About Calculus Through the Transfer Matrix Method of a Beam with Intermediate Support with Applications in Dental Restorations

Abstract

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.

1. Introduction

Interdisciplinary studies and research in many fields are very topical. This work presents an original and very interesting approach to a calculus problem involving beams with intermediate supports through the transfer matrix method (TMM). This is 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 consists of presenting a theoretical study about an indeterminate beam with an intermediate support. After this, it is particularized for application in a dental restoration case, with a dental bridge on three poles and with two missing teeth between the three poles. The dental bridge is assimilated to a simple static indeterminate beam resting on three supports—the three poles—with two openings, corresponding to the two missing teeth. 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.

The TMM is a method used in many fields, as will be presented below. Structural calculus with the TMM is presented in [1], and the classical analytical calculus for beams with different applications is given in [2,3]. Using the TMM, we can calculate different structural elements as cylinders, as presented in [4]. The TMM is used in orthodontics for mandible body bone calculus, and this is presented in [5]. Ref. [6] provides some practical notions for dental preparations for fixed single-dental prostheses. Ref. [7] shows figures of graphs partitioned by counting, sequence, and layer matrices. Ref. [8] provides a study on the ten-year survival of bridges placed in General Dental Services in England and Wales. Refs. [9,10] present studies about the bending fracture of Co-Cr dental bridges. Ref. [11] shows the integrated construction and simulation of tool paths for milling dental crowns and bridges. Refs. [12,13] provide studies about the influence of the fatigue of zirconia for dental bridge design. Research on the surface of dental alloys with a chrome cobalt base is presented in [14]. The effects of small grit grinding and glazing on the mechanical behavior and aging resistance of super-translucent dental zirconia are shown in [15]. Ref. [16] presents a study about the dimensional accuracy and surface roughness of polymeric dental bridges produced through different 3D printing processes. An unusual presentation of dental calculus is given in [17]. Ref. [18] presents zirconia as a dental biomaterial, and [19] provides studies about current and emerging applications of the 3D printing of dental prostheses. A comprehensive review of dental calculus is presented in [20,21], which assess the reliability and accuracy of the manual and electronic detection of subgingival calculus. An evaluation of a decision support system developed with a deep learning approach for detecting dental caries with cone-beam computed tomography imaging is presented in [22]. A comparison of the tensile bond strength of fixed–fixed versus cantilever single- and double-abutted resin-bonded bridges for dental prostheses is given in [23]. Ref. [24] presents a literature review about the detection, removal, and prevention of calculus. Ref. [25] provides studies reporting on periodontal regeneration versus extraction and dental implants or the prosthetic replacement of teeth being severely compromised by loss of attachment to the apex over 10 years. A practical guide for bonded orthodontic retention is given in [26]. Ref. [27] shows the current status of calculus detection technologies. A potential application of materials based on a polymer and CAD/CAM composite resins in prosthetic dentistry is presented in [28]. A review about a dental bridge procedure carried out to straighten loose teeth is given in [29]. Ref. [30] presents recent advances in the pathogenesis of and prevention strategies for dental calculus, and [31] describes the oral screening of dental calculus, gingivitis, and dental caries through segmentation on intraoral photographic images using deep learning. An in vitro study about the fracture resistance of CAD/CAM provisional crowns with two different designs is shown in [32]. A systematic review and meta-analysis for a biologically oriented preparation technique for tooth preparation are presented in [33,34], which also provide an in vitro study about the effects of different impression techniques on the marginal fit of restoration. A radiographic evaluation of the margins of clinically acceptable metal–ceramic crowns is given in [35]. The effect of preparation taper on the resistance to fracture of monolithic zirconia crowns is presented in [36,37], which also describe the restorative management of the posterior tooth after a pulpotomy. The effect of margin design on the fracture load of zirconia crowns is shown in [38,39], which also present the effect of endodontic access preparation on the fracture load of translucent versus conventional zirconia crowns with varying occlusal thickness.

The advantages of applying the TMM are the ease and elegance with which it can solve various problems. The TMM is a very easy method to program to quickly obtain good results. The application of TMM calculus can be performed very easily in the case of iterative problems, i.e., problems that require a large volume of repetitive calculus. The TMM lends itself very well to being programmed, which gives rise to immediate results with fast applicability in practice. The disadvantage is that there must be an adapted calculation code which is accessible to those who want to use this calculation method. The limitations of this proposal are given by the disadvantages mentioned.

Other original case studies with different modeling requirements will be presented in the future through the TMM.

2. Materials and Methods

2.1. Materials

Dental restorations for one or two missing teeth can be achieved through the use of dental crowns or dental bridges on one, two, or more pivots or abutments, depending on what the dentist has found to exist in situ, following a specialist consultation.

Dental crowns can be made of the following materials: full zirconia or stratified zirconia, i.e., zircon and ceramic material or metal and ceramic material, i.e., with a metal base and a ceramic coating or composite, which is used especially during prosthetic treatment as a temporary crown. Dental crowns can also be fixed on implants, which must be made of biocompatible materials.

The most suitable biocompatible materials for implants are titanium, zirconium, and ceramic materials, as these materials are resistant to pressure and daily wear and, at the same time, have a pleasant esthetic appearance.

Dental bridges can be fixed or mobile. They can be semi-physiognomic or totally physiognomic. Fixed bridges can be made entirely of zirconium, ceramic materials, acrylic materials, or metallic materials. Removable dentures can be made of metal—for additional structural support—or acrylic resins, which can incorporate ceramics for better esthetics.

2.2. Methods

Some beams with intermediate supports can be studied as statically indeterminate continuous beams. We consider a statically indeterminate continuous beam that is articulated at the left edge, with a simple support at the right edge and intermediate supports, as shown in Figure 1. The calculus is performed with the TMM [1].

2.2.1. Work Hypotheses

The beam with an intermediate support that will be studied is considered a statically indeterminate continuous beam that is articulated at the left edge, with a simple support at the right edge and i (i = 1, n − 1) intermediate simple supports (Figure 1).

The lengths of all parts of the beam are known.

The constant inertia of the entire beam is also considered.

It is considered a reference system, with the origin located at the left edge, i.e., the edge labeled 0.

External vertical forces are considered to act on the beam, which can be characterized by a charge density of q′(x) for the vertically concentrated loads, and q″(x) is the charge density for the force uniformly distributed over the entire beam.

Due to the external forces, unknown vertical reactions arise at each edge; there are (n + 1) vertical reactions. Further, these reactions will be considered external forces with a charge density of q‴(x).

2.2.2. Total Charge Density for a Continuous Beam

It is considered a statically indeterminate continuous beam that is articulated at the left edge, with a simple support at the right edge and n intermediate supports (Figure 1).

Thus, the total charge density, q(x), can be written as (1):

$$q\left(x\right)=q^{\prime }\left(x\right)+q^{″}\left(x\right)+q^{‴}(x),$$

(1)

where

• q(x) is the total charge density;
• q′(x) is the charge density for the exterior vertically concentrated loads;
• q″(x) is the charge density for the force uniformly distributed over the entire beam;
• q‴(x) is the density of charge corresponding to the vertical reactions at the edges.

The charge density corresponding to the exterior concentrated loads, q′(x), for j exterior loads, F_j, j = 1, k, acting on the point a_j, j = 1, k, can be written with Dirac’s and Heaviside’s functions and operators as (2):

$$q^{\prime }\left(x\right)=-\sum _{j=1}^k{F}_j\delta \left(x-a_j\right).$$

(2)

The charge density for the force uniformly distributed over the entire beam, q″(x), can be written with Dirac’s and Heaviside’s functions and operators as follows (3):

$$q^{″}(x)=-p(x)\delta (x),$$

(3)

and the charge density corresponding to the Vertical Reaction (VR_i, i = 0, n) at the edges, q‴(x), again using Dirac’s and Heaviside’s functions and operators, can be written as (4):

$$q^{‴}\left(x\right)=\sum _{i=0}^n{VR}_i\delta \left(x-l_i\right),$$

(4)

where

• j refers to the point on which the concentrated vertical force, F_j, acts;
• F_j, j = 1, k, are the exterior vertical concentrated loads;
• k is the total number of concentrated vertical forces;
• i refers to the edge number;
• n + 1 is the total number of edges;
• p(x) is the force uniformly distributed over the entire beam;
• VR_i is the Vertical Reaction at edge i.

Applying mathematical calculus with Dirac’s and Heaviside’s functions and operators for (2), we can obtain, successively, the relations given in (5):

$$\left\{\begin{array}{c}{q}_1^{\prime }\left(x\right)=-\sum _{j=1}^k{F}_{j}Y\left(x-a_j\right)\\ q_2^{\prime }\left(x\right)=-\sum _{j=1}^k{F}_j\left(x-a_j\right)Y\left(x-a_j\right)\\ q_3^{\prime }\left(x\right)=-\sum _{j=1}^k{F}_j{\displaystyle \frac{{\left(x-a_j\right)}^2}{2}}Y\left(x-a_j\right)\\ q_4^{\prime }\left(x\right)=-\sum _{j=1}^k{F}_j{\displaystyle \frac{{\left(x-a_j\right)}^3}{6}}Y\left(x-a_j\right)\end{array}\right.,$$

(5)

In the same way, this mathematical formalism will be applied to the expressions in (1), and the following relations (6) are obtained:

$$\left\{\begin{array}{c}{q}_1\left(x\right)=-{\displaystyle \sum _{j=1}^k}{F}_{j}Y\left(x-a_j\right)-p(x)xY\left(x\right)+{\displaystyle \sum _{i=1}^n}{VR}_{i}Y\left(x-l_i\right)\\ q_2\left(x\right)=-{\displaystyle \sum _{j=1}^k}{F}_j\left(x-a_j\right)Y\left(x-a_j\right)-p(x){\displaystyle \frac{x^2}{2}}Y\left(x\right)+{\displaystyle \sum _{i=1}^n}{VR}_i\left(x-l_i\right)Y\left(x-l_i\right)\\ q_3\left(x\right)=-{\displaystyle \sum _{j=1}^k}{F}_j{\displaystyle \frac{{\left(x-a_j\right)}^2}{2}}Y\left(x-a_j\right)-p(x){\displaystyle \frac{x^3}{6}}Y\left(x\right)+{\displaystyle \sum _{i=1}^n}{VR}_i{\displaystyle \frac{{\left(x-l_i\right)}^2}{2}}Y\left(x-l_i\right)\\ q_4\left(x\right)=-{\displaystyle \sum _{j=1}^k}{F}_j{\displaystyle \frac{{\left(x-a_j\right)}^3}{6}}Y\left(x-a_j\right)-p(x){\displaystyle \frac{x^4}{24}}Y\left(x\right)+{\displaystyle \sum _{i=1}^n}{VR}_i{\displaystyle \frac{{\left(x-l_i\right)}^3}{6}}Y\left(x-l_i\right)\end{array}\right.,$$

(6)

2.2.3. The State Vectors for Different Sections of the Continuous Beam

The state vectors for different sections of the continuous beam can be defined as follows.

The State Vector corresponding to section x is denoted as {SV(x)} at a point on the x-axis, with the abscissa being measured from the origin of the reference system, which is given by the left edge labeled 0, as (7):

$$\left\{SV\left(x\right)\right\}=\left\{\begin{array}{c}y\left(x\right)\\ \omega \left(x\right)\\ M\left(x\right)\\ T\left(x\right)\end{array}\right\}.$$

(7)

The elements of the state vector (7) are as follows:

• y(x) is the arrow in section x;
• ω(x) is the rotation in section x;
• M(x) is the bending moment in section x;
• T(x) is the cutting force in section x.

The state vector at the origin, {SV(0)} = ${\left\{SV\right\}}_0$, in section 0 is given as (8):

$$\left\{SV\left(0\right)\right\}=\left\{\begin{array}{c}y\left(0\right)\\ \omega \left(0\right)\\ M\left(0\right)\\ T\left(0\right)\end{array}\right\}={\left\{SV\right\}}_0=\left\{\begin{array}{c}{y}_0\\ {\omega }_0\\ M_0\\ T_0\end{array}\right\}.$$

(8)

The total length, l, of the continuous beam is given as (9):

$$l=\sum _{i=1}^n{a}_i,$$

(9)

where a_i, i = 1, n, is the distance between two consecutive supports.

The state vector in the last section, section n, {SV(l)} = ${\left\{SV\right\}}_l,$ for x = l is given as (10):

$$\left\{SV\left(l\right)\right\}=\left\{\begin{array}{c}y\left(l\right)\\ \omega \left(l\right)\\ M\left(l\right)\\ T\left(l\right)\end{array}\right\}={\left\{SV\right\}}_l=\left\{\begin{array}{c}{y}_l\\ {\omega }_l\\ M_l\\ T_l\end{array}\right\}.$$

(10)

2.2.4. The Transfer Matrix for Section x of the Continuous Beam

The connection between the state vector from the origin, section 0, and the state vector from some section x is made with relation (11):

$$\left\{\begin{array}{c}y\left(x\right)\\ \omega \left(x\right)\\ M\left(x\right)\\ T\left(x\right)\end{array}\right\}={\left[TM\right]}_x\left\{\begin{array}{c}y\left(0\right)\\ \omega \left(0\right)\\ M\left(0\right)\\ T\left(0\right)\end{array}\right\}+\left\{VEF\left(x\right)\right\},$$

(11)

where

• [TM]_x is the Transfer Matrix corresponding to section x;
• {VEF(x)} is the Vector for Exterior Forces at section x.

The Transfer Matrix, [TM]_x, is given as (12), from [1]:

$${\left[TM\right]}_x=\left[\begin{array}{cccc}1& x& {\displaystyle \frac{x^2}{2EI}}& -{\displaystyle \frac{x^3}{6EI}}\\ 0& 1& {\displaystyle \frac{x}{EI}}& -{\displaystyle \frac{x^2}{2EI}}\\ 0& 0& 1& -x\\ 0& 0& 0& 1\end{array}\right].$$

(12)

The vector for exterior forces in section x is given as (13):

$$\left\{VEF\left(x\right)\right\}=\left\{\begin{array}{c}-\sum _{j=1}^k{F}_j{\displaystyle \frac{{\left(x-a_j\right)}^3}{6}}Y\left(x-a_j\right)-p(x){\displaystyle \frac{x^4}{24}}Y(x)+\sum _{i=1}^n{VR}_i{\displaystyle \frac{{\left(x-l_i\right)}^3}{6}}Y\left(x-l_i\right)\\ -\sum _{j=1}^k{F}_j{\displaystyle \frac{{\left(x-a_j\right)}^2}{2}}Y\left(x-a_j\right)-p(x){\displaystyle \frac{x^3}{6}}Y(x)+\sum _{i=1}^n{VR}_i{\displaystyle \frac{{\left(x-l_i\right)}^2}{2}}Y\left(x-l_i\right)\\ -\sum _{j=1}^k{F}_j\left(x-a_j\right)Y\left(x-a_j\right)-p(x){\displaystyle \frac{x^2}{2}}Y(x)+\sum _{i=1}^n{VR}_i\left(x-l_i\right)Y\left(x-l_i\right)\\ -\sum _{j=1}^k{F}_{j}Y\left(x-a_j\right)-p(x)xY(x)+\sum _{i=1}^n{VR}_{i}Y\left(x-l_i\right)\end{array}\right\}.$$

(13)

The matrix relation (11) can be developed as (14):

$$\begin{gathered} \left\{\begin{array}{c}y\left(x\right)\\ \omega \left(x\right)\\ M\left(x\right)\\ T\left(x\right)\end{array}\right\}=\left[\begin{array}{cccc}1& x& {\displaystyle \frac{x^2}{2EI}}& -{\displaystyle \frac{x^3}{6EI}}\\ 0& 1& {\displaystyle \frac{x}{EI}}& -{\displaystyle \frac{x^2}{2EI}}\\ 0& 0& 1& -x\\ 0& 0& 0& 1\end{array}\right]\left\{\begin{array}{c}y\left(0\right)\\ \omega \left(0\right)\\ M\left(0\right)\\ T\left(0\right)\end{array}\right\}+ \\ \left\{\begin{array}{c}-{\displaystyle \sum _{j=1}^k}{F}_j{\displaystyle \frac{{\left(x-a_j\right)}^3}{6}}Y\left(x-a_j\right)-p(x){\displaystyle \frac{x^4}{24}}Y(x)+{\displaystyle \sum _{i=1}^n}{VR}_i{\displaystyle \frac{{\left(x-l_i\right)}^3}{6}}Y\left(x-l_i\right)\\ -{\displaystyle \sum _{j=1}^k}{F}_j{\displaystyle \frac{{\left(x-a_j\right)}^2}{2}}Y\left(x-a_j\right)-p(x){\displaystyle \frac{x^3}{6}}Y(x)+{\displaystyle \sum _{i=1}^n}{VR}_i{\displaystyle \frac{{\left(x-l_i\right)}^2}{2}}Y\left(x-l_i\right)\\ -{\displaystyle \sum _{j=1}^k}{F}_j\left(x-a_j\right)Y\left(x-a_j\right)-p(x){\displaystyle \frac{x^2}{2}}Y(x)+{\displaystyle \sum _{i=1}^n}{VR}_i\left(x-l_i\right)Y\left(x-l_i\right)\\ -{\displaystyle \sum _{j=1}^k}{F}_{j}Y\left(x-a_j\right)-p(x)xY(x)+{\displaystyle \sum _{i=1}^n}{VR}_{i}Y\left(x-l_i\right)\end{array}\right\} \end{gathered}$$

(14)

For the last section, for x = l, expression (12) can be written as (15):

$$\left\{\begin{array}{c}y\left(l\right)\\ \omega \left(l\right)\\ M\left(l\right)\\ T\left(l\right)\end{array}\right\}={\left[TM\right]}_l\left\{\begin{array}{c}y\left(0\right)\\ \omega \left(0\right)\\ M\left(0\right)\\ T\left(0\right)\end{array}\right\}+\left\{VEF\left(l\right)\right\},$$

(15)

Relation (15), with the developed transfer matrix, can be written as (16):

$$\begin{gathered} \left\{\begin{array}{c}y\left(l\right)\\ \omega \left(l\right)\\ M\left(l\right)\\ T\left(l\right)\end{array}\right\}=\left[\begin{array}{cccc}1& l& {\displaystyle \frac{l^2}{2EI}}& -{\displaystyle \frac{l^3}{6EI}}\\ 0& 1& {\displaystyle \frac{l}{EI}}& -{\displaystyle \frac{l^2}{2EI}}\\ 0& 0& 1& -l\\ 0& 0& 0& 1\end{array}\right]\left\{\begin{array}{c}y\left(0\right)\\ \omega \left(0\right)\\ M\left(0\right)\\ T\left(0\right)\end{array}\right\}+ \\ \left\{\begin{array}{c}-\sum _{j=1}^k{F}_j{\displaystyle \frac{{\left(l-a_j\right)}^3}{6}}Y\left(l-a_j\right)-p(l){\displaystyle \frac{l^4}{24}}+\sum _{i=1}^n{VR}_i{\displaystyle \frac{{\left(l-l_i\right)}^3}{6}}Y\left(l-l_i\right)\\ -\sum _{j=1}^k{F}_j{\displaystyle \frac{{\left(l-a_j\right)}^2}{2}}Y\left(l-a_j\right)-p(l){\displaystyle \frac{l^3}{6}}+\sum _{i=1}^n{VR}_i{\displaystyle \frac{{\left(l-l_i\right)}^2}{2}}Y\left(l-l_i\right)\\ -\sum _{j=1}^k{F}_j\left(l-a_j\right)Y\left(l-a_j\right)-p(l){\displaystyle \frac{l^2}{2}}+\sum _{i=1}^n{VR}_i\left(l-l_i\right)Y\left(l-l_i\right)\\ -\sum _{j=1}^k{F}_{j}Y\left(l-a_j\right)-p(l)l+\sum _{i=1}^n{VR}_{i}Y\left(l-l_i\right)\end{array}\right\}. \end{gathered}$$

(16)

In matrix relation (16), the conditions at the extreme edges can be placed, in section 0, as shown in (17):

$$\left\{\begin{array}{c}{y}_0=0\\ M_0=0\end{array}\right.,$$

(17)

and in section n, for x = l, the conditions are given in (18):

$$\left\{\begin{array}{c}y\left(l\right)=0\\ M\left(l\right)=0\end{array}\right.,$$

(18)

With the conditions given in (17) and (18), the matrix relation in (16) can be written as (19):

$$\begin{gathered} \left\{\begin{array}{c}0\\ \omega \left(l\right)\\ 0\\ T\left(l\right)\end{array}\right\}=\left[\begin{array}{cccc}1& l& {\displaystyle \frac{l^2}{2EI}}& -{\displaystyle \frac{l^3}{6EI}}\\ 0& 1& {\displaystyle \frac{l}{EI}}& -{\displaystyle \frac{l^2}{2EI}}\\ 0& 0& 1& -l\\ 0& 0& 0& 1\end{array}\right]\left\{\begin{array}{c}0\\ {\omega }_0\\ 0\\ T_0\end{array}\right\}+ \\ \left\{\begin{array}{c}-\sum _{j=1}^k{F}_j{\displaystyle \frac{{\left(l-a_j\right)}^3}{6}}Y\left(l-a_j\right)-p(x){\displaystyle \frac{l^4}{24}}+\sum _{i=1}^n{VR}_i{\displaystyle \frac{{\left(l-l_i\right)}^3}{6}}Y\left(l-l_i\right)\\ -\sum _{j=1}^k{F}_j{\displaystyle \frac{{\left(l-a_j\right)}^2}{2}}Y\left(l-a_j\right)-p(x){\displaystyle \frac{l^3}{6}}+\sum _{i=1}^n{VR}_i{\displaystyle \frac{{\left(l-l_i\right)}^2}{2}}Y\left(l-l_i\right)\\ -\sum _{j=1}^k{F}_j\left(l-a_j\right)Y\left(l-a_j\right)-p(x){\displaystyle \frac{l^2}{2}}+\sum _{i=1}^n{VR}_i\left(l-l_i\right)Y\left(l-l_i\right)\\ -\sum _{j=1}^k{F}_{j}Y\left(l-a_j\right)-p(x)l+\sum _{i=1}^n{VR}_{i}Y\left(l-l_i\right)\end{array}\right\}. \end{gathered}$$

(19)

Also, the conditions for the arrows in each intermediate support must be set; they must be null in each of the intermediate supports, as shown in (20):

$$y_i=0, \mathrm{f}\mathrm{o}\mathrm{r} i=1,n-1,$$

(20)

or (21):

$$\left\{\begin{array}{c}{y}_1=0\Rightarrow l_1{\omega }_0-l_1^3 {\displaystyle \frac{T_0}{6EI}}=-{\displaystyle \frac{q_4^{\prime }(l_1)}{6EI}}-{\displaystyle \frac{q_4^{”}(l_1)}{24EI}}\\ y_2=0\Rightarrow l_2{\omega }_0-l_2^3 {\displaystyle \frac{T_0}{6EI}}+{\left(l_2-l_1\right)}^3{\displaystyle \frac{{VR}_1}{6EI}}=-{\displaystyle \frac{q_4^{\prime }\left(l_2-l_1\right)}{6EI}}-{\displaystyle \frac{q_4^{”}\left(l_2-l_1\right)}{24EI}}\\ \dots \\ y_i=0\Rightarrow l_i{\omega }_0-l_i^3 {\displaystyle \frac{T_0}{6EI}}+{\displaystyle \sum _{i=1}^k}{(l_i-l_1)}^3{\displaystyle \frac{{VR}_i}{6EI}}=-{\displaystyle \sum _{i=1}^{n-1}}{\displaystyle \frac{q_4^{\prime }\left(l_i-l_1\right)}{6EI}}-{\displaystyle \sum _{i=1}^{n-1}}{\displaystyle \frac{q_4^{”}\left(l_i-l_1\right)}{24EI}}\\ \dots \\ y_{n-1}=0\Rightarrow l_{n-1}{\omega }_0-l_{n-1}^3 {\displaystyle \frac{T_0}{6EI}}+{\displaystyle \sum _{i=1}^{n-1}}{(l_i-l_1)}^3{\displaystyle \frac{{VR}_i}{6EI}}=-{\displaystyle \sum _{i=1}^{n-1}}{\displaystyle \frac{q_4^{\prime }\left(l_{n-1}-l_1\right)}{6EI}}-{\displaystyle \sum _{i=1}^{n-1}}{\displaystyle \frac{q_4^{”}\left(l_{n-1}-l_1\right)}{24EI}}\end{array}\right..$$

(21)

The conditions for the right end of the beam, for x = l, are given in (18), and these give rise to (22):

$$\left\{ \begin{array}{c}{ y}_n=0\Rightarrow l{\omega }_0-l^3 {\displaystyle \frac{T_0}{6EI}}+{\displaystyle \sum _{i=1}^n}{(l-l_i)}^3{\displaystyle \frac{{VR}_i}{6EI}}=-{\displaystyle \frac{q_4^{\prime }(l)}{6EI}}-{\displaystyle \frac{q_4^{”}(l)}{24EI}}\\ M_n=0\Rightarrow -l{T}_0+{\displaystyle \sum _{i=1}^n}{\left(l-l_i\right)}^3{VR}_i=-q_2^{\prime }\left(l\right)-q_2^{”}(l)\end{array}\right..$$

(22)

The relations in (21) and (22) can be written as a matrix relation like in (23), considering that ω₀, ${\displaystyle \frac{T_0}{6EI }},$ and $\sum _{i=1}^{n-1}{\displaystyle \frac{{VR}_i}{EI}}$ are unknown:

$$\left[\begin{array}{ccccccc}{l}_1& -l_1^3& 0& 0& 0& \cdots & 0\\ l_2& -l_2^3& ({l_2-l_1)}^3& 0& 0& \cdots & 0\\ l_3& -l_3^3& ({l_3-l_1)}^3& ({l_3-l_2)}^3& 0& \cdots & 0\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ l_i& -l_i^3& ({l_i-l_1)}^3& ({l_i-l_2)}^3& ({l_i-l_3)}^3& \cdots & 0\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ l& -l^3& ({l-l_1)}^3& ({l-l_2)}^3& ({l-l_3)}^3& \cdots & ({l-l_{n-1})}^3\\ 0& -l& l-l_1& l-l_2& l-l_3& \cdots & l-l_{n-1}\end{array}\right]\left\{\begin{array}{c}{\omega }_0\\ {\displaystyle \frac{T_0}{6EI}}\\ \begin{array}{c}{\displaystyle \frac{{VR}_1}{6EI}}\\ ⋮\\ \begin{array}{c}⋮\\ {\displaystyle \frac{{VR}_{n-2}}{6EI}}\\ {\displaystyle \frac{{VR}_{n-1}}{6EI}}\end{array}\end{array}\end{array}\right\}=\left\{\begin{array}{c}-{\displaystyle \frac{q_4^{\prime }(l_1)}{6EI}}-{\displaystyle \frac{q_4^{”}(l_1)}{24EI}}\\ -{\displaystyle \frac{q_4^{\prime }\left(l_2-l_1\right)}{6EI}}-{\displaystyle \frac{q_4^{”}\left(l_2-l_1\right)}{24EI}}\\ \begin{array}{c}⋮\\ -\sum _{i=1}^{n-1}{\displaystyle \frac{q_4^{\prime }\left(l_i-l_1\right)}{6EI}}-\sum _{i=1}^{n-1}{\displaystyle \frac{q_4^{”}\left(l_i-l_1\right)}{24EI}}\\ \begin{array}{c}⋮\\ -{\displaystyle \frac{q_4^{\prime }(l)}{6EI}}-{\displaystyle \frac{q_4^{”}(l)}{24EI}}\\ {-q}_2^{\prime }\left(l\right)-q_2^{”}(l)\end{array}\end{array}\end{array}\right\},$$

(23)

Solving the system of equations given by the matrix relation in (23) leads to the calculus of all unknowns. This can also be programmed very easily, thus quickly obtaining the values of the unknowns, and then, all the efforts and deformations in any section of the beam can be calculated.

3. Application and Results

3.1. Application for a Dental Restoration Case: Dental Bridge Assimilated as a Continuous Beam with Three Poles and Two Gaps Between the Three Poles

We consider a case with two missing teeth, as shown in Figure 2.

3.1.1. Conditions for Mathematical Calculus

A dental restoration with the following characteristics was proposed:

• The lengths of the two parts of the beam are known and equal to each other, denoted as l.
• The constant inertia of the entire beam is also considered.
• A reference system with the origin at the left edge, the edge labeled 0, is considered.
• The dental bridge should consist of three poles, which are the supports for the beam.
• The three poles could be natural teeth or implants, poles supporting the dental bridge.
• The left pole must be on a tooth that is stronger by one degree than that of the second pole.
• Between the three poles, there are two missing teeth.
• The distances between the middle of the left pole and the middle of the first missing tooth, between the middle of the first missing tooth and the middle of the second pole, between the middle of the second pole and the middle of the second missing tooth, and between the middle of the second missing tooth and the middle of the third pole are considered to be equal to each other.
• It is considered that a uniformly distributed vertical force acts on the dental bridge as a result of the action of the antagonistic teeth on the jaw, as shown in Figure 2 and Figure 3.

The dental bridge shown in Figure 3 can be assimilated as a continuous beam, with the following characteristics:

• The continuous beam is a statically indeterminate beam;
• The beam is considered articulated at the left edge (in the left pole), with a simple support at the right edge (in the right pole) and an intermediate support in the middle (as the middle pole of the bridge);
• The continuous beam is either a statically indeterminate or simply statically indeterminate beam;
• A uniformly distributed vertical load acts on the beam, as shown in Figure 2 and Figure 3;
• The distances between the supports are denoted as l.

3.1.2. Approach to Continuous Beam with an Intermediate Support by TMM

The dental bridge in Figure 2 is assimilated as the continuous beam in Figure 3, which is considered for this study.

It is a continuous beam that is articulated at the left edge, with a simple support at the right edge and a simple intermediate support. The simple support is considered an intermediate support because it is situated between the two ends of the continuous beam, i.e., between the articulated edge at the left side and the simple support at the right side, as shown in Figure 3.

The left edge is denoted as 0; i.e., the articulated edge is called the origin section, considering 0 as the origin of the reference system, Ox. The second edge, the simple intermediate support, is denoted as 1, and the right edge is denoted as 2.

The density charge, q″(x), corresponding to the exterior uniformly distributed vertical load for the continuous beam shown in Figure 3 is given as (24):

$$q″\left(x\right)=-p,$$

(24)

and by applying mathematical calculus with Dirac’s and Heaviside’s functions and operators and the matrix calculus presented in Section 2.2.4., for the case of Figure 3, we can obtain the matrix relation given in (25):

$$\left[\begin{array}{ccc}l& -l^2& 0\\ 2l& -{(2l)}^3& l^3\\ 0& -2l& l\end{array}\right]\left\{\begin{array}{c}{\omega }_0\\ {\displaystyle \frac{T_0}{6EI}}\\ {\displaystyle \frac{{VR}_1}{6EI}}\end{array}\right\}=\left\{\begin{array}{c}{\displaystyle \frac{p{l}^4}{24EI}}\\ {\displaystyle \frac{p{(2l)}^4}{24EI}}\\ \begin{array}{c}{\displaystyle \frac{p{(2l)}^2}{12EI}}\end{array}\end{array}\right\}$$

(25)

or (26):

$$\left[\begin{array}{ccc}l& -l^2& 0\\ 2l& -{8l}^3& l^3\\ 0& -2l& l\end{array}\right]\left\{\begin{array}{c}{\omega }_0\\ {\displaystyle \frac{T_0}{6EI}}\\ {\displaystyle \frac{{VR}_1}{6EI}}\end{array}\right\}=\left\{\begin{array}{c}{\displaystyle \frac{p{l}^4}{24EI}}\\ {\displaystyle \frac{2p{l}^4}{3EI}}\\ \begin{array}{c}{\displaystyle \frac{p{l}^2}{3EI}}\end{array}\end{array}\right\},$$

(26)

We can then write the matrix relation given in (26) in a more developed form as (27):

$$\left\{\begin{array}{c}l{\omega }_0-l^2{\displaystyle \frac{T_0}{6EI}}={\displaystyle \frac{p{l}^4}{24EI}}\\ 2l{\omega }_0{-8l}^3{\displaystyle \frac{T_0}{6EI}}+l^3{\displaystyle \frac{{VR}_1}{6EI}}={\displaystyle \frac{2p{l}^4}{3EI}}\\ \begin{array}{c}-2l{\displaystyle \frac{T_0}{6EI}}+l{\displaystyle \frac{{VR}_1}{6EI}}={\displaystyle \frac{p{l}^2}{3EI}}\end{array}\end{array}\right.,$$

(27)

or (28):

$$\left\{\begin{array}{c}{\omega }_0-l{\displaystyle \frac{T_0}{6EI}}={\displaystyle \frac{p{l}^3}{24EI}}\\ 2{\omega }_0{-4l}^2{\displaystyle \frac{T_0}{3EI}}+l^2{\displaystyle \frac{{VR}_1}{6EI}}={\displaystyle \frac{2p{l}^3}{3EI}}\\ \begin{array}{c}-T_0+{\displaystyle \frac{{VR}_1}{2}}=pl\end{array}\end{array}\right.$$

(28)

3.2. Results

The matrix relation given in (28) is a linear system of three equations with three unknowns. After applying calculus, the system solution is given as (29):

$$\left\{\begin{array}{c}{T}_0={\displaystyle \frac{3pl}{8}}\\ {VR}_1={\displaystyle \frac{5pl}{4}}\\ \begin{array}{c}{\omega }_0={\displaystyle \frac{p{l}^2}{3EI}}\end{array}\end{array}\right.$$

(29)

The values for T₀ and the reaction in the intermediate support, VR₁, are identical to those obtained when solving the problem with the statically indeterminate beam from Figure 3 with Clapeyron’s equation of three moments [2].

The results in (29) can be used to calculate the moments and shear forces in any section of the beam, as well as the deformations y and ω, and after this, the stresses in any section of the beam can be calculated.

4. Discussion

This work presents an original and very interesting approach to the calculus problem of beams with intermediate supports using the TMM. The TMM is a very easy method to program to quickly obtain good results. Some beams with intermediate supports can be studied as statically indeterminate continuous beams. In this work, we considered a statically indeterminate continuous beam articulated at the left edge, with a simple support at the right edge and intermediate supports. The theoretical approach is presented in Section 2.2. The applicability of this approach is presented through an example in dentistry, namely the calculus of a dental bridge on three poles. After this, the approach was particularized for application in a dental restoration case, with a dental bridge on three poles with two missing teeth between the three poles. The dental bridge was assimilated as a simple static indeterminate beam resting on three supports—the three poles—with two openings, corresponding to the two missing teeth, as described in Section 3.1. The results for this case study are presented in Section 3.2. The results were validated by comparison with those obtained by the classical calculation of the Resistance of Materials with Clapeyron’s equation of three moments [2].

5. Conclusions

This study is an original and interesting approach to the problem of intermediate supports for a statically indeterminate beam with the TMM. This is very important for many fields, both in industry and in other fields, such as medicine and dentistry, e.g., for dental restorations as dental bridges and/or crowns. The application of TMM calculus can be carried out very easily in the case of iterative problems, i.e., problems that require a large volume of repetitive calculus. The TMM lends itself very well to being programmed, which gives rise to immediate results with fast applicability in practice.

An interesting practical application is the study of dental crowns and bridges, such as the example presented in Section 3.1.2. 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. These will be presented in future research papers.