Research Article  Open Access
Thi My Dung Do, Thanh Quang Khai Lam, Thi Thu Nga Nguyen, Van Thuc Ngo, Hoang Hung Vu, Trong Chuc Nguyen, Van Duan Doan, "Initial Parameters Affecting the Multilayer Doubly Curved Concrete Shell Roof", Advances in Civil Engineering, vol. 2021, Article ID 7999103, 18 pages, 2021. https://doi.org/10.1155/2021/7999103
Initial Parameters Affecting the Multilayer Doubly Curved Concrete Shell Roof
Abstract
Doubly curved shell roof structures have been widely studied and applied in civil buildings, because of their compressive capacity. As a spatial structure, it should increase the space, reduce the thickness of the shell, and create the architecture for the building. In particular, reinforcement is needed to repair the shell surface, forming the multilayer curved shell roof structures. In this multilayer curved shell roof, it is necessary to investigate the influence of thickness of layers, the influence of the location of the steel fibres concrete layer, and the influence of steel fibres content contained in concrete on the state of stress and strain and build relationships, loadvertical displacement and stress in the x and y directions of the shell in the investigated cases. So, this paper presents an ANSYS numerical simulation study related to the state of stress and strain in doublelayer doubly curved concrete shell roof with the initial parameters being changed such as the thickness of the layers, the location of the steel fibres concrete layer in the structure (the steel fibres concrete layer that is placed above and below the normal concrete layer), and the steel fibres content contained in concrete shell with the size of 3000 × 3000 mm, which is simulated by ANSYS after being experimentally conducted on this curved shell roof; the results of experimental and simulation study are verified by each other. Research results show that the thickness of the steel fibres concrete layer is placed below the normal concrete layer, the percentage of steel fibres contained in the concrete is 2%, and the bearing capacity of the curved shell is optimal.
1. Introduction
Vlasov, who laid the groundwork in his research on shell types, proposed the shell theory, which is composed of thin shells. With this shell theory, Vlasov proved that these shells are mainly compressive or tensile, and the bending moment in the shell is very small and can be ignored in calculations. Vlasov [1] built a system of 2 equations with two stress and displacement ( and ) functions with the load distributed over the shell surface.
Since then, plenty of shell studies have occurred, including shell analysis with the different curvature or seismic response of smart nanocomposite cylindrical shell, and using HDQNewmark methods in conveying fluid flow [2], cylindrical and spherical lattice shell [3, 4], the strainstress state of concrete dome structure simulated by ANSYS [5], and doubly curved shells with square and rectangular plane was studied by the approximation method; this method is developed on the basis of finite difference method that studied and developed by Gabbasov. In the 1980s, the essence of this method was to solve general quadratic differential equations. To solve this general differential equation, Gabbasov used the meshing method and thereby established the relationship between the points; from there, the results were converted from general differential equations to the system of linear equations for each point on the grid [6].
In addition, the shell structures have been studied by many authors by experimental method, finite element method [7, 8], by Sap2000 software [9], and by numerical simulation method, ANSYS, for umbrella shell in vibration analysis [10]; many authors have studied various parameters such as repairing and strengthening of elliptical paraboloid reinforced concrete shells [11], RFunicularity [12], prediction of the buckling behavior of shell [13], funicular concrete shell foundation under ultimate loading [14], classical shell theory for instability analysis of concrete pipes [15], bearing capacity of the initial imperfect lattice shell [16], optimized highperformance concrete shells [17], stability of steel shell structures [18], including large scale shell [19], and investigation of the geometrically nonlinear behavior of thinshell structures [20]. Through the analyzed studies, the studies analyzed the state of stress and strain and considered the parameters affecting the shells. However, these studies only studied singlelayer shells.
According to research on curved shells, these shells are damaged during use, need to be repaired, or need to be waterproof, heat resistant, or strengthened for these shells to increase bearing capacity and increase the life of the buildings, so they are necessary to add one or more layers above or below the existing shell layer. Ambartsumyan [21] has also suggested the multilayer shell hypothesis. Many studies have been done on various types of shells such as cylindrical shells, spherical shells, doubly curved shells, and elliptical paraboloid shells using analytic methods, numerical methods, finite element methods, approximation methods, and other methods based on Ambartsumyan’s laminated shell theory. Ambartsumyan assumed that layers were referred to equivalent layers according to their modulus of elasticity, such as the layerwise theory of laminated doubly curved shells [22], analysis of laminated doubly curved shells in both static and dynamic conditions [23], the bending and free vibration analysis of laminated composite shells [24], multilayer shell element for shear walls in open sees [25], stressstrain state of threelayered shallow shells [26], and meshreinforced Sandwich concrete shell roof [27]. Many researches have used the multilayer shell element in the finite element (FE) model nonlinear analysis to approach nonlinear studies [28]. The multilayer shell structure is also an applicability analysis on the multilayer shell structure system in construction, to test the theoretical method with multilayer structures that have been tested by the above theoretical methods [29].
In analyzing the structures of multilayer doubly curved shell roofs, many studies have used analytical methods or numerical methods to solve these curved shell problems. However, with the use of these curved shells in civil buildings, the material of the shell must be analyzed. Currently, steel fibres concrete has the ability to increase bearing capacity and reduce cracks and has good impact resistance; in addition, steel fibres concrete can also work quite well when steel fibres concrete is located in the tension region. In the studies on these steel fibres concrete materials, many studies have used them in flexural multilayer structures such as doublelayer beams or threelayer beams. The performance of normal concrete and steel fibres concrete in these laminated beams is quite suitable when repairing damaged concrete beams. These studies have conducted experiments and simulated by ANSYS these multilayer flexural beams [30, 31]. However, there are other studies on steel fibres concrete materials with nanosilica that can also be applied in this doubly curved shell experiment and simulation [32, 33].
Recently, this multilayer doubly curved shell roof is further studied when analyzing the nonlinearity of steel fibres concrete materials and the change of steel fibres content in concrete by ANSYS. The study determined the state of stress and strain in the shell and the sliding ability between two different concrete layers in the shell [34]. However, the shell material must be investigated in both the experimental analysis and ANSYS simulation for the laminated doubly curved shells. Because shell’s compressive capacity and reinforcement are needed to repair the shell surface, it is necessary to investigate the influence of the thickness of layers, the influence of the location of the steel fibres concrete layer, and the influence of the steel fibres content contained in concrete on the state of stress and strain and build relationships, loadvertical displacement and stress in the x and y directions of the shell in the investigated cases. So, this paper presents an ANSYS numerical simulation study related to the state of stress and strain of doublelayer doubly curved concrete shell roof with the initial parameters being changed such as the thickness of the layers, location of the steel fibres concrete layer (the steel fibres concrete layer (SFCL) above and below the normal concrete layer (NCL)), and steel fibres content contained in concrete shell with the size of 3000 × 3000 mm, which is simulated by ANSYS after being experimentally conducted on this curved shell roof; the results of experimental and simulation study are verified by each other.
2. Materials and Methods
2.1. Vlasov Equations in Calculating the Shell Roof Structures
Vlasov [1] built a system of 2 equations with two stress and displacement ( and ) functions with the load distributed over the shell surface (equation (1)):where is the shell thickness, E is modulus of elasticity, k_{1} and k_{2} are main curvature in 2 directions, and D is the flexural stiffness of the shell.
With the above assumption, for doubly curved shell with rectangular plan, Ambartsumyan [21] has introduced a system of equations to solve the problem of multilayer doubly curved concrete shell with two functions to find for the vertical load component case z (equation (2)):where and are parameters that depend on the geometrical size of the shell.
2.2. The DoubleLayer Doubly Curved Concrete Shell Roof Was Built
Before investigating the initial parameters affecting the state of stress and strain of the shell, this study has been conducted to build a doublelayer doubly curved shell model and conduct this shell experiment. The ANSYS simulation and experimental structure design model are followed by TCVN 5574:2018 [35], which is presented in Tables 1 and 2.


The equation for the surface of the doubly curved shell is shown in equation (3):where f is the height of the shell at its highest point, ; and are the height in both directions; a and b are half the length of the shell’s rectangular plan edge.
Doublelayer curved concrete shell roof with a plan size of 3000 × 3000 mm, fixed joint with 4 vertical curved beams, and a constant crosssectional area of 150 × 200 mm are supported on 4 columns with a cross section of 300 × 300 × 1300 mm, as shown in Figure 1.
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2.3. Making the DoubleLayer Doubly Curved Concrete Shell Roof
The shell fabrication process consists of 3 steps: make the formwork according to the shape of the doubly curved shell (equation (3) and Figure 1(a)), the first layer of steel fibres concrete was poured, and then the normal concrete layer is poured last, as shown in Table 1 and Figure 2.
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2.4. Installation of Devices and Shell Experiment
Installation diagrams of strain gauges (tensiometers) are measured at the top and bottom surfaces and vertical displacement measuring devices (LVDTs) are measured at the bottom surface of the shell. Because of the influence of measuring devices, the number of measuring positions is arranged as shown in Figures 3(a) and 3(b).
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In the condition of using this curved shell roof in the buildings in Vietnam, the load acting on the shell includes the load on the shell itself and the live load on repairing the shell, so the total effective load is 5 kN/m^{2} or 45 kN of load acting on the shell surface (shell surface area: 9 m^{2}). The total load acting on the shell is 5 kN/m^{2}, and depending on the weight of the sand or small concrete blocks, it has been divided into 7 levels in this study.
Load acting on the shell consist of 7 levels: level 1 = 1.63 kN/m^{2}, level 2 = 2.17 kN/m^{2}, level 3 = 2.72 kN/m^{2}, level 4 = 3.26 kN/m^{2}, level 5 = 4.35 kN/m^{2}, level 6 = 5 kN/m^{2}, and level 7 = 5.44 kN/m^{2}. At the first 4 levels, the load is applied to the shell by small sand and then concrete blocks, as shown in Figure 3(c). All measuring devices at each level are linked to the computers, as shown in Figure 3(d).
In addition, in the process of pouring concrete layer by layer, it is necessary to check the thickness of the layers, and after experimenting, drilling holes are made to check the actual layers, as shown in Figure 3(e).
2.5. Finite Element Model for DoubleLayer Doubly Curved Concrete Shell Roof in ANSYS
2.5.1. Elements and Models
In numerical simulation by ANSYS, this study used solid65 element to simulate concrete; this is an 8node element and easy to analyze material nonlinearly and can simulate steel fibres in concrete with a concrete coefficient; this concrete coefficient is applied to the concrete according to the volumetric content that the distributed steel fibres. In analyzing steel fibres in concrete, ANSYS also asked to choose a model for them. Because the steel fibres are distributed in the concrete, this study chooses smeared model in three models of ANSYS, smeared, embedded, and discrete models, as shown in Figure 4.
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According to the perimeter of the shell boundary, there are steel bars (Figures 1(b) and 1(c)), so BEAM188 element was used in simulation analysis and this is an element consisting of 2 nodes.
Concrete is a brittle material, so a concrete cracking model must be selected in the analysis. There are two concrete cracking models, discrete and smeared, but the steel fibres are distributed in the concrete, so in this study, the smeared model was selected and shown in Figure 5.
Simulating with two different materials, normal concrete (NC) and steel fibres concrete (SFC), three models are used, continuous model, interface model, and thinlayer model. However, in this research, the NC is on steel fibres concrete, only in contact with each other, so the interface model was chosen, as shown in Figure 6.
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2.5.2. Material Properties
In compression result of two materials, SFC and NC, in the experiment (Figure 7), the stressstrain relationship of these two types of concrete is consistent with Kachlakev’s model. Therefore, in this study, Kachlakev’s model was used in the following models: Hognestad, Todeschini, Kent and Park, and Kachlakev.
In the simulation, the failure surface model is also built, with the 2 types of concrete in this research and the selected solid65 element; the failure criteria of Willam and Warnke are appropriate, as shown in Figure 8.
Meshing: the meshing of elements in the shell and the nodes of the mesh must be together; if the mesh is too thick, the calculation time will be slow. With this thinshell structure, the mesh shape is divided by spatial tetrahedron, and the element size is half the thickness of the thinnest shell concrete layer as shown in Figure 9(a).
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Fixed link shell with vertical curved beams: the applied load is distributed on the top surface of the shell at the nodes of the tetrahedron mesh, by a compressive force P uniformly distributed over the shell surface, as shown in Figure 9(b).
3. Results and Discussion
3.1. Evaluation of Research Results
The experimental results of the shell roof are compared with ANSYS and Sap2000 software by values of vertical displacement and stresses; total effective load is 5 kN/m^{2}.(i)Load and vertical displacement relationship of the shell in methods is shown in Figure 10. Discussion of Result. The study results in Figure 10 show that ANSYS and Sap2000 are similar (here are the numeric methods), so when these numeric methods are compared to the experimental method, there is a difference near the shell boundary and at the top of the shell. There is a larger vertical displacement value near the shell boundary, but a smaller vertical displacement value at the top of the shell, and the value difference between the methods is very small (0.05 mm). As a result, the vertical displacement values of all three methods are similar.(ii)Load and stress relationship of the shell in methods is shown in Figure 11. Discussion of Result. In Figure 11, with the ANSYS and Sap2000 methods, the value of the stresses is similar. However, the value of the experimental method is changed at the location (x = 1.2 m) of the stress and at the location (x = 0.6 m) of the stress , compared with ANSYS and Sap2000 methods from 15 to 20 N/cm^{2}. These values may be affected by measuring devices, material homogeneity, and so on; given this difference between experimental and numerical methods, further studies are needed, but this change is very small and can be accepted in this study.
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The results of this experimental study show that this multilayer doubly curved shell roof from location 0.6 m to 1.2 m tends to be affected by boundary conditions, so it also needs further studies on the influence of these boundary conditions.
With the use of this shell roof under the conditions of use in Vietnam, cracks have not appeared in the shell, and it is necessary to continue research in ANSYS with larger load values.
3.2. Investigate the Effect of Thickness of Each Layer
In experimental research and simulated by ANSYS with SFCL (h_{1}), thickness of 2 cm, under NCL (h_{2}), thickness of 3 cm, and relationships loadvertical displacement and loadstress in the x and y directions have been built (Case 1). To investigate the influence of thickness of layers on the stressstrain state in the shell, this study continues with 2 investigated cases, Case 2 and Case 3, by ANSYS , as presented in Table 3.
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Discussion of Result(i)Case 1. With the load acting on the shell surface according to the conditions of use in Vietnam of 5 kN/m^{2}, the shell has no cracks. Experimentally, it is difficult to study the shell further, and ANSYS will simulate the shell at larger load levels. So, in the period when cracks appear in the concrete, P = 14 kN/m^{2}, stress 133.8 N/cm^{2}, and the first cracks appeared at the SFC layer and along the shell boundary with a vertical displacement at the top of the shell of 0.17 mm (Figure 12).(ii)Case 2. Similarly, in the period when cracks appear in the concrete, P = 12.5 kN/m^{2}, stress 139 N/cm^{2}, and the first cracks appeared at the SFC layer and along the shell boundary with a vertical displacement at the top of the shell of 0.187 mm (Figure 13).(iii)Case 3. Similarly, in the period when cracks appear in the concrete, P = 17.5 kN/m^{2}, stress 127.6 N/cm^{2} and the first cracks appeared at the SFC layer and along the shell boundary with a vertical displacement at the top of the shell of 0.207 mm
The results of vertical displacements and stresses in the 3 investigated cases are shown in Figure 14.
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Vertical displacement and stresses values are presented in Table 4.

Discussion of Result. In Figure 14 and Table 4,(i)Case 1 and Case 3: the total thickness is equal but the SFCL is larger, and the vertical displacement and stresses are smaller in the 3 investigated cases. In Figure 14(a), the thickness of the SFCL is increased, and the vertical displacement is reduced; that is, the larger the thickness of the SFCL is, the better the shell works.(ii)The stage where the concrete cracks appear: in Case 3, the underlying SFCL is thicker, so the cracks appear more slowly. In Case 2 (minimum thickness), cracks appear earlier than in other cases. That is, the underlying SFCL will resist the appearance of cracks in the shell when the load is increased.(iii)The research results show that the cracks in the three investigated cases are in the SFCL below it. That is, at the shell boundary, the lower layer of concrete is tensile, while the upper layer of normal concrete is compressive.
3.3. Investigate the Effect of the Location of the SFCL
The use of an SFCL under an NCL was studied. The SFCL is put on top of the NCL in other cases (such as when fixing the shell roof's surface). This study conducted two experiments to see how the location of the SFCL influences the state of stress and strain of the shell; the SFCL is located above and below the NCL, with the thickness of the layers in the shell being the same, as presented in Table 5.

The results of vertical displacements and stresses are shown in Figure 15.
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Vertical displacement and stresses values are presented in Table 6.

Discussion of Result.
In Figure 15 and Table 6, the concrete begins to show cracks; in Case 4, load P = 11.5 kN/m^{2}, stress is 102.7 N/cm^{2}, the first cracks appear in the shell along the boundary of the lower NCL, and the maximum vertical displacement at the top of the shell is 0.174 mm, while in Case 2, it is P = 12.5 kN/m^{2}; this means that the SFCL below NCL will crack later than it does on top, and in both Case 2 and Case 4, the concrete layer below will be cracked first.
When the SFCL is located in the compression zone of the shell, the stress near the shell boundary is reduced by 15%, and the stress is reduced by 52%. The results of this study show that the lower layer is the tensile layer, and the upper layer is compressive. Therefore, strengthening the SFCL located in the tensile zone will be more effective when it is located above.
3.4. Investigate the Influence of the Percentage of Steel Fibres Contained in Concrete
In Table 3, the percentage of steel fibres contained in concrete is 2%, and here it is changed to 0% and 3% for all 4 investigated cases . Case 1. This study investigates the change of the percentage of steel fibres contained in concrete, 0%, 2%, and 3%, as shown in Figure 16: Discussion of Result. Direction of movement of the layers in the shell: the upper layer displaces towards the center, while the lower layer is directed away from the center of the shell when the shell is under load (Figure 16(b)). In Figure 16(c), the change of vertical displacement value is 0.05 mm, and this value is very small and considered the same. Stress in the x and y directions (Figures 16(d), and 16(e)): these stresses are the same for the percentage of steel fibres in concrete 0% and 3% and vary from 9 N/cm^{2} to 13 N/cm^{2} when the percentage of steel fibres is 2%. Not using this percentage of steel fibres or using so much the percentage of steel fibres in concrete is inefficient and counterproductive, that is, 3% steel fibres that has surpassed the limit of the percentage of steel fibres in concrete [35]. Case 2. This study investigates the change of the percentage of steel fibres in concrete, 0%, 2%, and 3%, as shown in Figure 17: Discussion of Result. In Figure 17, similar to Case 1, vertical displacement and stresses and with 2% the percentage of steel fibres are the most optimal percentage of the studied percentage of steel fibres. The vertical displacement at the percentage of steel fibres in concrete is 2%, from 0% to 3% of 0.05 mm, which is very small and is considered unaffected. There is also the stress difference in the directions from 10 to 13 N/cm^{2} at the location near the shell boundary and at the top of the shell. The two analyzed cases (Case 1 and Case 2) showed that the locations near the boundary and at the top of the shell are the locations that will be sensitive to the load. When designing this shell, attention should be paid to these 2 locations. Case 3. This study investigates the change of the percentage of steel fibres in concrete, 0%, 2%, and 3%, as shown in Figure 18. Discussion of Result. In Figures 16–18, all 3 cases (Case1, Case 2, and Case 3) have the largest vertical displacement value at the top of the shell, and the difference is 0.9 mm. Stresses and are also the highest at the top of the shell and range from 29 to 39 N/cm^{2}. Among the percentage of steel fibres in concrete, the 2% steel fibres content is still the most optimal in all the investigated cases. The locations near the boundary and top of the shell are locations that are very sensitive to the load. Case 4. This study investigates the percentage of steel fibres in concrete, 0%, 2%, and 3%, as shown in Figure 19. Discussion of Result. In Figure 19, the vertical displacement of the shell is similar to the three cases above where the SFCL is placed on top of NCL. However, the stress values and start to change in the direction of increasing the stress value, and the stress at the top of the shell increases from 39 to 50 N/cm^{2}. The percentage of steel fibres in concrete with 2% is still the most optimal in all cases; however, the difference in stress between 2%, 0%, and 3% is 12 to 20 N/cm^{2}. The 4 cases that have been investigated show that the above SFCL will make the shell increase the stresses values in the directions.
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4. Conclusion
The results of the study lead to the following conclusions:(1)Research results of vertical displacement and stresses and in all 3 methods, experimental, ANSYS simulation, and Sap2000 with similar shapes: however, the experimental results are affected by boundary conditions, material homogeneity, and so on, further research is needed at the location from 0.6 m to 1.2 m to clarify these difference values. However, these difference values are not large and can be accepted in this study.(2)The influence of the thickness of the layers shows that the thickness of the SFCL is increased, and the vertical displacement is reduced; that is, the larger the thickness of the SFCL, the better the shell works. At the shell boundary, the lower layer of concrete is tensile, while the upper layer of NC is compressive.(3)The influence of the location of the SFCL in the shell: the first cracks appear in the shell along the boundary of the lower NCL, and the SFCL below NCL will crack later than it does on top, and in both Case 2 and Case 4, the concrete layer below will be cracked first. Therefore, strengthening the SFCL located in the tensile zone will be more effective when it is located above.(4)The influence of the percentage of steel fibres in concrete: not using this percentage of steel fibres, or using so much the percentage of steel fibres in concrete is inefficient and counterproductive. Among the percentage of steel fibres in concrete, the 2% steel fibres content is still the most optimal in all the investigated cases. The locations near the boundary and top of the shell are locations that are very sensitive to the load. The 4 cases that have been investigated show that the above SFCL will make the shell increase the stresses values in the directions.
Abbreviations
FE:  Finite element 
NC:  Normal concrete 
NCL:  Normal concrete layer 
SFC:  Steel fibre concrete 
SFCL:  Steel fibre concrete layer. 
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conﬂicts of interest regarding the publication of this paper.
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Copyright © 2021 Thi My Dung Do et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.