Porous implants must satisfy several physical and b iological requirements in order to be promising materials for orthopedic application: they should have the proper levels of stiffness, permeability, and fatigue resistance and in proximity to how much they are in bone tissues. In recent years, several exper imental, numerical, and analytical studies have been carried out on the influence of u nit cell geometry on such properties. Even though experimental and numerical techniques c an effectively study and predict the behavior of different micro-structure, they lac k the ease the analytical relationships provide for such predictions. Even though it is wel l-known that Timoshenko beam theory gives much better accuracy in predicting the deformation of a beam (and as a result lattice structures), many of the already-exi sting relationships in the literature have been derived based on Euler-Bernoulli beam the ory. The question that arises here is that can there be a convenient way to convert th e already-existing relationships based on Euler-Bernoulli to relationships based on Timoshenko beam theory without the need to rewrite all the derivations from the st arting point. In this paper, this question is addressed and answered, and a simple ap proach is presented. This technique is applied to six unit cells for which Eu ler-Bernoulli analytical relationships could be found in the literature, but Timoshenko th eories could not be found: BCC, hexagonal packing, rhombicuboctahedron, diamond, tr uncated cube, and truncated octahedron. The results of this study demonstrated that converting analytical relationships based on Euler-Bernoulli to equivalen t Timoshenko ones can decrease the difference between the analytical and numerical val ues for one order of magnitude which is a significant improvement in accuracy of t he analytical formulas. The methodology presented in this study is not only ben eficial to the already-existing analytical relationships but also facilitates deriv ation of accurate analytical relationships for other, yet unexplored, unit cell types. 3 Keywords: Mechanical properties; Euler-Bernoulli beam theory ; Timoshenko beam theory; analytical relationship; Finite element met hod 1. INTRODUCTION Recently, partially or fully porous load-bearing im plants have been proposed to replace the traditionally solid implants in case of large b one defects. While metallic foams manufactured by traditional techniques such as powd er metallurgy [1], investment casting [2], and space-holder [3] have found their way in this field, they all lack a good controllability over the microstructural geometry o f the implant which affects its static mechanical properties [4], fatigue resistance [5, 6 ], and biological response [7]. Recent blast in the application of additive manufacturing (AM) in this field has opened the possibility of manufacturing porous implants with a rbitrary micro-architecture. AM makes it possible to manufacture open-cell (i.e. pe rmeable) porous materials with precisely designed microstructure both in micro- an d macro- scales [8]. The porous implants should satisfy many physical an d biological requirement in order to be in optimal state for biomedical application: they should have the right levels of stiffness, permeability, and fatigue resistance and in proximity to how much they are in bone tissues. This is especially crucial to avoi d undesired consequences of very stiff solid metals which can cause problems such as stres s shielding [9]. The regular porous biomaterials are usually made of repeating elements known as unit cells. The mechanical, physical, and biological pro perties of the implant are determined by three main characteristics of the unit cell they are made of: the shape of the cells, their size, and their relative density (which is de fined as the ratio of space occupied by the solid material). In recent years, several experimental [10-14], nume rical [15-20], and analytical [21-25] studies have been dedicated to studying the influen ce of unit cell shape on such properties. Even though experimental and numerical techniques can effectively study 4 and predict the behavior of different micro-structu re, they lack the ease the analytical relationships provide for such predictions. Analyti cal relationships have several benefits: they can be used for validation of numeri cal or experimental results, they can be implemented in Machine Learning or Artificial In telligence algorithms to construct optimally-designed patient-specific implants, and t hey can give a clear and quick indication of what properties have the most contrib ution in each of the mechanical properties of a regularly repeated lattice structur e. Previously, analytical relationships for elastic pr operties (elastic modulus, Poisson’s ratio, and yield stress) of different unit cells su ch as body-centred cubic (BCC) [26], cube [27, 28], diamond [21], hexagonal packing [29] , iso-cube [30], octahedral [22], rhombic dodecahedron [23, 25], rhombicuboctahedron [24], truncated cube [27], truncated cuboctahedron [31], and truncated octahed ron [32] have been derived. In the literature, relationships for octahedral [22], rhom bic dodecahedron [25], and truncated cuboctahedron [31] have been derived based on both Euler-Bernoulli and Timoshenko beam theories. However, for the rest of the above-m entioned geometries the relationships are only derived based on Euler-Berno ulli beam theory. The question that arises here is whether or not there can be a conven ient way to convert the already- existing relationships based on Euler-Bernoulli to relationships based on Timoshenko beam theory without the need to rewrite all the der ivations from the starting point. In this study, we try to answer that question by pr esenting a technique to convert the already-existing Euler-Bernoulli analytical relatio nships to equivalent Timoshenko ones. This technique is used to convert Euler-Berno ulli elastic relationships including elastic modulus, Poisson’s ratio, and yield stress of any porous material ( ,  , ). The efficiency of this technique is evaluated in six un it cells: BCC, hexagonal packing, rhombicuboctahedron, diamond, truncated cube, and t runcated octahedron. Numerical simulations are also carried out using finite eleme nt (FE) modelling to evaluate if the 5 new technique gives better accuracy in the analytic al results. Moreover, experimental data points from previous studies are used for vali dation of the proposed technique. 2. MATERIALS AND METHODS 2.1. Euler-Bernoulli beam theory For a beam under no distributed load, the Euler–Ber noulli beam equation can be written as   =0 (1) where  is the deflection. The solution to this differenti al equation can be expressed as: = +  +  +   (2) where constants −  must be determined by applying certain boundary co nditions. For a cantilever Euler-Bernoulli beam with a point load  at its end, we have = 3   = 2 (3) and for a cantilever beam with a concentrated momen t at its end, the displacement and rotation are obtained as =  2   =  (4) In beams that the angle of the free end does not ch ange during the deformation (e.g. the beams of the lattice structure considered in th is study), the rotations produced by the lateral load  and moment must be equal and opposite from which the value of can be identified:  2=   → = 2 (5) While force  tends to increase the lateral displacement, moment tends to reduce the deflection. The total deflection caused by forc e  and moment is then = 3−! 2" 2 = 12  (6) 6 Rewriting (6) as a function of  gives (see Figure 2a) =12    (7) According to Eq. (5), we will have =6% 2 . Similarly, the axial force required to displace the end of a rod for & is ' &/ (see Figure 2c). The equations for a cantilever beam with rotation but with no displacement in the end can be obtained in a similar manner and the results are shown in Figure 2b. 2.2. Timoshenko beam theory Now, we try to find how the moments and forces show n in Figure 2a-b are changed due to change in the beam theory. The Timoshenko be am theory takes into account shear deformation and rotational inertia effects, m aking it suitable for describing the behaviour of short beams. For a homogenous beam of constant cross-section, the Timoshenko beam governing equations are:   !)  "=*+ ,,-   =)−1 .' /  !)  " (8) where ) is the angle of rotation of the normal to the mid- surface of the beam,  is the displacement of the mid-surface in the z-direction, and . is the shear coefficient factor. The . coefficient is a dimensionless quantity, dependent on the shape of the cross- section, which is introduced to account for the fac t that the shear stress and strain are distributed not-uniformly over the cross-section [33]. In a linear elastic Timoshenko beam, the bending moment 00 and the shear force 10 are related to the angle of rotation ) and the displacement  by 00 =−2) 2 10=.' /+−)+2 2 - (9) 7 a) A cantilever beam with lateral displacement but with no rotation in the end (Figure 3a): Since the distributed load (force per length) *+ - is zero, the solution to the first differential equation of Eq. (8) can b e expressed as: )  =*+ -=0 →)  =3 → )  =3 +3 →) =3 2 +3 +3 (10) By applying the boundary condition of no rotation a t the root of the cantilever beam, Eq. (10) gives 3= 0. Similarly, applying the boundary condition of no rotation ( ) = 0) at the end of the beam ( = ) to Eq. (10) gives 3= −456 . Substituting 3 and 3 in the second line of Timoshenko beam theory (Eq. ( 8)) gives the deflection of the beam as follows: =3 27  3−  28− .' /3 2+2 −-+3 (11) The two other constants ( 3 , 3) can be found from the boundary conditions in = 0 and = . At the fixed end ( = 0), the deflection equals zero, hence: 3=−3  2.' / (12) Moreover, the deflection at the end of the beam equ als  which gives: 3 =−  12 + .' / (13) By substituting 3 , 3 , 3 , 3 in the first line of Eq. (8) and in Eq. (11), the T imoshenko beam theory for a cantilever beam with lateral disp lacement but with no rotation could be obtained as follows: )=−  6+2 .'/ + − -  =−  12 + .' /71 27  3−   28−1 .' / 8 (14) 8 Now it is possible to find the moments and forces s hown in Figure 3a. As mentioned above, the bending moment 00 and the shear force 10 are related to the angle of rotation ) and the displacement  by Eq. (9). Therefore, by substituting = and 10=  at =  in respectively the first and second lines of Eq. (9), we have: =  6+2 .'/   =  12 + .' / (15) b) A cantilever beam with rotation but with no late ral displacement in the end (Figure 3b): Since the distributed load *+ - and the boundary condition of the beam at the clamped side ( = 0) for this case is similar to the previous case (Ca se a), the relationship for angle of rotation ) is the same as that given in Eq. (10), and also 3= 0. By considering  = 0 at both = 0 and = , the constants 3 and 3 can be found through Eq. (11) as: 3=3 !2 .' /− 3" 3=3 .' / (16) The beam at the free side ( = ) has a rotation with the known value of ) = which by substituting it in Eq. (10), we would have: 3 =  6+2 .' / (17) Therefore, the Timoshenko beam theory governing equ ations for a cantilever beam with rotation but with no lateral displacement at t he end could be obtained: )=3 +23 2 =1 273  3+3 8−3 .' / (18) 9 By considering 00 = and 10=  at =  in Eq. (9), the relationship of bending moment and shear force can be found as follows =92 .' /+2 3  6+2 .' /: (19) =  6+2 .'/  (20) 2.3. From Euler-Bernoulli to Timoshenko The deformation of any arbitrary strut of an open-c ell lattice structure can be decomposed into four basic deformations (Figure 1). According to relationships obtained for Euler-Bernoulli and Timoshenko beam th eories in Sections 2.1 and 2.2, the resultant relationships for elastic modulus and Poisson’s ratio of any structure based on Euler-Bernoulli theory can be converted in to Timoshenko beam theory relationships by making the changes suggested in Ta ble 1 into the stiffness matrix or in the derivation formulas. In the following, it is explained, how the changes have been made for each of the six unit cell types BCC, hexag onal packing, rhombicuboctahedron, diamond, truncated cube, and t runcated octahedron (Figure 4- 5) in order to convert their Euler-Bernoulli formul as [21, 24, 26, 27, 29, 34] to corresponding Timoshenko ones. It must be noted that the commonly known geometries for the six noted unit cell shapes are presented in Figure 4 and the actual geo metries used for both the analytical and numerical analyses are given in Figure 5. The r eason for such conversion of unit cell shape for BCC is obvious as the molecular stru cture of BCC (Figure 4a) is composed of spheres rather than struts (Figure 5a). For the case of diamond, the unit cell shown in Figure 4b is rotated, and the unit ce ll in a specific direction (for which all the struts have similar angles with respect to horizontal plane) has been considered for analysis (Figure 5b). For the other cases, the unit cell conversion has been made 10 in order to avoid neighbour cells having adjacent s truts. If the neighbour cells share an edge, the analytical relationships obtained for unit cell does not represent that of a lattice structure. More explanation regarding this can be found in Section 2.1.2 of [27]. The conversion procedure from Euler-Bernoulli beam theory into Timoshenko beam theory for each unit cell is described as follows: a) BCC The relationships for elastic modulus and Poisson’s ratio of BCC unit cell have been presented in equations “17” and “18” of [26]. In these equations, the axial extension ( ;<= 6) and lateral bending (< => 6?) terms of Euler-Bernoulli theory can be identified easily. By substituting the term < => 6? with @ A? BCD =EFA GHI =J, the relationships for elastic modulus and Poisson’s rat io can be obtained. b) Diamond For transforming relationships of elastic modulus a nd Poisson’s ratio from Euler-Bernoulli into Timoshenko beam theory, the ba sic equations “7” and “8” of [35] for elastic modulus and “29” and “30” of [3 5] for Poisson’s ratio were considered. Axial extension ( ;<= 6) and lateral bending ( < => 6?) terms of Euler- Bernoulli theory can be identified easily. By subst ituting the term < => 6? with @ A? BCD =EFA GHI =J, the relationships for elastic modulus and Poisson ’s ratio based on Timoshenko beam theory were obtained. c) Hexagonal packing For this unit cell, since the analytical relationsh ips for mechanical properties of structure have not been derived based on Euler-Bern oulli theory in literature [29], the analytical relationships for Euler-Bernou lli and Timoshenko beam theories have been derived in this study. The detai led derivation can be found in Appendix C. 11 d) Rhombicuboctahedron Since the final relationships for elastic modulus a nd Poisson’s ratio of this unit cell have been presented in K 6 terms, and the stiffness matrix of unit cell conta ins ;<= 6, < => 6?, L=M 6 and N<=> 6C, recognition of terms provided in Table 1 in the f inal equations “29” and “33” of [36] is not possible. Th erefore, by substituting the term < => 6? with @ A? BCD =EFA GHI =J and N<=> 6C with @ AC OD=EFC GHI =J in the stiffness matrix (equation “27”of [36]) and solving the system of eq uations, the relationships based on Timoshenko theory were obtained. e) Truncated cube The procedure for truncated cube unit cell is very similar to what was described above for rhombicuboctahedron unit cell. The final stiffness matrix of unit cell in equation “36” of [37] contains ;<= 6 and < => 6? terms. Therefore, by substituting the term < => 6? with @ A? BCD =EFA GHI =J, the stiffness matrix based on Timoshenko beam theory can be derived. Afterwards, by solving the s ystem of equations, the mechanical properties relationships based on Timosh enko beam theory were obtained. f) Truncated octahedron The relationships for elastic modulus and Poisson’s ratio of truncated octahedron unit cell have been presented in equatio ns “15” and “17” of [38]. Since the deformation of this unit cell includes ax ial extension and lateral bending, the terms ;<= 6 and < => 6? can be extracted from these equations. By substitution the term <> 6? with @ A? BCD =EFA GHI =J, the relationships presented for elastic modulus and Poisson’s ratio can be transfor med into corresponding Timoshenko ones. 12 It is worth noting that the relationships for norma lized yield stress for 5 unit cells of six noted unit cells (BCC, hexagonal packing, rhomb icuboctahedron, diamond, and truncated octahedron) based on Euler-Bernoulli and Timoshenko beam theories have been obtained separately in another study and could be found in [39]. 2.4. Numerical analysis The 3D CAD representation of the unit calls used fo r numerical modelling and analysis are demonstrated in Figure 5. The actual FE models of the unit cells along with the boundary conditions are demonstrated in Figure B in Appendix B. The struts of the unit cells were discretized using Timoshenko beam e lements (element type 189 in ANSYS) and each strut was discretised using five be am elements. The beam elements were rigidly connected to each other at their share d vertices, and they were not allowed to rotate in any direction in the connecting point. The mechanical properties of the titanium alloy Ti-6Al-4V-ELI were used for modellin g the local behaviour of the material in the FE models. A linear elastic materia l model with elastic modulus and Poisson’s ratio of 113.8 GPa and 0.342 was implemen ted in the FE model. Since this type of element uses linear interpolation and takes transverse shear deformation into account, it is expected that the numerical results will be closer to the Timoshenko analytical solution. In the FE models of BCC, diamo nd, truncated cube, and truncated octahedron, a single unit cell with periodic bounda ry condition has been analysed under compressive loading (Figure Ba,b,e,f). For hexagona l packing and rhombicuboctahedron topologies, lattice structures consisting of 11×11×11 unit cells was used for numerical modelling. This was due to t he complexity of modelling the repetitive boundary conditions in these two unit ce lls, as in these two cases, each side of the unit cell is composed of two strut types (ra ther than one in the case of other unit cell types) which have non-symmetrical displac ement under compressive loading (demonstrated as strut types A and B in Figure 5c-d ). 13 In all the FE models, the lowermost nodes of the un it cell were fixed in the direction parallel to the loading direction and were not allo wed to rotate in any direction. For the case of FE models made out of single unit cells (Figure B1a,b,e,f in Appendix B), the side vertices of the unit cell were constrained rotationally as they were shared with adjacent unit cells (repetitive boundary condition) . In all the FE models, a downward displacement was applied to the uppermost node(s) o f the single unit cell/lattice structure to induce axial deformation. The uppermos t nodes were not allowed to rotate in any directions. Mechanical properties of the FE models have been ca lculated based on basic definition of elastic modulus, Poisson’s ratio, and yield stre ss: • Elastic modulus: The formula P4 =QRS TRS ;RS URS was used for calculating numerical elastic modulus, where VP4 is the structure length in the direction parallel to loading direction, 'P4 is the cross-sectional area of the structure in th e direction perpendicular to the loading direction, P4 is the downward displacement applied to the uppermost nodes, and P4 is obtained by summing the reaction forces of the lowermost nodes. • Poisson’s ratio: The formula WP4 = −XC XB= −TBUC TCUB was used for obtaining Poisson’s ratio. In this formula,  and V are the downward displacement applied to the uppermost nodes and unit cells lengt h in the direction parallel to loading direction, respectively. Parameters  and V are respectively the lateral displacement of the side nodes and the structure le ngth in the direction perpendicular to loading direction. • Yield stress: The formula YZ YZ= =QRS ;RS Y[\] was used to calculate normalized yield stress. In this formula, P4 is obtain from the sum of reaction forces of the lowermost nodes, 'P4 is the structures cross-sectional area in the dire ction perpendicular to the loading direction, and ^_0 is the maximum von Mises 14 stress experienced in the most critical point of th e structure. The critical points of each unit cell can be seen in Figure 9. For the cases which a lattice model was implemented for calculating numerical results of mechanical properties (rhombicuboctahedron and h exagonal packing), all the presented terms denoted by `3 should be denoted by V,,a b and the lattice structures dimensions should be used for calculation. 3. RESULTS The transformed elastic modulus, Poisson’s ratio, a nd yields stress relationships for the six geometries are presented in Table 2-4. For simplifying the equations, the terms c =;<= 6 , d = A? BCD =EFA GHI = , ` =CD=E GHI =AFCA ? AC OD=EFC GHI =, e = AC OD=EFC GHI =, and f =L=M 6 and have been used in the noted tables for the relationships based on the Timoshenko beam theory. For four of the geometries (BCC [26], rhombicuboctahedron [2 4], truncated cube [27], and truncated octahedron [34]), the original Euler-Bern oulli relationships have been presented in Table 2-3. In the case of hexagonal pa cking, the original Euler-Bernoulli relationships [29] for elastic modulus and Poisson’ s ratio have been improved and adjusted (the description on how and why can be fou nd in Appendix C), and the improved relationships are presented in Table 2-3. For the case of diamond unit cell, the original relationships for elastic modulus and Poisson’s ratio have been conserved but stated as a function of g/ rather than relative density, h. As for the yield stress (Table 4), the analytical Euler-Bernoulli presented in our other paper [39] have been presented. In order to compare the already-existing [21, 24, 2 6, 27, 29, 34] or improved Euler- Bernoulli analytical relationships with the newly t ransformed Timoshenko analytical relationships, the analytical relationships for bot h the Euler-Bernoulli and Timoshenko 15 beam theory have been compared with numerical and e xperimental results in Figure 6-8. The relative density relationships for the six geometries are presented in Table 5. For all the geometries, the newly transformed Timos henko relationships have exceptionally good overlapping with the numerical r esults for all the mechanical properties: elastic modulus, Poisson’s ratio, and y ield stress (Figure 6-8). As for the previously obtained formulas obtained in the litera ture or newly adjusted Euler- Bernoulli formulas, the maximum difference between Euler-Bernoulli analytical elastic modulus and corresponding numerical values for BCC, diamond, Hexagonal packing, rhombicuboctahedron, truncated cube, and truncated octahedron (at relative density of h = 0.5) are respectively 21.34%, 57.71%, 20.21%, 14.52%, 14.98% and 45.54%. However, the corresponding difference between the Timoshenko analytical relationships and the numerical values for the note d geometries are respectively 1.13%, 2.21%, 8.29%, 2.97%, 0.43% and 3.15%. Similarly, the maximum difference between Euler-Bernoulli analytical Poisson’s ratio and corresponding numerical values for BCC, diamond , Hexagonal packing, rhombicuboctahedron, truncated cube, truncated octa hedron are respectively 3.07%, 27.69%, 28.00%, 21.54%, 73.58% and 40.51%, while th e corresponding difference between the Timoshenko analytical relationships and the numerical values for the noted geometries are respectively 0.133%, 0.826%, 0 .899%, 6.24%, 1.36% and 0.498%, which shows a significant improvement in the accura cy of the analytical relationships. As it can be seen, the difference of the Timoshenko analytical and numerical values has improved significantly and to values around 1/1 0 of how much it was for the case of Euler-Bernoulli beam theories. As for the yield stress (Figure 8), the analytical Euler-Bernoulli and Timoshenko relationships are identical for BCC and diamond. Th e reason is explained in Section 4.2. However, for the three other geometries, namel y hexagonal packing, rhombicuboctahedron, and truncated octahedron, the analytical relationships based on 16 Euler-Bernoulli differ from the relationships obtai ned based on Timoshenko beam theory. Nevertheless, for all the cases, the Timosh enko analytical yield stress curve has exceptionally good agreement with numerical results (Figure 8). For BCC, diamond and truncated cube unit cells (the geometries which have the same yield stress analytical relationships for Euler-Bernoulli and Ti moshenko beam theories), the maximum difference between analytical and numerical values is less than 0.25%. As for the three other geometries, the maximum differe nce between Euler-Bernoulli analytical normalized yield stress and correspondin g numerical values for hexagonal packing, rhombicuboctahedron and truncated octahedr on are respectively 21.25%. 3.02% and 7.52%. However, such differences for the analytical relationships based on Timoshenko beam theory and numerical values are res pectively 9.49%, 0.09% and 0.162% which shows a significant improvement in the accuracy of the analytical relationships. As for the proximity of the analytical/numerical el astic modulus and yield stress values to the experimental data, the experimental data poi nts have been presented in Figure 6 and 8 for rhombicuboctahedron, truncated cube and diamond unit cells only, as in the literature, there is experimental measurements for such unit cells only [40]. As for the elastic modulus, converting the analytical rela tionships from Euler-Bernoulli to Timoshenko has led to closer proximity of analytica l and experimental values (Figure 6b,d,e). As for the yield stress, for the three geo metries for which experimental data points are available (diamond, rhombicuboctahedron, and truncated cube), the analytical relationships based on Euler-Bernoulli a nd Timoshenko are identical or almost overlapping, and both are in good agreement with experimental data points (Figure 8b,d,e). 4. DISCUSSIONS 4.1. Why the new relationships give much better acc uracy? 17 In this paper, new relationships based on Timoshenk o beam theory have been derived for elastic modulus, Poisson’s ratio, and yield str ength of several lattice structures. In addition to considering of Timoshenko beam theories , some adjustments in deriving analytical relationships based on Euler-Bernoulli b eam theory have been implemented for hexagonal packing unit cell (see Appendix C). T imoshenko beam theory takes into account shear deformation and rotational bending ef fects, making it suitable for describing the behaviour of thick beams. As a resul t, the new relationships based on Timoshenko beam theory give much better accuracy ev en at larger relative densities of lattice structures. This improvement is signific ant for analytical/experimental agreement and exceptional for analytical/numerical agreement. To give a more physically tangible understanding, it must be noted that taking into account the shear mechanisms of deformation effectively increases the flexibility of the beam which effectively leads to larger deflections of the stru ts (and therefore the whole lattice structure) under an imposed load. This leads to res pectively lower and higher elastic modulus and Poisson’s ratio of the structure for Ti moshenko theory as compared to Euler-Bernoulli theory. 4.2. Yield strength As mentioned in the Results section, the normalized yield stress relationships based on Timoshenko and Euler-Bernoulli theories are identic al for BCC, diamond, and truncated cube structures and both theories have go od agreement with the numerical results. However, for the hexagonal packing, trunca ted octahedron, and rhombicuboctahedron structures, the Euler-Bernoulli and Timoshenko yield strength results are quite different (particularly for hexag onal packing case) and the Timoshenko theory shows much better overlapping wit h numerical results as compared to how Euler-Bernoulli analytical curve does. The r eason why the Timoshenko and Euler-Bernoulli yield strengths are identical for s ome geometries and different for some other is described extensively in [39], but it is e xplained here briefly. As introduced 18 above, d =< => 6?, and e =N<=> 6C for Euler-Bernoulli beam theory, and d = A? BCD =EFA GHI = and e = AC OD=EFC GHI = for Timoshenko beam theory. For both the theories, k l=6 , hence as the analytical yield strength relationship for both the diamond and BCC in the final stage of derivation is a mere function of k l, both the theories give the same result for yield strength relationship in these two unit cells. As f or the truncated cube case, the normalized yield stress in critical strut could be obtained from forces and moments equilibrium and the strain causing the damage is me rely due to axial loading rather than bending, hence the Euler-Bernoulli and Timoshe nko theories give the same results. Nevertheless, for the three other unit cel ls, on the one hand, the displacement caused by bending needs to be taken into account in the calculations, and on the other hand, the formulas in the final stages of derivatio n are not a mere function of k l, and the V and T terms are present independently and not in a fraction form in such a way that one is a factor of the other. Therefore, the r esults differ for Euler-Bernoulli and Timoshenko theories for the three other unit cells. 4.3. Some points regarding experimental data points Both analytical and numerical techniques over-predi ct the experimental elastic modulus of the diamond and truncated cube porous st ructures, while they under- predict their experimental yield stress. The random irregularities and imperfections created during the AM process diminish the elastic modulus of the lattice structures. These defects create weak links in the structure th at lower the mechanical properties of structures. On the other hand, in the case of yield stress, wh en the initial regions of the lattice structures in the critical points (w hich experience the highest stress levels in the whole lattice structure) are yielded, their local yield stress increases due to strain-hardening phenomenon, and the structure i s still able to remain almost intact as the strain-hardening strengthens the structure a t the initial damage points. However, when the external load is increased even f urther, the second (and possibly 19 the third) critical points are damaged and together with the initial points, the structure is unable to keep its integrity as it was before, a s now damage and softening has propagated throughout the whole structure. That is why in practice, the yielding in the structure as a whole usually occurs for the ext ernal loads which are between the external loads which theoretically cause the first and second set of critical points become locally yielded. This was shown in [31]. 4.4. Application to biomedical implants There are several applications for cellular materia ls including heat exchangers, filters, load bearing components, and biomedical implants. L attice structures can improve the implants’ performance significantly, from both mech anical and biological points of view. Obtaining the accurate characteristics and mechanic al properties of the lattice structures are necessary to improve their use in or thopaedic implants. The mechanical properties of lattice structures depend mainly on t he following three parameters: the material they are made of, the cell topology, and t he relative density. Obtaining accurate analytical relationships for different cel l topologies is a crucial factor in developing computational tools for designing patien t-specific implants with non- uniform mechanical property distribution. In this p aper, we presented a new and convenient method to transform properties of a latt ice structure from Euler-Bernoulli theory into Timoshenko theory for both the already- existing Euler-Bernoulli relationships and for future studies. Using the tra nsformation relationships presented in this paper (Table 1), developing new analytical relationships based on Timoshenko beam theory becomes easier than how it has been bef ore. 5. CONCLUSIONS In this paper, a new methodology to convert analyti cal relationships based on Euler- Bernoulli to Timoshenko has been presented. Six uni t cells for which Euler-Bernoulli analytical relationships could be found in the lite rature, but Timoshenko theories could not be found have been considered: BCC, hexagonal p acking, rhombicuboctahedron, 20 diamond, truncated cube, and truncated octahedron [ 21, 24, 26, 27, 29, 34]. The results of this study demonstrated that converting analytic al relationships based on Euler- Bernoulli to equivalent Timoshenko ones can decreas e the difference between the analytical and numerical values for one order of ma gnitude which is a significant improvement in accuracy of the analytical formulas. The methodology presented in this study is not only beneficial to the already-ex isting analytical relationships but also to facilitate derivation of analytical relationship s for other, yet unexplored, unit cell types.