This work presents an original and very interesting approach to a calculus problem involving beams with intermediate supports through the transfer-matrix method, a very easy method to program to quickly obtain good results. To exemplify the applicability of this approach in dentistry, the calculus of a dental bridge on three poles is explored. Dental restorations are very important for improving a person’s general state of health as a result of improving mastication and esthetic appearance. The approach used in this study consists of presenting a theoretical study about an indeterminate beam with an intermediate support and then particularizing it for application in a dental restoration case, with a dental bridge on three poles and two missing teeth between the three poles. The bridge is assimilated to a simple static indeterminate beam. This paper is unique in that it involves the application of the transfer-matrix method for a case study in dental restoration. The assimilation of a dental bridge with a statically undetermined beam, resting on the extremities and on an intermediate support, is an original approach. The results obtained in the presented case study were validated by comparison with those obtained through the classical calculation of the Resistance of Materials, with Clapeyron’s equation of three moments. Due to the ease and elegance of solving various problems with the TMM, this approach will continue to be relevant to other original case studies with different modeling requirements, and these applications will be presented in future research.

The Transfer-Matrix Method (TMM) is an original and relatively simple mathematical approach for the calculus of thin-walled cylindrical tubes presented in this work. Calculation with TMM is much less used than calculation with the Finite Elements Method (FEM), even though it is much easier to apply in different fields. That is why it was considered imperative to present this original study. The calculus is based on Dirac’s and Heaviside’s functions and operators and on matrix calculation. The state vectors, the transfer-matrix, and the vector corresponding to the external efforts were defined, which were then used in the calculations. A matrix relation can be written, which gives the state vector of the last section depending on the state vector of the first section, a relation in which the conditions of the two end supports can be set. As an application, a heat exchanger was studied, with a large cylinder subjected to a uniformly distributed internal load, and from the inner cylinder bundle, a cylinder subjected to both uniform internal and external loads was considered. For the second cylinder, two possibilities of action for the external forces were considered, a successive action and a simultaneous action, achieving the same results in both situations. The TMM is intended to be used for iterative calculus in optimization problems where rapid successive results are required. In the future, we want to expand this method to other applications, and we want to develop related programs. This is an original theoretical study and is a complement to the research in the field on thin-walled cylinder tubes and their applications in heat exchangers.

This work presents a new, original and very interesting approach to a calculus pro-blem of beams with intermediate supports by Transfer-Matrix Method, a very easy method to program to quickly obtain good results. To exemplify the applicability of this approach is given in dentistry, the calculus of a dental bridge on three poles. The dental restorations are very important for improving the general state of health as a result of improving mastication and aesthetic appearance. The approach of this study consists of to present a theoretical study about an indeterminate beam with intermediate support and after, to particularize for an application of a dental restoration case, with a dental bridge on three poles and with two missing teeth between the three poles, assimilated to a simple static indeterminate beam. The results obtained in the presented case study were validated with those obtained by the classical calculation of the Resistance of Materials, with the equation of the three moments of Clapeyron. Due to the ease and elegance of solving various problems with TMM, their approach will continue with other new and original case studies with different modeling and requests, and they will be presented in future research.

This article presents the Transfer Matrix Method as a mathematical approach for the calculus of different structures that can be discretized into elements using an iterative calculus for future applications in the vehicle industry. Plate calculus is important in construction, medicine, orthodontics, and many other fields. This work is original due to the mathematical apparatus used in the calculus of long rectangular plates embedded in both long borders and required by a uniformly distributed force on a line parallel to the long borders. The plate is discretized along its length in unitary beams, which have the width of the rectangular plate. The unitary beam can also be discretized into parts. As applications, the long rectangular plates embedded on the two long borders and charged with a vertical uniform load that acts on a line parallel to the long borders are studied. A state vector is associated with each side. For each of the four cases studied, a matrix relationship was written for each side, based on a transfer matrix, the state vector corresponding to the origin side, and the vector due to the action of external forces acting on the considered side. After, it is possible to calculate all the state vectors for all sides of the unity beam. Now, the efforts, deformations, and stress can be calculated in any section of the beam, respectively, for the long rectangular plate. This calculus will serve as a calculus of resistance for different pieces of the components of vehicles.

This work presents the Transfer-Matrix Method as a mathematical approach for the calculus of different structures that can be discretized into elements, using an iterative calculation for future applications in vehicle industry. Plate calculus is important in construction, in medicine, in orthodontics and in many other fields too. This work is original and new. The plate is discretized along its length in unitary beams, which have the width of the rectangular plate. The unitary beam can also be discretized into parts. As applications, they are studied long rectangular plates, embedded on the two long borders and charged with a vertical uniform load that acts on a line parallel to the long borders. It is associated with each side a state vector. For each of the four cases studied, a matrix relationship was written for some side, based on a transfer matrix, the state vector corresponding to origin side and the vector due to the action of external forces acting to the considered side. After, it is possible to calculate all the state vectors for all sides of the unity beam. Now, the efforts, deformations and stress can be calculated in any section of the beam, respectively for the long rectangular plate. This calculus will serve as calculus of resistance for different pieces of components of vehicles.

The aim of this work is to present an original, relatively simple, and elegant approach to the analysis of long rectangular plates subjected to uniformly distributed vertical loads acting on various surfaces. Plate analysis is important in many fields, especially where components are either rectangular plates or can be approximated as such. The Transfer Matrix Method is increasingly used in research, as evidenced by the references cited. The advantages of this method lie in the simplicity of its algorithm, the ease of implementation in programming, and its straightforward integration into optimization software. The approach consists of discretizing the rectangular plate by sectioning it with planes parallel to the short sides—i.e., perpendicular to the two long edges. This results in a set of beams, each with a length equal to the width of the plate, a height equal to the plate’s thickness, and a unit width. Each unit beam has support at its ends that replicate the edge conditions of the plate along its long sides. In the study presented, the rectangular plate is embedded along its two long edges, meaning the unit beam is considered embedded at both ends. The Transfer Matrix Method is particularly valuable because it lends itself well to iterative calculations, making it easy to develop software capable of analyzing long rectangular plates quickly. This makes it especially useful for shape optimization applications, which we intend and hope to pursue in future studies.