A Mathematical Model for Non-Linear Structural Analysis Reinforced Beams of Composite Materials

Partial or total contact between the two elements occurs when two elements that can withstand bending moments are elastically coupled at the interface. If the elastic connection is flexible, there will be differential axial strains at the common interface that cause slip, and there may also be differential deflections that cause uplift between the two parts. The majority of composite structures are dimensionally reducible, and utilizing lower-dimensional structural models, their analysis can be accelerated [1]. This article develops a composite beam element that can be used to simulate the nonlinear behavior of composite beams. It was discovered that at the same load level, increasing the cover plate's thickness increased the final load while decreasing the maximum slide [2]. One-dimensional beam models, for instance, can depict thin structural elements where the length is substantially more than the cross-sectional dimension. Although beam theory has been available for many centuries, it wasn't until about 1985 that academics started to concentrate largely on developing beam models for structures with arbitrary cross-sections and composite materials [3]. The efficient high-fidelity approach developed by Hodges and his coworkers stands out significantly in the quest for effective yet accurate models appropriate for nonlinear analysis of composite beams and is gaining acceptance in both industry and the scientific community [4]. Both in elastic and inelastic areas, fiber-reinforced plastic composites (FRP) have physically nonlinear features. If lamina shear stressors reach a 5 reasonably substantial ratio to longitudinal tensile stresses, they are known to be the main cause of elastic nonlinearity. In this case, the mechanical performance of the composites is dominated by the resin matrix. The in-plane shear responses of FRP plies are nonlinear over the whole range investigated because the shear stress-strain responses of polymer resins are nonlinear over the entire strain range and at extremely low strain levels [5, 6]. Able to predict failure brought on by stress concentrations with accuracy [7, 8]. The Ramberg-Osgood equation [9], which is also common in investigations of metal fatigue, is another widely used model. The use of mathematical curve fitting functions provides a more adaptable description methodology [10-12]. Results from Matrix 3D are contrasted with those from the reputable commercial structural analysis program SAP2000. When compared to SAP2000, Matrix 3D's non-linear analysis solver performed better when computing a given structure with hinges that were modelled at element ends [13]. The usage of sophisticated and complicated finite element structural analysis computer programs has been made possible by the consistent advancement of computer technology over the past ten years. Commercial and non-commercial structural analysis programs generally fall into two categories [14-17]. It is important to take convergence issues into account when using finite element programs for sophisticated non-linear calculations. When non-linear constitutive relations, in particular softening ones, are added to FE non-linear investigations, the software solver encounters convergence issues in addition to explicit numerical approaches. In order to obtain any findings from the problematic FE system, users typically turn to explicit dynamic analysis [18, 19]. Contrary to non-linear static and implicit dynamic analyses, which are iterative, explicit dynamic analyses do not converge to solution in iterations. Therefore, in order to examine the outcomes of explicit dynamic analysis, extra steps are necessary. In this regard, it is advantageous for the field of non-linear structural analysis to investigate whether computation convergence may be enhanced using non-linear static or implicit dynamic approaches [20, 21]. The publications by Najem et al. [22], Avalon and Donaldson [23], Vershchaka et al. [24], Ranz et al. [25], Vereshaka and Karash [26] appear to be the only ones for the case of thin-walled curved beams made of isotropic materials that fully take into account shear deformability. Palanim and Rajasekaran [27] has created a stability study for laminated, curved thin-walled beams without taking shear effects into account. In any case, relatively few studies have taken into account the flexibility caused by warping shear in addition to flexural shear when studying the mechanics of curved thin-walled beams made of composite materials. In order to assess the mechanics of out-of-plane movement. [28, 29] developed a model that takes into account the full shear flexibility in curved beams made of laminated composite materials. The paradigm given by Cortínez et al. [28] and Liu et al. [29] is only truly applicable in balanced symmetrical or particularly orthotropic situations. In a variety of industries, including aeronautics, the marine sector, energy, and civil construction, composite materials with geometry that incorporates significant bending radii are frequently used in engineering constructions [30]. Interlayer delamination results from the breakdown of these components, which is mostly caused by interlinear tensile strength [31] In order to obtain an effective design, it is essential to determine interlinear tensile strength (ILTS). To determine ILTS, a variety of experimental techniques are available.

In this article, nine mathematical models of beams will be designed and made up of different composite materials and in different shapes (straight, twisted, arched), in order to compare between deformations, stresses and strains when a high load is placed on them using the ANSYS-15.0 program, and this study differs from previous studies because it takes Linear and other non-linear models have the same dimensions, and the results are compared for all models. The last paragraphs of the article are as follows: The material used and the mathematical model, Results and Discussion, Conclusions, Acknowledgment, future studies and references.

Nine mathematical models were created for three distinct forms and three different composite material mixtures, and they were tested in the ANSYS 15.0 program with a load of (10 MN) applied to the upper surface of each model while the sides of the models remained fixed while the load was shed, to determine the deformations, stresses, and strains that these models are subjected to when the load is applied, and the following findings were made:

3.1 Models (M-1, M-2, M-3)

The three models (1, 2, and 3), which are composed of various composite materials but have the same geometric design, were tested, and the findings are shown in the Figures 3-15. The results of the deflection (δ) for these models are shown in Figure 3, and the results show that the deflection values for the three models are, as is abundantly obvious (δ₁=72.77 mm, δ₂=153.11 mm, and δ₃= 351.79 mm). The results of the component of the displacement in a direction (x), for these models are shown in Figure 4, The displacement outcomes are shown in the figure as follows, (Ux₁=- 26.24 to 26.24 mm, Ux₂=- 8.39 to 8.41 mm, and Ux₃=- 25.97 to 29.99 mm). Figure 5 shows the models' results for the component of displacement in a direction (y), the figure suggests that the results' values are, (Uy₁=72.77 mm, Uy₂=153.11 mm, and Uy₃= 351.79 mm).  Figure 6 shows the components of these models' displacements in a direction (z), according to the figure, displacement values might be either positive or negative are (Uz₁=- 0.0046 to 0.0327 mm, Uz₂=- 0.0187 to 0.1923 mm, and Uz₃=- 0.0058 to 0.0496 mm). The results of the normal stress (σ_x) for these models are shown in Figure 7. The outcomes of this stresses show the values (σ_x1=- 4029.13 to 2739.6 MPa, σ_x2=- 9029.29 to 4261.27 MPa, and σ_x3=- 3913.65 to 2517.31 MPa). The results of the typical stress (σ_y), as shown Figure 8, the figure displays the effects of this stress for several models, with values ranging from $\begin{aligned} & \left(\sigma_{y_1}=-2906.21 \text { to } 756.97 \mathrm{MPa}, \sigma_{y_2}=\right. \left.-3633.4 \text { to } 1103.9 \mathrm{MPa} \text {, and } \sigma_{y_3}=-2616.15 \text { to } 917.49 \mathrm{MPa}\right) \text {. }\end{aligned}$ Figure 9 displays the shear stress (τ_xy) findings for these models, the figure demonstrates that the shear stresses values range from $\begin{aligned} & \quad\left(\tau_{x y_1}=-1928.51 \text { to } 1929.26 \mathrm{MPa}, \tau_{x y_2}=\right.-1695.37 \text { to } 1696.03 \mathrm{MPa} \text {, and } \tau_{x y_3}= -1750.01 \text { to } 1750.51 \mathrm{MPa}) .\end{aligned}$ The greatest stress intensity (σ_int.) values for the three models are shown in figure 10 results of the stresses intensity (σ_int.), and they were $\begin{aligned} & \left(\sigma_{\text {int.}_1}=5466.95 \mathrm{MPa}, \sigma_{\text {int.}_2}=9518.33 \mathrm{MPa} \text {, and } \sigma_{\text {int. }_3}=\right. 5130.09 \mathrm{MPa}) .\end{aligned}$ The results of the normal strain (ε_x), for three models, are shown in Figure 11, and the figure demonstrates that this strain's value fall within the following range $\begin{gathered}\left(\varepsilon_{x_1}=-0.0117 \text { to } 0.0113, \varepsilon_{x_2}=-0.0285 \text { to } 0.0168 \text { and } \varepsilon_{x_3}\right. =-0.0635 \text { to } 0.0485) .\end{gathered}$ Figure 12 shows the outcomes of these models' normal strains (ε_y), and the figure shows that the values of this strain range between the following values $\begin{aligned} \left(\varepsilon_{y_1}=-0.0161 \text { to } 0.0105, \varepsilon_{y_2}=\right. \left.-0.1328 \text { to } 0.0962 \text {, and } \varepsilon_{y_3}=-0.1961 \text { to } 0.0785\right)\end{aligned}$. The results of the normal strain (ε_z), for three models, are shown in Figure 13, the values of the results show that the strains have values between $\begin{aligned} & \quad\left(\varepsilon_{z_1}=-0.0043 \text { to } 0.0099, \varepsilon_{z_2}=\right.  \left.-0.0345 \text { to } 0.1233 \text {, and } \varepsilon_{z_3}=-0.0210 \text { to } 0.1011\right) .\end{aligned}$ Figure 14 displays the results of the shear strain (ε_xy) for three models, this strain's value, according to the figure, fall within the following range $\begin{aligned} & \left(\varepsilon_{x y_1}=-0.0273 \text { to } 0.0274, \varepsilon_{x y_2}=\right. \left.-0.4256 \text { to } 0.4258 \text {, and } \varepsilon_{x y_3}=-0.4357 \text { to } 0.4388\right)\end{aligned}$. Figure 15 displays the findings of the models (1, 2, and 3) for the intensity strain (ε_xy); the figure reveals that the three models' maximum strain values were $\left(\varepsilon_{\text {int. } 1}=0.0376, \varepsilon_{i n t .2}=0.4424\right.$, and $\left.\varepsilon_{\text {int }_{\cdot 3}}=0.4599\right)$.

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Figure 3. Results of the deflection (δ), for models (1, 2, and 3)

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Figure 4. Results of the component of the displacement in a direction (Ux), for models (1, 2, and 3)

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Figure 5. Results of the component of the displacement in a direction (Uy), for models (1, 2, and 3)

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Figure 6. Results of the component of the displacement in a direction (Uz), for models (1, 2, and 3)

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Figure 7. Results of the normal stress (σ_x), for models (1, 2, and 3)

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Figure 8. Results of the normal stress (σ_y), for models (1, 2, and 3)

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Figure 9. Results of the shear stress (τ_xy), for models (1, 2, and 3)

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Figure 10. Results of the intensity stress (σ_int.), for models (1, 2, and 3)

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Figure 11. Results of the normal strain (ε_x), for models (1, 2, and 3)

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Figure 12. Results of the normal strain (ε_y), for models (1, 2, and 3)

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Figure 13. Results of the normal strain (ε_z), for models (1, 2, and 3)

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Figure 14. Results of the shear strain (ε_xy), for models (1, 2, and 3)

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Figure 15. Results of the intensity strain (ε_int.)), for models (1, 2, and 3)

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Figure 16. Compares the results of the relationship between displacement (U_x) and distance on a linear path (A-A), for models (1, 2, and 3)

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Figure 17. Compares the results of the relationship between displacement (U_y) and distance on a linear path (A-A), for models (1, 2, and 3)

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Figure 18. Compares the results of the relationship between displacement (U_sum) and distance on a nonlinear path (A-A), for models (1, 2, and 3)

The results in the Figures 16-27 compare the deformations, stresses, and strains that the beams are subjected to when they are fixed to the sidewalls and a load is applied to them in the middle of the surface of the models using ANSYS – 15.0 program. From the start of the models to their end, as indicated by the path (A - A) in the Figure 2. The findings indicate that: Figure 16 compares the findings of different models for the relationship between displacement (U_x) and distance on a linear path (A-A). The results of various models for the relationship between displacement (U_x) and distance on a linear path (A-A) are compared in Figure 16, the findings show that the highest displacement values were in the model (M-3), where the highest displacement negative value (25.968 mm) was at distance (1230.8 mm) and the highest displacement positive value (25.941 mm) was at distance (3692.3 mm). Figure 17 compares the findings for various models' linear route (A-A) relationship between displacement (Uy) and distance, the figure demonstrates that the model (M-3) experienced the greatest displacement, with the greatest downward displacement of (341.04 mm) occurring at the distance (2540 mm). The relationship between displacement (U_sum) and distance on a linear path (A-A), for three models, is compared in Figure 18, According to the data shown in the figure, the distance (2538.5 mm) recorded the highest values of displacement (U_sum), and its value was (341.04 mm) for the model (M-3). Figure 19 compares the findings for three models for the relationship between normal stress (σ_x) and distance on a linear path (A-A), and the figure shows that the highest normal stress values were recorded for the model (M-2), and the stress value (2629.5 MPa) was at the distance (2500 mm). Figure 20 compares the outcomes for different models for the relationship between normal stress (σ_y) and distance on a linear path (A-A), The highest normal stress (σ_y) was for the model (M-2), and its highest value was measured at the distance (2500 mm), The figure indicated that, and its value was (226.74 MPA). Figure 21 illustrates the comparison of the outcomes for different models for the relationship between shear stress (τ_xy) and distance along a linear path (A-A), and the figure depicts the maximum shear stress value (τ_xy) recorded for the model (M-1), where the value of this negative shear stress (600 MPa) was at the distance (0 mm), while the highest positive shear stress values were at the distance (5,000 mm), where its value was (600 MPa). According to the statistics in Figure 22, the second model had the highest stress intensity value (σ_int.), which compares the results of the relationship between distance on a linear path (A-A) and stress intensity (σ_int.) for three models, with the first value (3950 MPa) at the distance of (0 mm), the second value (2,630 MPa) at the distance of (2,500 mm), and the third value at the distance of (5,000 mm) and its value (3,950 MPa). Figure 23 compares the outcomes of the relationship between normal strain (ε_x) and distance on a linear path (A-A), for three models, and the figure demonstrates that the model (M-3) had values of the largest strain, with the highest negative strain value being recorded at distances of (0 and 5,000 mm), and its value being (0.04859), while the greatest positive strain measurements were (0.0419) and at a distance (2,500 mm). Figure 24 Compare the results of the relationship between normal strain (ε_y) and distance on a linear path (A-A), for these models, and the figure demonstrates that the model (M-3) recorded the largest strain value, and the stress value was (0.0283) at the distance (2,500 mm). Figure 25 compares the findings for three models' normal strain (ε_z) and distance on a linear path (A-A) relationships, and the findings of this strain in this figure demonstrate that the second model had the maximum negative strain, with a value of (0.0312) at the two distances (0, 5,000 mm), but the third model had the maximum negative strain, and its value (0.0158 mm) when viewed from a distance (2,500 mm). Figure 26 Compare the results of the relationship between shear strain (ε_xy) and distance on a linear path (A-A), for these three models, and the data in the figure, the third model had the maximum shear strain (ε_xy) and that its maximum negative shear strain value was (0.0815) at a distance (0 mm), in contrast, the greatest positive shear strain measurement (0.0815) was at a distance (5,000 mm). Figure 27 compares the outcomes of the relationship between distance on a linear path (A-A) and intensity strain (ε_int.), for the models (M-1, M-2, M-3), and the figure demonstrates that the second model had the maximum strain intensity values, with its value (0.0839) at a distance (0 mm).

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Figure 19. Compares the results of the relationship between normal stress (σ_x) and distance on a linear path (A-A), for models (1, 2, and 3)

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Figure 20. Compares the results of the relationship between normal stress (σ_y) and distance on a linear path (A-A), for models (1, 2, and 3)

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Figure 21. Compares the results of the relationship between shear stress (τ_xy) and distance on a nonlinear path (A-A), for models (1, 2, and 3)

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Figure 22. Compares the results of the relationship between stress intensity (σ_intesity) and distance on a linear path (A-A), for models (1, 2, and 3)

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Figure 23. Compares the results of the relationship between normal strain (ε_x) and distance on a linear path (A-A), for models (1, 2, and 3)

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Figure 24. Compare the results of the relationship between normal strain (ε_y) and distance on a linear path (A-A), for models (1, 2, and 3)

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Figure 25. Compare the results of the relationship between normal strain (ε_z) and distance on a linear path (A-A), for models (1, 2, and 3)

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Figure 26. Compare the results of the relationship between shear strain (ε_xy) and distance on a linear path (A-A), for models (1, 2, and 3)

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Figure 27. Compare the results of the relationship between intensity strain (ε_int.) and distance on a linear path (A-A), for models (1, 2, and 3)

3.2 Models (M-4, M-5, M-6)

The ANSYS 15.0 program was used to test the three models (4, 5, and 6), which are made of different composite materials but have the same geometric design. The results are displayed in the Figures 28-40. The results of the deflection (δ) for these models are shown in Figure 28, and the results show that the deflection values for the three models are, as is abundantly obvious $\left(\delta_4=108.315 \mathrm{~mm}, \delta_5=209.32 \mathrm{~mm}\right.$, and $\left.\delta_6=381.75 \mathrm{~mm}\right)$. The results of the component of the displacement in a direction (x), for these models are shown in Figure 29, The displacement outcomes are shown in the figure as follows, $\begin{aligned} & \quad\left(\mathrm{Ux}_4=-12.291 \text { to } 12.634 \mathrm{~mm}, \mathrm{Ux}_5=\right.  \left.-22.277 \text { to } 15.045 \mathrm{~mm}, \text { and } \mathrm{Ux}_6=-42.614 \text { to } 34.993 \mathrm{~mm}\right) .\end{aligned}$ Figure 30 shows the models' results for the component of displacement in a direction (y), the figure shows that the maximum results' values are, $\left(\mathrm{Uy}_4=108.296 \mathrm{~mm}, \mathrm{Uy}_5=209.267 \mathrm{~mm}, \text { and } \mathrm{Uy}_6=\right. 381.732 \mathrm{~mm})$. Figure 31 shows the components of these models' displacements in a direction (z), according to the figure, displacement values might be either positive or negative ar $\begin{aligned} & \left(\mathrm{Uz}_4=-0.3022 \text { to } 2.043 \mathrm{~mm}, \mathrm{Uz}_5=\right. \left.-0.1634 \text { to } 0.1729 \mathrm{~mm} \text {, and } \mathrm{Uz}_6=-0.0363 \text { to } 0.0537 \mathrm{~mm}\right) \text {. }\end{aligned}$ The results of the normal stress (σ_x) for these models are shown in Figure 32, the outcomes of this stresses show the values $\begin{aligned} & \quad\left(\sigma_{x_4}=-4450.28 \text { to } 1771.53 \mathrm{MPa}, \sigma_{x_5}=\right. -8693.53 \text { to } 3401.96 \mathrm{MPa} \text {, and } \sigma_{x_6}= -3927.74 \text { to } 2160.15 \mathrm{MPa}) .\end{aligned}$ The results of the typical stress (σ_y), as shown Figure 33, the figure displays the effects of this stress for several models, with values ranging from $\begin{aligned} & \left(\sigma_{y_4}=-3663.68 \text { to } 1588.22 \mathrm{MPa}, \sigma_{y_5}=\right. -5930.38 \text { to } 1722.76 \mathrm{MPa}, \text { and } \sigma_{y_6}= -3025.11 \text { to } 1072.91 \mathrm{MPa}) .\end{aligned}$ The results of the shear stress (τ_xy), for these models, are shown in Figure 34, the figure demonstrates that the shear stresses values range from ($\tau_{x y_1}=-1928.51$ to $1929.26 \mathrm{MPa}$, $\tau_{x y_2}=-1695.37$ to $1696.03 \mathrm{MPa}$ and $\tau_{x y_3}=-1750.01$ to $1750.5$) The greatest stress intensity (σ_int.) values for the three models are shown in Figure 35 results of the stresses intensity (σ_int.), and they were $\left(\sigma_{\text {int. }{ }_4}=7369.18 \mathrm{MPa}, \sigma_{\text {int. } 5}=9057.31 \mathrm{MPa}\right.$, and $\sigma_6=$ 4869.94 MPa). The results of the normal strain (ε_x), for three models, are shown in Figure 36, and the figure demonstrates that this strain's value fall within the following range $\begin{aligned} & \left(\varepsilon_{x_4}=-0.0334 \text { to } 0.0155, \varepsilon_5=\right. \left.-0.0857 \text { to } 0.0496, \text { and } \varepsilon_{x_6}=-0.1384 \text { to } 0.0470\right) .\end{aligned}$ Figure 37 shows the outcomes of these models' normal strains (ε_y), and the figure shows that the values of this strain range between the following values $\begin{aligned} \left(\varepsilon_{y_4}-0.0239 \text { to } 0.0109, \varepsilon_{y_5}=\right. \left.-0.0584 \text { to } 0.0496 \text {, and } \varepsilon_{y_6}=-0.2055 \text { to } 0.0805\right).\end{aligned}$ The results of the normal strain (ε_z), for three models, are shown in Figure 38, the values of the results show that the strains have values between $\begin{aligned} & \quad\left(\varepsilon_4=-0.0039 \text { to } 0.0123, \varepsilon_{z_5}=\right. \left.-0.0201 \text { to } 0.1163 \text {, and } \varepsilon_{z_6}=-0.0164 \text { to } 0.0958\right) \text {. }\end{aligned}$ Figure 39 displays the results of the shear strain (ε_xy) for three models, this strain's value, according to the figure, fall with in the following range $\begin{aligned} & \left(\varepsilon_{x y_4}=-0.0473 \text { to } 0.0505, \varepsilon_{x y_5}=\right.  \left.-0.4439 \text { to } 0.3196, \text { and } \varepsilon_{x y_6}=-0.4443 \text { to } 0.3435\right).\end{aligned}$ The results of the models (4, 5, and 6) for the intensity strain (ε_xy) are shown in Figure 40, and the figure reveals that the three models' maximum strain values were $\left(\varepsilon_{\text {int }{ }_{.4}}=0.0581, \varepsilon_{\text {int. } .5}=0.4471\right.$, and $\left.\varepsilon_{\text {int. } 6}=0.4473\right)$.

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Figure 28. Results of the deflection (δ), for models (4, 5, and 6)

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Figure 29. Results of the displacement component in a direction (x), for models (4, 5, and 6)

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Figure 30. Results of the displacement component in a direction (y), for models (4, 5, and 6)

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Figure 31. Results of the displacement component in a direction (z), for models (4, 5, and 6)

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Figure 32. Results of the normal stress (σ_x), for models (4, 5, and 6)

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Figure 33. Results of the normal stress (σ_y), for models (4, 5, and 6)

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Figure 34. Results of the shear stress (τ_xy), for models (4, 5, and 6)

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Figure 35. Results of the intensity stress (σ_int.), for models (4, 5, and 6)

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Figure 36. Results of the normal strain (ε_x), for models (4, 5, and 6)

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Figure 37. Results of the normal strain (ε_y), for models (4, 5, and 6)

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Figure 38. Results of the normal strain (ε_z), for models (4, 5, and 6)

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Figure 39. Results of the shear strain (ε_xy), for models (4, 5, and 6)

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Figure 40. Results of the intensity strain (ε_int.), for models (4, 5, and 6)

The ANSYS-15.0 program's results are shown in the Figures 41-52 and compare the deformations, stresses, and strains that the beams experience when they are fastened to the sidewalls and a load is applied to them in the center of the surface of the models. The path (B - B) in the figure represents the path from the beginning of the models to their end (2). The results show that: The results of various models for the relationship between displacement (U_x) and distance on a nonlinear path (B - B) are compared in Figure 41, the findings show that the highest displacement values were in the model (M-5), where the highest displacement negative value (14.146 mm) was at distance (1820 mm) and the highest displacement positive value (8.581 mm) was at distance (4280 mm). Figure 42 compares the findings for various models' nonlinear route (B - B) relationship between displacement (U_y) and distance, the figure demonstrates that the model (M-6) experienced the greatest displacement, with the greatest downward displacement of (371.77 mm) occurring at the distance (3050 mm). The relationship between displacement (U_sum) and distance on a nonlinear path (B - B), for three models, is compared in Figure 43, According to the data shown in the figure, the distance (3053.8 mm) recorded the highest values of displacement (U_sum), and its value was (371.8 mm) for the model (M-6). Figure 44 compares the findings for three models for the relationship between normal stress (σ_x) and distance on a nonlinear path (B - B), and the figure shows that the highest normal stress values were recorded for the model (M-4), and the stress value (2130.8 MPa) was at the distance (3053.8 mm). Figure 45 compares the outcomes for different models for the relationship between normal stress (σ_y) and distance on a nonlinear path (B - B), The highest normal stress (σ_y) was for the model (M-4), and its highest value was measured at the distance (4900 mm), according to the figure, and its value was (2284.9 MPa). Figure 46 illustrates the comparison of the outcomes for different models for the relationship between shear stress (τ_xy) and distance along a nonlinear path (B - B), and the figure depicts the maximum shear stress value (τ_xy) recorded for the model (M-6), where the value of this negative shear stress (783.66 MPa) was at the distance (776.92 mm), while the highest positive shear stress values were at the distance (5053.8 mm), where its value was (759.52 MPa). Figure 47 compares the outcomes of the relationship between distance on a nonlinear path (B - B) and stress intensity (σ_int.) for three models, and the data in the figure clearly show that the second model had the highest value of stress intensity (σ_int.), with the first value (2789.9 MPa) at the distance of (776.92 mm), the second value (2755.2 MPa) at the distance of (5053.8 mm). The results of the relationship between normal strain (ε_x) and distance on a nonlinear path (B - B) for three models are compared in Figure 48. The figure shows that the model (M-6) had values of the largest strain, with the highest negative strain value being recorded at distances of (776.62 and 5053.8 mm) and its value being (0.01085), while the greatest positive strain measurements were (0.0432 mm) and at a distance of (5053.8 mm) (3053.8 mm). Figure 49 Compare the results of the relationship between normal strain (ε_y) and distance on a nonlinear path (B - B), for these models, and the figure demonstrates that the model (M-6) recorded the largest strain value, and the stress value was (0.0375) at the distance (776.92 mm). Figure 50 compares the results for the relationships between normal strain (ε_z) and distance on a nonlinear path (B - B) for three models. The results show that the model (M-6) had the maximum negative strain, with a value of (0.0522) at the two distances (776.92, 3,053.8 mm), but the model (M-6) had the maximum negative strain, and its value (0.0134) when viewed from a distance (3053.8 mm). Figure 51 For these three models, compare the results of the relationship between shear strain (ε_xy) and distance on a nonlinear path (B - B) with the data in the figure, and the sixth model had the highest shear strain (ε_xy), and its maximum negative shear strain value was (0.181) at a distance (776.92 mm), while the highest positive shear strain measurement (0.1467) was at a distance (5,053.8 mm). Figure 52 contrasts the results of the relationship between distance on a nonlinear path (B - B) and intensity strain (ε_int.), for the models (M-4, M-5, M-6), and shows that the sixth model had the highest values of strain intensity, with its value (0.2225) at a distance (776.92 mm) and high value (0.2048) at distance (5,053.8).

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Figure 41. Compare the results of the relationship between displacement (U_x) and distance on a nonlinear path (B-B), for models (M-4, M-5, M-6)

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Figure 42. Compare the results of the relationship between displacement (U_y) and distance on a nonlinear path (B-B), for models (M-4, M-5, M-6)

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Figure 43. Compare the results of the relationship between displacement (U_sum) and distance on a nonlinear path (B-B), for models (4, 5, and 6)

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Figure 44. Compare the results of the relationship between normal stress (σ_x) and distance on a nonlinear path (B-B), for models (4, 5, and 6)

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Figure 45. Compare the results of the relationship between normal stress (σ_y) and distance on a nonlinear path (B-B), for models (4, 5, and 6)

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Figure 46. Compare the results of the relationship between shear stress (τ_xy) and distance on a nonlinear path (B-B), for models (4, 5, and 6)

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Figure 47. Compare the results of the relationship between stress intensity (σ_int.) and distance on a nonlinear path (B-B), for models (4, 5, and 6)

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Figure 48. Compare the results of the relationship between normal strain (ε_x) and distance on a nonlinear path (B-B), for models (4, 5, and 6)

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Figure 49. Compare the results of the relationship between normal strain (ε_y) and distance on a nonlinear path (B-B), for models (4, 5, and 6)

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Figure 50. Compare the results of the relationship between normal strain (ε_z) and distance on a nonlinear path (B-B), for models (4, 5, and 6)

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Figure 51. Compare the results of the relationship between shear strain (ε_xy) and distance on a nonlinear path (B-B), for models (4, 5, and 6)

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Figure 52. Compare the results of the relationship between intensity strain (ε_intensity) and distance on a nonlinear path (B-B), for models (4, 5, and 6)

3.3 Models (7, 8, and 9)

The three models (7, 8, and 9), which are composed of various composite materials but have the same geometric design, were tested using the ANSYS 15.0 program, and the Figures 53-65 show the outcomes. The results of the deflection (δ) for these models are shown in Figure 53, and the results show that the deflection values for the three models are, as is abundantly obvious $\left(\delta_7=29.67 \mathrm{~mm}, \delta_8=47.7441 \mathrm{~mm}\right.$, and $\left.\delta_9=95.2507 \mathrm{~mm}\right)$. The results of the component of the displacement in a direction (x), for these models are shown in Figure 54, The displacement outcomes are shown in the figure as follows, $\begin{aligned} & \left(\mathrm{Ux}_7=-4.034 \text { to } 4.0626 \mathrm{~mm}, \mathrm{Ux}_8=\right.  \left.-3.9636 \text { to } 5.175 \mathrm{~mm}, \text { and } U x_9=-10.0258 \text { to } 9.4659 \mathrm{~mm}\right) .\end{aligned}$ Figure 55 shows the models' results for the component of displacement in a direction (y), the figure shows that the maximum results' values are, $\begin{aligned} & \left(\mathrm{Uy}_7=29.67 \mathrm{~mm}, \mathrm{Uy}_8=47.7441 \mathrm{~mm}, \text { and } \mathrm{Uy}_9=\right.  95.2503 \mathrm{~mm}) .\end{aligned}$ Figure 56 shows the components of these models' displacements in a direction (z), according to the figure, displacement values might be either positive or negative are $\begin{aligned} & \left(\mathrm{Uz}_7=-0.0594 \text { to } 0.0094 \mathrm{~mm}, \mathrm{Uz}_8=\right.  \left.-0.0666 \text { to } 0.2114 \mathrm{~mm} \text {, and } \mathrm{Uz}_9=-0.1162 \text { to } 0.0048 \mathrm{~mm}\right)\end{aligned}$. The results of the normal stress (σ_x) for these models are shown in Figure 57, the outcomes of this stresses show the values $\begin{aligned} & \quad\left(\sigma_{x_7}=-2115.99 \text { to } 811.667 \mathrm{MPa}, \sigma_{x_8}=\right. -6259.01 \text { to } 1752.3 \mathrm{MPa} \text {, and } \sigma_{x_9}=-3071.03 \text { to } 863.097 \mathrm{MPa}) .\end{aligned}$ The results of the typical stress (σ_y), as shown Figure 58, the figure displays the effects of this stress for several models, with values ranging from $\begin{aligned} & \left(\sigma_{y_7}=-1915.14 \text { to } 384.78 \mathrm{MPa}, \sigma_{y_8}=\right.-2069.49 \text { to } 692.133 \mathrm{MPa}, \text { and } \sigma_{y_9}=-1498.68 \text { to } 210.28 \mathrm{MPa}) .\end{aligned}$ The results of the shear stress (τ_xy), for these models, are shown in Figure 59, the figure demonstrates that the shear stresses values range from $\begin{aligned} & \left(\tau_{x y_7}=-1061.82 \text { to } 1058.43 \mathrm{MPa}, \tau_{x y_8}=\right.-1331.41 \text { to } 1345.32 \mathrm{MPa} \text {, and } \tau_{x y_9}=-1261.63 \text { to } 1266.82 \mathrm{MPa}) .\end{aligned}$ The greatest stress intensity (σ_int.) values for the three models are shown in Figure 60 results of the stresses intensity (σ_int.), and they were $\begin{aligned} & \left(\sigma_{\text {int. }}=3068.31 \mathrm{MPa}, \sigma_{\text {int. }}=6604.36 \mathrm{MPa} \text {, and } \sigma_9=\right. 3746.87 \mathrm{MPa}) .\end{aligned}$ The results of the normal strain (ε_x), for three models, are shown in Figure 61, and the figure demonstrates that this strain's value fall within the following range $\begin{aligned} & \quad\left(\varepsilon_{x_7}=-0.0146 \text { to } 0.0070, \varepsilon_{x_8}=\right. \left.-0.0378 \text { to } 0.0250 \text {, and } \varepsilon_{x_9}=-0.0519 \text { to } 0.0184\right) .\end{aligned}$ Figure 62 shows the outcomes of these models' normal strains (ε_y), and the figure shows that the values of this strain range between the following values $\begin{aligned} & \left(\varepsilon_{y_7}=-0.0037 \text { to } 0.0032, \varepsilon_{y_8}=-0.0651 \text { to } 0.0399 \text {, and } \varepsilon_{y_9}=\right.  -0.0903 \text { to } 0.0360) \text {. }\end{aligned}$ The results of the normal strain (ε_z), for three models, are shown in Figure 63, the values of the results show that the strains have values between $\begin{aligned} & \quad\left(\varepsilon_7=-0.0017 \text { to } 0.0062, \varepsilon_{z_8}=\right. \left.-0.0154 \text { to } 0.0729, \text { and } \varepsilon_{z_9}=-0.0072 \text { to } 0.0620\right) .\end{aligned}$ Figure 64 displays the results of the shear strain (ε_xy) for three models, this strain's value, according to the figure, fall within the following range $\begin{aligned} & \left(\varepsilon_{x y_7}=-0.0147 \text { to } 0.0154, \varepsilon_{x y_8}=\right.  \left.-0.3367 \text { to } 0.3112 \text {, and } \varepsilon_{x y_9}=-0.3238 \text { to } 0.3060\right) .\end{aligned}$ The results of the models (4, 5, and 6) for the intensity strain (ε_xy) are shown in Figure 65, and the figure reveals that the three models' maximum strain values were $\left(\varepsilon_{\text {int. } .7}=0.0247, \varepsilon_{\text {int. } .8}=0.3387\right.$, and $\left.\varepsilon_{\text {int. } .9}=0.3252\right)$.

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Figure 53. Results of the deflection (δ), for models (7, 8, and 9)

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Figure 54. Results of the displacement in a direction component (U_x), for models (7, 8, and 9)

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Figure 55. Results of the displacement in a direction component (U_y), for models (7, 8, and 9)

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Figure 56. Results of the displacement in a direction component (U_z), for models (7, 8, and 9)

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Figure 57. The results of normal stress (σ_x), for models (7, 8, and 9)

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Figure 58. The results of normal stress (σ_y), for models (7, 8, and 9)

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Figure 59. The results of shear stress (τ_xy), for models (7, 8, and 9)

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Figure 60. The results of intensity stress (σ_int.), for models (7, 8, and 9)

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Figure 61. The results of normal strain (ε_x), for models (7, 8, and 9)

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Figure 62. The results of normal strain (ε_y), for models (7, 8, and 9)

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Figure 63. The results of normal strain (ε_z), for models (7, 8, and 9)

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Figure 64. The results of shear strain (ε_xy), for models (7, 8, and 9)

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Figure 65. The results of intensity strain (ε_int.), for models (7, 8, and 9)

The results of using the ANSYS-15.0 program on the three models (7, 8, and 9) are shown in the Figures 66-77, which compare the deformations, stresses, and strains that the beams experience when they are fastened to the sidewalls and a load is applied to them in the middle of the surface. The path (A - A) in the figure represents the path from the beginning of the models to their end (2). The results show that: Figure 66 compares the results of various models for the relationship between displacement (U_x) and distance on a nonlinear path (C-C), and the results show that the model (M-9) had the highest displacement values, with the highest displacement positive value (9.5312 mm) being at distance and the highest displacement negative value (7.013 mm) being at distance (1,652.7 mm) (3,735 mm). The results for the nonlinear path (C-C) relationship between displacement (U_y) and distance for different models are compared in Figure 67. The figure shows that the model (M-9) experienced the most displacement, with the greatest downward displacement of (91.625 mm) happening at the distance (2,563.4 mm). Figure 68 compares the distance on a nonlinear path (C-C) and the displacement (U_sum) for three different models, and the displacement (U_sum) statistics in the figure indicate that the distance (2,563.4 mm) recorded the greatest values for the model (M-9), with the model's value being (91.834 mm). Figure 69 compares the results for three models for the relationship between normal stress (σ_x) and distance on a nonlinear path (C-C). The figure demonstrates that the model (M-9) had the highest normal stress values, the distance (4,578 mm) had the highest negative normal stress value (1,520 MPa), and the distance (2,599.8 mm) had the highest positive normal stress value (867.6 MPa). The results of various models for the connection between normal stress (σ_y) and distance on a nonlinear path (C-C) are compared in Figure 20, according to the figure, the model (M-7) had the highest positive normal stress (σ_y), and its maximal positive value was determined at a distance of (2,563.4 mm) (86.266 MPA). The maximum shear stress value (τ_xy) recorded for the model (M-9) is shown in Figure 71, which compares the results for various models for the relationship between shear stress (τ_xy) and distance along a nonlinear path (C-C), and the highest shear stresses were for the ninth model. The negative shear stress's value, which was (359.26 Mpa), was at the distance (528.86 mm), while the highest positive shear stress values were at the distance (4,620.3 mm), where they were at (811.09 MPa). Figure 72 compares the results of the relationship between distance along a nonlinear path (C-C) and stress intensity (σ_int.) for three different models. The data clearly demonstrate that the models (M-8, and M-9) had the highest value of stress intensity (σ_int.), with the first value (941.2 MPa) at the distance of (528 mm) for the ninth model, the second value (893 MPa) at the distance of (2,818.4 mm), and the third value for the ninth model at a distance of (4620.3 mm) and its value (1954.3 MPa). The results of the relationship between normal strain (ε_x) and distance on a nonlinear path (C-C) for three models are compared in Figure 73. The figure shows that the model (M-9) had values of the largest strain (ε_x), with the greatest positive strain measurements being (0.0175) and at a distance of (0.0175 mm), and the highest negative strain value being recorded at a distance of (4,578 mm) and its value being (0.0428). Figure 74 compares the findings for different models for the relationship between normal strain (ε_y) and distance on a nonlinear path (C-C). The figure shows that the model (M-8) recorded the highest strain value, and the normal strain value was (0.0428) at the distance (4,535.7 mm). Figure 75 compares the results for three models' normal strain (ε_z) and distance on a nonlinear path (C-C) relationship. The results show that the ninth model, when viewed from a distance (2,599.8 mm), had the maximum negative strain, with a value of (0.007), while the maximum positive strain, with a value of (0.0214) at the distance (4,620.3 mm). Figure 76 Compare the results of the relationship between shear strain (ε_xy) and distance on a nonlinear path (C-C) for these three models and the data in the figure. The ninth model had the highest shear strain (ε_xy), and its maximum negative shear strain value was (0.0634) at a distance (528.8 mm), in contrast to the highest positive shear strain measurement (0.1569), which was at a distance of (4,620.3 mm). For the models (7, 8, and 9), Figure 77 compares the results of the association between distance on a nonlinear path (C-C) and intensity strain (ε_int.). The figure shows that the ninth model had the highest strain intensity values, with its value (0.1593) at a distance (4,620.3 mm).

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Figure 66. Compare the results of the relationship between displacement (U_x) and distance on a nonlinear path (C-C), for models (7, 8, and 9)

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Figure 67. Compare the results of the relationship between displacement (U_y) and distance on a nonlinear path (C-C), for models (7, 8, and 9)

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Figure 68. Compare the results of the relationship between displacement (U_sum) and distance on a nonlinear path (C-C), for models (7, 8, and 9)

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Figure 69. Compare the results of the relationship between normal stress (σ_x) and distance on a nonlinear path (C-C), for models (7, 8, and 9)

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Figure 70. Compare the results of the relationship between normal stress (σ_y) and distance on a nonlinear path (C-C), for models (7, 8, and 9)

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Figure 71. Compare the results of the relationship between shear stress (τ_xy) and distance on a nonlinear path (C-C), for models (7, 8, and 9)

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Figure 72. Compare the results of the relationship between stress intensity (σ_int.) and distance on a nonlinear path (C-C), for models (7, 8, and 9)

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Figure 73. Compare the results of the relationship between normal strain (ε_x) and distance on a nonlinear path (C-C), for models (7, 8, and 9)

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Figure 74. Compare the results of the relationship between normal strain (ε_y) and distance on a nonlinear path (C-C), for models (7, 8, and 9)

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Figure 75. Compare the results of the relationship between normal strain (ε_z) and distance on a nonlinear path (C-C), for models (7, 8, and 9)

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Figure 76. Compare the results of the relationship between shear strain (ε_xy) and distance on a nonlinear path (C-C), for models (7, 8, and 9)

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Figure 77. Compare the results of the relationship between intensity strain (ε_int.) and distance on a nonlinear path (C-C), for models (7, 8, and 9)

In future studies, we suggest using different forces such as (20 MN, 30 MN, ........) to simulate different working conditions and reveal the effect of forces on the models, as well as another study using different sizes and the effect of the size of the model on the resistance of different materials, as well as experimental models can be built, tested and compared with the results obtained.