used, is not enough to evaluate thixotropy. The model is based on the assumption that viscosity is a direct measure of the structural level of the paint. The model depends on two equations: the Cross-Carreau equation to describe the equilibrium viscosity and a second order kinetic equation to express the time dependence of viscosity. Two characteristic thixotropic times are differentiated: one for the net structure breakdown, which is defined as a power law function of shear rate, and an other for the net structure buildup, which is not dependent on the shear rate. The knowledge of both kinetic processes can be used to improve the quality and applicability of paints. Five representative commercial protective marine paints are tested. They are based on chlorinated rubber, acrylic, alkyd, vinyl, and epoxy resins. The temperature dependence of the rheological behavior is also studied with the temperature ranging from 5 ºC to 35 ºC. It is found that the paints exhibit both shear thinning and thixotropic behavior. The model fits satisfactorily the thixotropy of the studied paints. It is also able to predict the thixotropy dependence on temperature. Both viscosity and the degree of thixotropy increase as the temperature decreases.

Local thermodynamics and the generalized Gibbs-Duhem equation in systems with long-range interactions

The local thermodynamics of a system with long-range interactions in d dimensions is studied using the mean-field approximation. Long-range interactions are introduced through pair interaction potentials that decay as a power law in the interparticle distance. We compute the local entropy, Helmholtz free energy, and grand potential per particle in the microcanonical, canonical, and grand canonical ensembles, respectively. From the local entropy per particle we obtain the local equation of state of the system by using the condition of local thermodynamic equilibrium. This local equation of state has the form of the ideal gas equation of state, but with the density depending on the potential characterizing long-range interactions. By volume integration of the relation between the different thermodynamic potentials at the local level, we find the corresponding equation satisfied by the potentials at the global level. It is shown that the potential energy enters as a thermodynamic variable that modifies the global thermodynamic potentials. As a result, we find a generalized Gibbs-Duhem equation that relates the potential energy to the temperature, pressure, and chemical potential. For the marginal case where the power of the decaying interaction potential is equal to the dimension of the space, the usual Gibbs-Duhem equation is recovered. As examples of the application of this equation, we consider spatially uniform interaction potentials and the self-gravitating gas. We also point out a close relationship with the thermodynamics of small systems.

Effective Electrodiffusion equation for non uniform nanochannels

We derive a one dimensional formulation of the Planck-Nernst-Poisson equation to describe the dynamics of of a symmetric binary electrolyte in channels whose section is of nanometric section and varies along the axial direction. The approach is in the spirit of the Fick-Jacobs di fusion equation and leads to a system of coupled equations for the partial densities which depends on the charge sitting at the walls in a non trivial fashion. We consider two kinds of non uniformities, those due to the spatial variation of charge distribution and those due to the shape variation of the pore and report one and three-dimensional solutions of the electrokinetic equations.

Vuokratyöntekijän työsuhdeturva: Sääntelyn haasteita ja tulkintoja

Tutkimus pyrkii selvittämään, vastaako vuokratyöntekijän työsuhdeturva työlainsäädännön tavoitteita ja niiden taustalla vaikuttavaa työntekijän suojelun periaatetta. Lainsäädännössä vuokratyöntekijän työsuhdeturvan kohdalla olevat keskeisimmät aukkokohdat ja tulkinnat esitetään juridisen argumentaation avulla. Tilanteen tekee ongelmalliseksi erityisesti se, että nykylainsäädäntö ei varsinaisesti estä käyttämästä vuokratyössä toistuvia, kunkin toimeksiannon mittaisia määräaikaisuuksia. Tulokset osoittavat, että sekä kollektiivinen että individuaaliperusteinen irtisanomissuoja on käytännössä heikko. Lisäksi työsuhdeturvan ja joustavuuden tasapaino työmarkkinoilla näyttää olevan melko vaikea yhtälö ratkaistavaksi. Eräänä mahdollisena ratkaisuna tutkimuksen lopussa esitetään Ruotsissa käytössä oleva malli, jossa työntekijät ovat toistaiseksi voimassa olevissa työsuhteissa, ja saavat palkkaa myös toimeksiantojen väliseltä ajalta. Järjestelmään liittyy vahvasti myös alan järjestäytyminen ja eettisten pelisääntöjen määritteleminen liittojen välisin sopimuksin. Tällä tavalla vuokratyöntekijän työsuhdeturvaan liittyvät ongelmakohdat ratkaistaan, mutta toisaalta vuokratyön työllisyyttä ja kilpailukykyä tukeva vaikutus menetetään. Vaihtoehtoisina keinoina on esitetty lukuisia kevyempiä keinoja yksittäisten ongelmakohtien ratkaisuksi.

Receiver operating characteristic (ROC) curve for classification of18F-NaF uptake on PET/CT

Abstract Objective: To assess the cutoff values established by ROC curves to classify18F-NaF uptake as normal or malignant. Materials and Methods: PET/CT images were acquired 1 hour after administration of 185 MBq of18F-NaF. Volumes of interest (VOIs) were drawn on three regions of the skeleton as follows: proximal right humerus diaphysis (HD), proximal right femoral diaphysis (FD) and first vertebral body (VB1), in a total of 254 patients, totalling 762 VOIs. The uptake in the VOIs was classified as normal or malignant on the basis of the radiopharmaceutical distribution pattern and of the CT images. A total of 675 volumes were classified as normal and 52 were classified as malignant. Thirty-five VOIs classified as indeterminate or nonmalignant lesions were excluded from analysis. The standardized uptake value (SUV) measured on the VOIs were plotted on an ROC curve for each one of the three regions. The area under the ROC (AUC) as well as the best cutoff SUVs to classify the VOIs were calculated. The best cutoff values were established as the ones with higher result of the sum of sensitivity and specificity. Results: The AUCs were 0.933, 0.889 and 0.975 for UD, FD and VB1, respectively. The best SUV cutoffs were 9.0 (sensitivity: 73%; specificity: 99%), 8.4 (sensitivity: 79%; specificity: 94%) and 21.0 (sensitivity: 93%; specificity: 95%) for UD, FD and VB1, respectively. Conclusion: The best cutoff value varies according to bone region of analysis and it is not possible to establish one value for the whole body.

Older consumers adopting information and communication technology: Evaluating opportunities for health care applications

The purpose of this dissertation is to analyse older consumers' adoption of information and communication technology innovations, assess the effect of aging related characteristic, and evaluate older consumers' willingness to apply these technologies in health care services. This topic is considered important, because the population in Finland (as in other welfare states) is aging and thus offers a possibility for marketers, but on the other hand threatens society with increasing costs for healthcare. Innovation adoption has been under research from several aspects in both organizational and consumer research. In the consumer behaviour, several theories have been developed to predict consumer responses to innovation. The present dissertation carefully reviews previous research and takes a closer look at the theory of planned behaviour, technology acceptance model and diffusion of innovations perspective. It is here suggested that there is a possibility that these theories can be combined and complemented to predict the adoption of ICT innovations among aging consumers, taking the aging related personal characteristics into account. In fact, there are very few studies that have concentrated on aging consumers in the innovation research, and thus there was a clear indent for the present research. ICT in the health care context has been studied mainly from the organizational point of view. If the technology is thus applied for the communication between the individual end-user and service provider, the end-user cannot be shrugged off. The present dissertation uses empirical evidence from a survey targeted to 55-79 year old people from one city in Southern-Carelia. The empirical analysis of the research model was mainly based on structural equation modelling that has been found very useful on estimating causal relationships. The tested models were targeted to predict the adoption stage of personal computers and mobile phones, and the adoption intention of future health services that apply these devices for communication. The present dissertation succeeded in modelling the adoption behaviour of mobile phones and PCs as well as adoption intentions of future services. Perceived health status and three components behind it (depression, functional ability, and cognitive ability) were found to influence perception of technology anxiety. Better health leads to less anxiety. The effect of age was assessed as a control variable, in order to evaluate its effect compared to health characteristics. Age influenced technology perceptions, but to lesser extent compared to health. The analyses suggest that the major determinant for current technology adoption is perceived behavioural control, and additionally technology anxiety that indirectly inhibit adoption through perceived control. When focusing on future service intentions, the key issue is perceived usefulness that needs to be highlighted when new services are launched. Besides usefulness, the perception of online service reliability is important and affects the intentions indirectly. To conclude older consumers' adoption behaviour is influenced by health status and age, but also by the perceptions of anxiety and behavioural control. On the other hand, launching new types of health services for aging consumers is possible after the service is perceived reliable and useful.

The performance and the characteristic field of a centrifugal compressor

Tässä työssä tarkastellaan radiaalikompressorin puristusprosessiin liittyvä peruslaskenta. Ilmankosteuden vaikutukset laskentaan huomioidaan ja kompressorin tunnuskenttä esitellään tässä työssä. Esimerkkinä lasketaan pisteitä kompressorin tunnuskentästä.

Predicting overall service quality: a structural equation modelling approach

In this article, the results of a modified SERVQUAL questionnaire (Parasuraman et al., 1991) are reported. The modifications consisted in substituting questionnaire items particularly suited to a specific service (banking) and context (county of Girona, Spain) for the original rather general and abstract items. These modifications led to more interpretable factors which accounted for a higher percentage of item variance. The data were submitted to various structural equation models which made it possible to conclude that the questionnaire contains items with a high measurement quality with respect to five identified dimensions of service quality which differ from those specified by Parasuraman et al. And are specific to the banking service. The two dimensions relating to the behaviour of employees have the greatest predictive power on overall quality and satisfaction ratings, which enables managers to use a low-cost reduced version of the questionnaire to monitor quality on a regular basis. It was also found that satisfaction and overall quality were perfectly correlated thus showing that customers do not perceive these concepts as being distinct

Minimization of the experimental workload for the prediction of pollutants propagation in rivers. Mathematical modelling and knowledge re-use

The present thesis in focused on the minimization of experimental efforts for the prediction of pollutant propagation in rivers by mathematical modelling and knowledge re-use. Mathematical modelling is based on the well known advection-dispersion equation, while the knowledge re-use approach employs the methods of case based reasoning, graphical analysis and text mining. The thesis contribution to the pollutant transport research field consists of: (1) analytical and numerical models for pollutant transport prediction; (2) two novel techniques which enable the use of variable parameters along rivers in analytical models; (3) models for the estimation of pollutant transport characteristic parameters (velocity, dispersion coefficient and nutrient transformation rates) as functions of water flow, channel characteristics and/or seasonality; (4) the graphical analysis method to be used for the identification of pollution sources along rivers; (5) a case based reasoning tool for the identification of crucial information related to the pollutant transport modelling; (6) and the application of a software tool for the reuse of information during pollutants transport modelling research. These support tools are applicable in the water quality research field and in practice as well, as they can be involved in multiple activities. The models are capable of predicting pollutant propagation along rivers in case of both ordinary pollution and accidents. They can also be applied for other similar rivers in modelling of pollutant transport in rivers with low availability of experimental data concerning concentration. This is because models for parameter estimation developed in the present thesis enable the calculation of transport characteristic parameters as functions of river hydraulic parameters and/or seasonality. The similarity between rivers is assessed using case based reasoning tools, and additional necessary information can be identified by using the software for the information reuse. Such systems represent support for users and open up possibilities for new modelling methods, monitoring facilities and for better river water quality management tools. They are useful also for the estimation of environmental impact of possible technological changes and can be applied in the pre-design stage or/and in the practical use of processes as well.

On the Application of Beam on Elastic Foundation Theory to the Analysis of Stiffened Plate Strips

The main objective of this thesis is to show that plate strips subjected to transverse line loads can be analysed by using the beam on elastic foundation (BEF) approach. It is shown that the elastic behaviour of both the centre line section of a semi infinite plate supported along two edges, and the free edge of a cantilever plate strip can be accurately predicted by calculations based on the two parameter BEF theory. The transverse bending stiffness of the plate strip forms the foundation. The foundation modulus is shown, mathematically and physically, to be the zero order term of the fourth order differential equation governing the behaviour of BEF, whereas the torsion rigidity of the plate acts like pre tension in the second order term. Direct equivalence is obtained for harmonic line loading by comparing the differential equations of Levy's method (a simply supported plate) with the BEF method. By equating the second and zero order terms of the semi infinite BEF model for each harmonic component, two parameters are obtained for a simply supported plate of width B: the characteristic length, 1/ λ, and the normalized sum, n, being the effect of axial loading and stiffening resulting from the torsion stiffness, nlin. This procedure gives the following result for the first mode when a uniaxial stress field was assumed (ν = 0): 1/λ = √2B/π and nlin = 1. For constant line loading, which is the superimposition of harmonic components, slightly differing foundation parameters are obtained when the maximum deflection and bending moment values of the theoretical plate, with v = 0, and BEF analysis solutions are equated: 1 /λ= 1.47B/π and nlin. = 0.59 for a simply supported plate; and 1/λ = 0.99B/π and nlin = 0.25 for a fixed plate. The BEF parameters of the plate strip with a free edge are determined based solely on finite element analysis (FEA) results: 1/λ = 1.29B/π and nlin. = 0.65, where B is the double width of the cantilever plate strip. The stress biaxial, v > 0, is shown not to affect the values of the BEF parameters significantly the result of the geometric nonlinearity caused by in plane, axial and biaxial loading is studied theoretically by comparing the differential equations of Levy's method with the BEF approach. The BEF model is generalised to take into account the elastic rotation stiffness of the longitudinal edges. Finally, formulae are presented that take into account the effect of Poisson's ratio, and geometric non linearity, on bending behaviour resulting from axial and transverse inplane loading. It is also shown that the BEF parameters of the semi infinite model are valid for linear elastic analysis of a plate strip of finite length. The BEF model was verified by applying it to the analysis of bending stresses caused by misalignments in a laboratory test panel. In summary, it can be concluded that the advantages of the BEF theory are that it is a simple tool, and that it is accurate enough for specific stress analysis of semi infinite and finite plate bending problems.