The Design and Analysis of Internally Stiffened GFRP Tubular Decks—A Sustainable Solution

: The aim of this paper was to ﬁnd an optimal stiffener conﬁguration of thin-wall tubular panels made by glass ﬁber reinforced polymer (GFRP) composite material, which is a low carbon emission, low life cycle cost, and sustainable material. Finite-element analysis (FEA) was used to investigate the ﬂexural and torsional stiffness of various internally stiffened sections of thin-wall GFRP decks. These decks consist of internally stiffened tubular proﬁles laid side by side and bonded together with epoxy to ensure the panel acts as an assembly. Three-dimensional models of the seven proposed decks were assembled with tubular proﬁles of different stiffener patterns. First, the non-stiffened tube proﬁle was tested experimentally to validate the parameters used in the subsequent numerical analysis. Then, the ﬁnite element software, ANSYS, was used to simulate the ﬂexural and torsional behavior of the decks with different stiffener patterns under bending and torsional loads. The decks with stiffener patterns such as “O” type, “V” type, and “D” type were found to be the most effective in bending. For torsion, there was a distinct tendency for deck panels with closed shaped stiffener patterns to perform better than their counterparts. Overall, the “O” type deck panel was an optimal stiffener conﬁguration.


Introduction
With exposure to saline conditions in coastal areas, the use of de-icing chemicals in cold regions, and the frequent occurrence of earthquakes in some regions, the aging process of infrastructures accelerates. As a result, the loading capacities of reinforced concrete bridges often are not sufficient to carry the loads imposed upon them. In the last two decades, civil engineers have considered the use of alternative materials, such as fiber reinforced polymer (FRP) composite materials, to rehabilitate existing infrastructures or rebuild new ones. FRP composite materials are noted for their anti-corrosion, lightweight, low carbon emissions, low thermal conductivity, and excellent weather resistance. Additionally, they are also attractive for their other beneficial mechanical properties, such as high strength-to-weight ratio, and high modulus of elasticity. In the meantime, FRP composite materials can offer a longer maintenance-free service life and low life cycle cost. Due to the lightweight nature of FRP material, the modular FRP components could be shipped to the field for assembling; the time and cost for field construction can thus be significantly reduced.
Regarding carbon emission, Li et al. [1] calculated that the all-GFRP pedestrian bridge reduced total carbon emission by about 43% and 19% compared to a reinforced concrete pedestrian bridge and a steel pedestrian bridge, respectively. Another study showed that for a GFRP composite superstructure of a 12-m long road bridge, carbon emissions were reduced by 48% and 57% compared with a concrete bridge and steel bridge [2]. The FRP composite materials have a lower environmental cost than GFRP tubular panels. In this study, the finite-element analysis (FEA) software, ANSYS, was used to simulate the bending and torsional behaviors of the different proposed deck panels under static loading, and then to evaluate the bending and torsional performance of these deck panels. The paper begins with an introduction of the material properties of FRP composites. Then, the procedures towards the finite element simulation are discussed. Finally, the paper summarizes the findings of the finite element analysis.

FRP Composite Material
FRP is a composite material which mainly owes it properties to the nature of the resin, the type and orientation of the reinforcing fibers, the fiber-to-matrix ratio, the fiber-matrix interface properties, and the manufacturing process used to produce the profiles. The rule of mixture is usually used to evaluate the physical properties of the FRP material. Assuming the fibers are aligned regularly in one direction and perfect bonds exists between the fibers and the matrix, the longitudinal (along the direction of the fibers) Young's modulus can be evaluated as follows: where E z is the longitudinal Young's modulus of the composite material, V f and V m are the volume fractions of the fiber and the matrix respectively, and E f , E m are the Young's moduli of the fiber and the matrix, respectively. Similarly, the major Poisson's ratio is evaluated as: where ν f and ν m are the Poisson's ratio of the fiber and the matrix, respectively. However, the same rule does not apply to the material properties in the transverse direction given that the material is orthotropic. The Young's modulus in the transverse and the shear modulus in the X-Y direction of the composite are simply taken as the corresponding values for the matrix itself. In fact, the FRP composites are anisotropic with the Young's moduli and Poisson's ratio at different directions in each orientation. The pultruded orthotropic thin-wall GFRP tube profiles made with glass fiber and epoxy resin are considered and were used to simulate the behavior of the proposed deck panels. When force is applied in the same direction as that of the fibers (i.e., the longitudinal direction), the stress-strain diagram can be represented by Figure 1. When force is applied perpendicular to the direction of the fibers (i.e., the transverse direction), the stress-strain diagram can be represented by Figure 2. Table 1 lists the material properties provided by GFRP manufacturer. static loading, and then to evaluate the bending and torsional performance of these deck panels. The paper begins with an introduction of the material properties of FRP composites. Then, the procedures towards the finite element simulation are discussed. Finally, the paper summarizes the findings of the finite element analysis.

FRP Composite Material
FRP is a composite material which mainly owes it properties to the nature of the resin, the type and orientation of the reinforcing fibers, the fiber-to-matrix ratio, the fiber-matrix interface properties, and the manufacturing process used to produce the profiles. The rule of mixture is usually used to evaluate the physical properties of the FRP material. Assuming the fibers are aligned regularly in one direction and perfect bonds exists between the fibers and the matrix, the longitudinal (along the direction of the fibers) Young's modulus can be evaluated as follows: where Ez is the longitudinal Young's modulus of the composite material, Vf and Vm are the volume fractions of the fiber and the matrix respectively, and Ef, Em are the Young's moduli of the fiber and the matrix, respectively. Similarly, the major Poisson's ratio is evaluated as: where νf and νm are the Poisson's ratio of the fiber and the matrix, respectively. However, the same rule does not apply to the material properties in the transverse direction given that the material is orthotropic. The Young's modulus in the transverse and the shear modulus in the X-Y direction of the composite are simply taken as the corresponding values for the matrix itself. In fact, the FRP composites are anisotropic with the Young's moduli and Poisson's ratio at different directions in each orientation.
The pultruded orthotropic thin-wall GFRP tube profiles made with glass fiber and epoxy resin are considered and were used to simulate the behavior of the proposed deck panels. When force is applied in the same direction as that of the fibers (i.e., the longitudinal direction), the stress-strain diagram can be represented by Figure 1. When force is applied perpendicular to the direction of the fibers (i.e., the transverse direction), the stress-strain diagram can be represented by Figure 2. Table  1 lists the material properties provided by GFRP manufacturer.

Geometries of Tubular Decks
Seven proposed tubular decks were considered in this study. Each consisted of three 7.5 × 7.5 cm pultruded square tube profiles with thickness ×0.6 cm featuring different stiffener configurations that are assumed to be flawlessly bonded together to form tubular decks. The stiffened tubes were generated by adding various stiffener patterns including "Plus" type (+ type), "V" type, "X" type, "Diamond" type (D type), "Y" type, "H" type, and "O" type to the square tube profile. The square tube is shown in Figure 3a. The three square tube assembly shown in Figure 3b was used as a benchmark deck to evaluate the flexural and torsional stiffness. Figure 4 and Table 2 present the cross sections and dimensions of the proposed tubular decks.

Geometries of Tubular Decks
Seven proposed tubular decks were considered in this study. Each consisted of three 7.5 × 7.5 cm pultruded square tube profiles with thickness ×0.6 cm featuring different stiffener configurations that are assumed to be flawlessly bonded together to form tubular decks. The stiffened tubes were generated by adding various stiffener patterns including "Plus" type (+ type), "V" type, "X" type, "Diamond" type (D type), "Y" type, "H" type, and "O" type to the square tube profile. The square tube is shown in Figure 3a. The three square tube assembly shown in Figure 3b was used as a benchmark deck to evaluate the flexural and torsional stiffness. Figure 4 and Table 2 present the cross sections and dimensions of the proposed tubular decks.  ᵃ Z direction or XZ and YZ plane (parallel to the pultrusion fiber axis); ᵇ X and Y direction or XY plane (perpendicular to the pultrusion fiber axis).

Geometries of Tubular Decks
Seven proposed tubular decks were considered in this study. Each consisted of three 7.5 × 7.5 cm pultruded square tube profiles with thickness ×0.6 cm featuring different stiffener configurations that are assumed to be flawlessly bonded together to form tubular decks. The stiffened tubes were generated by adding various stiffener patterns including "Plus" type (+ type), "V" type, "X" type, "Diamond" type (D type), "Y" type, "H" type, and "O" type to the square tube profile. The square tube is shown in Figure 3a. The three square tube assembly shown in Figure 3b was used as a benchmark deck to evaluate the flexural and torsional stiffness. Figure 4 and Table 2 present the cross sections and dimensions of the proposed tubular decks.     Figure 4).

Finite-Element Model of the GFRP Tube
The geometric models of the proposed panels were first generated with CAD software, then exported to ANSYS workbench to conduct finite-element analysis. The material properties in the finite element analysis are defined in Table 1. The deck panels were modeled as being perfectly glued together. Such an assumption is even more valid if the deck panels are intended to be covered up with an outer wrap of FRP using vacuum-assisted resin transfer molding (VARTM) or if pre-stressing is to be used to enhance the panel action of the tube assemblies as suggested by Wu et al. [12]. Moreover, Li [21] also mentioned that the observed failure mode of similar tubular decks was not localized in the adhesive joint region but rather at the tube section flange, which justifies the assumption made above. A SOLID186 element type was used, which is a 20-noded brick element.

Experimental Validation of the Finite-Element Model
To ensure the accuracy of the finite element model presented in the previous section, a validation test was performed in which a 7.5 × 7.5 cm square tube profile with thickness 0.5 cm was subjected to a three-point bending test performed on the thin-wall GFRP tube. The three-point test was performed   Figure 4).

Finite-Element Model of the GFRP Tube
The geometric models of the proposed panels were first generated with CAD software, then exported to ANSYS workbench to conduct finite-element analysis. The material properties in the finite element analysis are defined in Table 1. The deck panels were modeled as being perfectly glued together. Such an assumption is even more valid if the deck panels are intended to be covered up with an outer wrap of FRP using vacuum-assisted resin transfer molding (VARTM) or if pre-stressing is to be used to enhance the panel action of the tube assemblies as suggested by Wu et al. [12]. Moreover, Li [21] also mentioned that the observed failure mode of similar tubular decks was not localized in the adhesive joint region but rather at the tube section flange, which justifies the assumption made above. A SOLID186 element type was used, which is a 20-noded brick element.

Experimental Validation of the Finite-Element Model
To ensure the accuracy of the finite element model presented in the previous section, a validation test was performed in which a 7.5 × 7.5 cm square tube profile with thickness 0.5 cm was subjected to Sustainability 2018, 10, 4538 6 of 15 a three-point bending test performed on the thin-wall GFRP tube. The three-point test was performed according to the ASTM D790 test method [22]. The illustration figure of the three-point bending test is shown in Figure 5. The geometry of the profile utilized in the validation test was similar to that of the non-stiffened tube profile. Figure 6 shows a picture of the experimental setup of the three-point bending test performed on the GFRP tube profile. The dial gauge (DDP-30A, Tokyo Sokki Kenkyujo Co., Ltd., Tokyo, Japan) was placed below the specimen to measure mid-span displacement of the specimen and a load cell (WF 17120, Wykeham Farrance, Milan, Italy) with 50 tons capacity was utilized. In the meantime, a high precision data logger (KL-10, Geomaster Group, Tianjin, China) was used to record the force and displacement data and a sampling rate of 1 data/sec was set to record the force-displacement information.
Sustainability 2018, 10, x FOR PEER REVIEW 6 of 15 according to the ASTM D790 test method [22]. The illustration figure of the three-point bending test is shown in Figure 5. The geometry of the profile utilized in the validation test was similar to that of the non-stiffened tube profile. Figure 6 shows a picture of the experimental setup of the three-point bending test performed on the GFRP tube profile. The dial gauge (DDP-30A, Tokyo Sokki Kenkyujo Co., Ltd., Tokyo, Japan) was placed below the specimen to measure mid-span displacement of the specimen and a load cell (WF 17120, Wykeham Farrance, Milan, Italy) with 50 tons capacity was utilized. In the meantime, a high precision data logger (KL-10, Geomaster Group, Tianjin, China) was used to record the force and displacement data and a sampling rate of 1 data/sec was set to record the force-displacement information.  The material properties of the specimen employed in the validation test are identical to those presented in Table 1. The load-displacement relationship obtained from the experiment was approximately linear throughout the earlier stage. The maximum load recorded shortly before failure was 12.40 kN with a corresponding deflection of 11.4 mm at mid-span of the specimen. The experiment was then simulated by finite element analysis. The geometry of the specimen was first according to the ASTM D790 test method [22]. The illustration figure of the three-point bending test is shown in Figure 5. The geometry of the profile utilized in the validation test was similar to that of the non-stiffened tube profile. Figure 6 shows a picture of the experimental setup of the three-point bending test performed on the GFRP tube profile. The dial gauge (DDP-30A, Tokyo Sokki Kenkyujo Co., Ltd., Tokyo, Japan) was placed below the specimen to measure mid-span displacement of the specimen and a load cell (WF 17120, Wykeham Farrance, Milan, Italy) with 50 tons capacity was utilized. In the meantime, a high precision data logger (KL-10, Geomaster Group, Tianjin, China) was used to record the force and displacement data and a sampling rate of 1 data/sec was set to record the force-displacement information.  The material properties of the specimen employed in the validation test are identical to those presented in Table 1. The load-displacement relationship obtained from the experiment was approximately linear throughout the earlier stage. The maximum load recorded shortly before failure was 12.40 kN with a corresponding deflection of 11.4 mm at mid-span of the specimen. The experiment was then simulated by finite element analysis. The geometry of the specimen was first The material properties of the specimen employed in the validation test are identical to those presented in Table 1. The load-displacement relationship obtained from the experiment was approximately linear throughout the earlier stage. The maximum load recorded shortly before failure was 12.40 kN with a corresponding deflection of 11.4 mm at mid-span of the specimen. The experiment Sustainability 2018, 10, 4538 7 of 15 was then simulated by finite element analysis. The geometry of the specimen was first generated and the finite element model was properly restricted from translation along the y axis (vertical direction) at the supports. Figure 7 illustrates the y-direction displacement contour for the first selected load value namely 1.24 kN while Figure 8 compares load-displacement relationships of the experiment and finite-element analysis. The finite-element numerical results were in good agreement with the experimental data.
Sustainability 2018, 10, x FOR PEER REVIEW 7 of 15 generated and the finite element model was properly restricted from translation along the y axis (vertical direction) at the supports. Figure 7 illustrates the y-direction displacement contour for the first selected load value namely 1.24 kN while Figure 8 compares load-displacement relationships of the experiment and finite-element analysis. The finite-element numerical results were in good agreement with the experimental data.  generated and the finite element model was properly restricted from translation along the y axis (vertical direction) at the supports. Figure 7 illustrates the y-direction displacement contour for the first selected load value namely 1.24 kN while Figure 8 compares load-displacement relationships of the experiment and finite-element analysis. The finite-element numerical results were in good agreement with the experimental data.  To further confirm the accuracy of the finite element model, the numerical and analytical results of the three-point bending test for an applied load of 9.045 kN at mid-span of the profile were examined. The Euler-Bernoulli Beam Theory (EBT) and the Timoshenko Beam Theory (TBT) shown in Equations (3) and (4) were utilized. The TBT was also used to account for the influence of the shear deformation on deflection.
In Equations (3) and (4), δ max represents the maximum deflection of the square tube profile, while P is the concentrated load at mid-span (7.485 kN), L is the clear span length (100 cm), I is the section moment of inertia (119.9 cm 4 ), G is the shear modulus (3150 MPa), A is the cross sectional area (14.2 cm 2 ), and ν is the Poisson's ratio (0.33). A value of 29,584 MPa was computed from the experimental results and used for the Young's modulus E. The shear coefficient, κ =0.437, for a thin-walled tube was obtained by Cowper [23] as: Table 3 features the maximum deflection obtained from the experiment for a central load of 7.485 kN and those obtained from the finite element analysis, the EBT and TBT. It was shown that the deflection values calculated from the TBT and FEA are much closer to the experimental data than that of the EBT.

Finite-Element Analysis of Internally Stiffened GFRP Tubular Decks
Decks are subject to various types of loads, including dead load, traffic loads, earthquake loads, and wind loads. The combination of these loads induces bending and torsional loads on the decks. Bending and flexural rigidities are therefore important considerations during the design process of the decks. The bending and torsional deformability of a GFRP deck is mainly dependent on its bending and torsional stiffness. This, in turn, depends on the material properties, the cross section, and the length of the component. To increase the stiffness while maintaining the economic feasibility, choosing an efficient cross section is one of the best ways to proceed, and hence is the main object of this study.
After the validation of the finite element model established in the previous section, a mesh convergence study with the deflection as the criterion was performed on each of the specimens. Also, an appropriate number of elements was selected to model each deck panel to manage accuracy and computational efficiency. Figure 9 shows the result of one mesh convergence study (representative of the others) that was performed on the benchmark deck panel. For this case, based on evidence of the slope of the convergence curve being less than 5 × 10 −8 , the selected number of 29,640 elements was deemed sufficient to model the deck panel. The same process was followed to determine a suitable number of elements for each of the seven other deck panels. Figure 10 illustrates the discretization of each deck panel with the corresponding number of elements used, based on the results of each mesh convergence study. suitable number of elements for each of the seven other deck panels. Figure 10 illustrates the discretization of each deck panel with the corresponding number of elements used, based on the results of each mesh convergence study. After the accuracy of the finite element model was ensured through the convergence study performed on the benchmark deck panel, each of the seven proposed tube assemblies was then used to simulate the actual behavior of each of the tubular decks. The specimen was first tested in bending under the same experimental setting as the one in the validation test. The tubular decks were simply supported and subjected to three-point bending. A length of the tubular decks of 110 cm was used for the validation test specimen and the distance between the supports was 90 cm. The finite element models were properly restrained from translation in the y-direction at the supports. Then, a concentrated load was applied at the mid-span. Figure 11 illustrates the analysis setting for the bending test of the FRP deck. The resulting deflection values at the mid-span of each deck panel were recorded with the help of a y-direction deformation probe fitted at the center of the bottom face of the finite element models. suitable number of elements for each of the seven other deck panels. Figure 10 illustrates the discretization of each deck panel with the corresponding number of elements used, based on the results of each mesh convergence study. After the accuracy of the finite element model was ensured through the convergence study performed on the benchmark deck panel, each of the seven proposed tube assemblies was then used to simulate the actual behavior of each of the tubular decks. The specimen was first tested in bending under the same experimental setting as the one in the validation test. The tubular decks were simply supported and subjected to three-point bending. A length of the tubular decks of 110 cm was used for the validation test specimen and the distance between the supports was 90 cm. The finite element models were properly restrained from translation in the y-direction at the supports. Then, a concentrated load was applied at the mid-span. Figure 11 illustrates the analysis setting for the bending test of the FRP deck. The resulting deflection values at the mid-span of each deck panel were recorded with the help of a y-direction deformation probe fitted at the center of the bottom face of the finite element models. The specimens were then tested in torsion. The span of the tubular decks was again set to 100 cm, the tubular decks assemblies were cantilevered at one end and moments of increasing values were applied to the other end. Figure 12 illustrates the analysis setting for the torsion test of the GFRP tubular deck. The average rotation values were obtained by calculating the average values of the rotation about the z axis. The specimens were then tested in torsion. The span of the tubular decks was again set to 100 cm, the tubular decks assemblies were cantilevered at one end and moments of increasing values were applied to the other end. Figure 12 illustrates the analysis setting for the torsion test of the GFRP tubular deck. The average rotation values were obtained by calculating the average values of the rotation about the z axis.
The specimens were then tested in torsion. The span of the tubular decks was again set to 100 cm, the tubular decks assemblies were cantilevered at one end and moments of increasing values were applied to the other end. Figure 12 illustrates the analysis setting for the torsion test of the GFRP tubular deck. The average rotation values were obtained by calculating the average values of the rotation about the z axis.

Numerical Analysis Results and Discussion
Based on the finite-element numerical analysis results, we are able to distinguish which internally stiffened sections have better bending or torsional rigidity. Then, we can incorporate these GFRP deck sections in the engineering design and applications.

Numerical Analysis of the Bending Stiffness
In this analysis, eight tubular decks were evaluated including the non-stiffened tubular deck and seven proposed tubular decks with stiffeners. The loads applied to each tubular deck ranged from 0 to 46 kN. Figure 13 shows the plot of the applied loads against the corresponding deflection values recorded at the mid-span of each tubular deck. The bending stiffness was defined as the force divided by the displacement, i.e., the slopes of the curves shown in Figure 13; and the values are listed in Table 4. Table 4 lists the moment of inertia about the x axis as well as the increase of bending stiffness in each tubular deck, defined as the percentage increase of the stiffness of the tubular deck above the stiffness of the non-stiffened tubular deck.
An analysis of the results revealed that the tubular deck with the "O" type stiffener pattern performed the best in the bending test with an increase in bending stiffness of 44.39%. It is worth mentioning that other tubular decks, namely the "V" type and the "D" type tubular deck, share the characteristics of vertical support at their midsection and also possess diagonal components that enhance the overall diagonal rigidity of the structure. These performed much better than other tubular decks that only have either one of the two characteristics just mentioned. A previous study of the load transfer between the flanges and the webs, conducted by Valbona [24] on a deck with similar characteristics, revealed that due to the triangular configuration of the deck, the forces were mainly transferred through the web by truss action. This load transfer mechanism, contributing to reducing the global deflection of the tubular deck, could be a rational reason to explain the superior performances in the bending of the three best performing tubular decks compared to the four others. It is moreover important to notice that despite its significantly high moment of inertia, the "H" type tubular deck performed poorly as shown in Table 4.
to 46 kN. Figure 13 shows the plot of the applied loads against the corresponding deflection values recorded at the mid-span of each tubular deck. The bending stiffness was defined as the force divided by the displacement, i.e., the slopes of the curves shown in Figure 13; and the values are listed in Table 4. Table 4 lists the moment of inertia about the x axis as well as the increase of bending stiffness in each tubular deck, defined as the percentage increase of the stiffness of the tubular deck above the stiffness of the non-stiffened tubular deck.  An analysis of the results revealed that the tubular deck with the "O" type stiffener pattern performed the best in the bending test with an increase in bending stiffness of 44.39%. It is worth mentioning that other tubular decks, namely the "V" type and the "D" type tubular deck, share the characteristics of vertical support at their midsection and also possess diagonal components that enhance the overall diagonal rigidity of the structure. These performed much better than other tubular decks that only have either one of the two characteristics just mentioned. A previous study Figure 13. Plot of the force-displacement relationships at the mid-span of tubular decks.

Numerical Analysis of the Torsional Stiffness
The same eight FRP tubular decks were then evaluated in torsion by fixing them at one end and applying pure moments varying in magnitude from 0 to 500 N-m at the free end. Figure 14 shows the plot of the applied twisting moments versus the corresponding rotation angle values recorded at the free end of each tubular deck. The torsional stiffness was defined as the twisting moment divided by the corresponding rotation angle; and the values are listed in Table 5. Table 5 lists the polar moment of inertia about the z-axis together with the torsional stiffness increase of each tubular deck.
As was the case for the bending analysis, the "O" type deck panel demonstrated a superior performance as compared to other tubular decks considered in this study. This was evident by a torsional stiffness increase of about 80.08%. Aside from the "O" type deck panel, another tubular deck that stood out in terms of performance was the "D" type tubular deck, which also offered a torsional stiffness increase of around 71.29%. The superior performance of the "O" type and "D" type tubular deck in torsion can be ascribed to the circular pattern or closed shape, which allows a smooth shear flow through the section of the tubular deck.
Furthermore, the same observation in the bending analysis was made for the "H" type tubular deck concerning the disproportion that existed between its polar moment of inertia, which is the highest, and its relative torsional stiffness increase. The performance was the worst among the seven stiffened tubular decks considered in this study. The polar moment of inertia was a factor in the torsional rigidity of a panel, defined as the product of the polar moment of inertia (also referred to as torsion constant, J) and the modulus of rigidity (also called the shear modulus, G). The low shear stiffness of the section could most likely be the main contributor to the poor performance of the "H" type tubular deck as compared to the other stiffened tubular decks. tubular deck performed poorly as shown in Table 4.

Numerical Analysis of the Torsional Stiffness
The same eight FRP tubular decks were then evaluated in torsion by fixing them at one end and applying pure moments varying in magnitude from 0 to 500 N-m at the free end. Figure 14 shows the plot of the applied twisting moments versus the corresponding rotation angle values recorded at the free end of each tubular deck. The torsional stiffness was defined as the twisting moment divided by the corresponding rotation angle; and the values are listed in Table 5.  Figure 14. Plot of the twisting moment-rotation angle relationships at the free end of tubular decks. As was the case for the bending analysis, the "O" type deck panel demonstrated a superior performance as compared to other tubular decks considered in this study. This was evident by a torsional stiffness increase of about 80.08%. Aside from the "O" type deck panel, another tubular deck that stood out in terms of performance was the "D" type tubular deck, which also offered a torsional stiffness increase of around 71.29%. The superior performance of the "O" type and "D" type

Conclusions
From the above results, the following conclusions could be drawn: (1) For the 7.5 × 7.5 cm square tube profile with thickness 0.5 cm subjected to the three-point bending test, the finite-element numerical results were in good agreement with the experimental results.
The deflection values of various tubular decks calculated from the TBT are much closer to the experimental data than that of the EBT. Therefore, the shear deformation needs to be considered. (2) For the bending analysis, the tubular decks with stiffener patterns of the "O" type, "V" type, and "D" type show superior performance compared to others. Their bending stiffness compared to the non-stiffened benchmark tubular deck increased by 44.39%, 33.29%, and 33.26%, respectively. (3) For the torsional analysis, the tubular decks with stiffener patterns that form circular patterns or closed shapes inside each profile performed better than their counterparts. The tubular decks with stiffener patterns of the "O" type, and "D" type show superior performance compared to others. Their torsional stiffness compared to the non-stiffened benchmark tubular deck increased by 80.08%, 71.29%, and 33.26%, respectively.
(4) The "O" type tubular deck clearly showed excellent performance under the bending and torsional loadings. Compared to the non-stiffened benchmark tubular deck, the "O" type tubular deck showed an increase of 44.39% and 80.08% in bending and torsional stiffness, respectively. (5) Based on the finite-element numerical analysis results, it is possible to distinguish which internally stiffened sections have better bending or torsional rigidity. However, before extensively using the internally stiffened sections in engineering applications, these finite-element models should be carefully checked.