Elaboration of Design and Optimization Methods for a Newly Developed CFRP Sandwich-like Structure Validated by Experimental Measurements and Finite Element Analysis

Nowadays, the application of composite materials and light-weight structures is required in those industrial applications where the primary design aims are weight saving, high stiffness, corrosion resistance and vibration damping. The first goal of the study was to construct a new light-weight structure that utilizes the advantageous characteristics of Carbon Fiber Reinforced Plastic (CFRP) and Aluminum (Al) materials; furthermore, the properties of sandwich structures and cellular plates. Thus, the newly constructed structure has CFRP face sheets and Al stiffeners, which was manufactured in order to take experimental measurements. The second aim of the research was the elaboration of calculation methods for the middle deflection of the investigated sandwich-like structure and the stresses that occurred in the structural elements. The calculation methods were elaborated; furthermore, validated by experimental measurements and Finite Element analysis. The third main goal was the elaboration of a mass and cost optimization method for the investigated structure applying the Flexible Tolerance optimization method. During the optimization, seven design constraints were considered: total deflection; buckling of face sheets; web buckling in stiffeners; stress in face sheets; stress in stiffeners; eigenfrequency of the structure and constraints for the design variables. The main added values of the research are the elaboration of the calculation methods relating to the middle deflection and the occurred stresses; furthermore, elaboration of the optimization method. The primary aim of the optimization was the construction of the most light-weighted structure because the new light-weight sandwich-like structure can be utilized in many industrial applications, e.g., elements of vehicles (ship floors, airplane base-plate); transport containers; building constructions (building floors, bridge decks).


Introduction
Increasing market competition, fast changing customers' demands and the pandemic situation results that companies have to put emphasis on the application of advanced materials, innovative structures and modern manufacturing technologies in order to maintain their competitiveness.
• FRP composites have many advantageous properties compared to metal materials, which are the following: low density, high strength, high vibration damping, chemical and corrosion resistance, high bending stiffness, good thermal insulation, advantageous design versatility, etc. [1,2].
One of the most important characteristics of the FRP materials is their low density, which causes significant weight saving compared to traditional materials, such as steel. Therefore, the FRP composites are widely used in those industrial applications where the primary design aim is weight saving, e.g., structural elements of transport vehicles; transport containers; chemical vessels; building constructions (building floors, elements of bridges or warehouses, etc.) [3][4][5].
A new construction was developed, which consists of Al square hollow section stiffeners and CFRP face sheets. The face sheets are riveted to the stiffeners. The newly constructed structure was manufactured in order to take experimental measurements (the developed new sandwich-like structure is introduced in Section 2).
(2) The second aim of the research was the elaboration of calculation methods on the one hand for the middle deflection of the new investigated sandwich-like structure; on the other hand, for the stresses that occurred in the structural elements. Before the elaboration of the calculation methods relating to the sandwich-like structure, preliminary calculations and experimental measurements had to be achieved relating to the different structural elements (face sheets, stiffeners and rivets). These preliminary calculations and measurements were discussed in Section 2.1, Section 2.2, Section 2.3.
After it, calculation methods were elaborated for the middle deflection of the investigated new sandwich-like structure and the stresses that occurred in the structural elements (Section 3.1). Then experimental measurements (Section 3.2) and Finite Element analysis (Section 3.3) were carried out. The comparison of the results of the calculations, measurements and Finite Element analysis showed good agreements which confirmed the correctness of the elaborated calculation methods (Section 3.4).
(3) The third main goal was the elaboration of an optimization method for the investigated newly developed light-weight sandwich-like structure.
In Section 4, the elaborated optimization method is introduced. The optimization was achieved by the Flexible Tolerance optimization method (Section 4.4). The elaborated mass and cost objective functions were applied during the optimization (Sections 4.1 and 4.2). Furthermore, the following 7 design constraints were also considered (Section 4.3) during the optimization: (1.) total deflection; (2.) buckling of face sheet; (3.) web buckling in stiffeners; (4.) stress in the CFRP face sheets; (5.) stress in the stiffeners; (6.) eigenfrequency of the construction; (7.) size constraints relating to the design variables.
The cost objective function was elaborated based on the experiences gained during the manufacturing of the investigated structure. The elaborated calculation methods were applied during the formulation of the following design constraints: total deflection, stress in the face sheets and stress in the stiffeners.
• The main added value and novelty of the research are the following: (1) Calculation methods were elaborated on the one hand for the middle deflection of the newly constructed sandwich-like structure; on the other hand, for the stresses that occurred in the structural elements (CFRP face sheets and Al stiffeners). Then experimental measurements and Finite Element analysis were carried out which validated the correctness of the elaborated calculation methods.
(2) The other novelty of my research is that a mass and cost optimization method was elaborated relating to the newly constructed sandwich-like structure. The optimization was achieved by the Flexible Tolerance optimization method considering seven design constraints. The primary aim of the optimization was to construct a minimal weight structure.
It can be concluded that there is a gap in the existing literature, because there cannot be found any publication which discusses either the above-mentioned elaborated new calculation methods (middle deflection; stresses occurred in the structural elements) or the elaborated optimization method for the newly constructed sandwich-like structure.
Therefore, the results of the study fill the gap in the recent literature which provides new theoretical information for the researchers. At the same time the newly constructed light-weight composite sandwich-like structure can be widely used in the practice (e.g., element of vehicles; transport containers; building constructions) which can be utilized by the end users.

Materials and Methods-Structural Components of the Analyzed Sandwich-like Structure
The newly constructed sandwich-like structure consists of CFRP face sheets and two Al square hollow section stiffeners. The face sheets are riveted to the stiffeners ( Figure 1). of sandwich structures); furthermore, the stiffeners are made of metal (as in the case of cellular plates).
The elaborated single-cellular sandwich-like structure is shown in Figure 1. The face sheets are manufactured from 8 laminae, with the following layer sequence [0, +45, −45, 0]s ( Figure 2). The face sheets are riveted to the Aluminum square hollow section (SHS) stiffeners (30 mm × 30 mm × 2 mm).
The investigated structure (L = 1200 mm, bc = 220 mm) is simply supported. The load (F = 500 N) acts on the middle line of the construction as a uniformly distributed line-load.  Due to the complexity of the investigated structure, the mechanical properties of the structure are difficult to define applying numerical and analytical approximations. Therefore, experimental measurements and Finite Element simulations were carried out to validate the calculation methods and their results.

CFRP Face Sheets as Structural Elements
The newly developed sandwich-like structure consists of 3 structural elements: (1.) CFRP face sheets; (2.) two Al square hollow section stiffeners and (3.) rivets for joining the CFRP face sheets and the Al stiffeners.

CFRP Face Sheets-Calculated Elasticity Modulus and Flexural Modulus
The CFRP face sheets are manufactured from eight laminas with the following layer sequence [0, +45, −45, 0]s. The volume fraction of the carbon fiber is 55%, while the volume This structure is the combination of different materials (CFRP and Al) and structural elements (elements of sandwich structures and cellular plates). This construction is a sandwich-like structure, because the face sheets are made of FRP composite (as in the case of sandwich structures); furthermore, the stiffeners are made of metal (as in the case of cellular plates).
The elaborated single-cellular sandwich-like structure is shown in Figure 1. The face sheets are manufactured from 8 laminae, with the following layer sequence [0, +45, −45, 0] s ( Figure 2). The face sheets are riveted to the Aluminum square hollow section (SHS) stiffeners (30 mm × 30 mm × 2 mm). elements (elements of sandwich structures and cellular plates). This construct sandwich-like structure, because the face sheets are made of FRP composite (as in of sandwich structures); furthermore, the stiffeners are made of metal (as in the cellular plates).
The elaborated single-cellular sandwich-like structure is shown in Figure 1. sheets are manufactured from 8 laminae, with the following layer sequence [0, + 0]s ( Figure 2). The face sheets are riveted to the Aluminum square hollow sectio stiffeners (30 mm × 30 mm × 2 mm).
The investigated structure (L = 1200 mm, bc = 220 mm) is simply supported. T (F = 500 N) acts on the middle line of the construction as a uniformly distributed li  Due to the complexity of the investigated structure, the mechanical properti structure are difficult to define applying numerical and analytical approximations fore, experimental measurements and Finite Element simulations were carried ou idate the calculation methods and their results.

CFRP Face Sheets as Structural Elements
The newly developed sandwich-like structure consists of 3 structural eleme CFRP face sheets; (2.) two Al square hollow section stiffeners and (3.) rivets for joi CFRP face sheets and the Al stiffeners. The investigated structure (L = 1200 mm, b c = 220 mm) is simply supported. The load (F = 500 N) acts on the middle line of the construction as a uniformly distributed line-load.
Due to the complexity of the investigated structure, the mechanical properties of the structure are difficult to define applying numerical and analytical approximations. Therefore, experimental measurements and Finite Element simulations were carried out to validate the calculation methods and their results.

CFRP Face Sheets as Structural Elements
The newly developed sandwich-like structure consists of 3 structural elements: (1.) CFRP face sheets; (2.) two Al square hollow section stiffeners and (3.) rivets for joining the CFRP face sheets and the Al stiffeners.

CFRP Face Sheets-Calculated Elasticity Modulus and Flexural Modulus
The CFRP face sheets are manufactured from eight laminas with the following layer sequence [0, +45, −45, 0] s . The volume fraction of the carbon fiber is 55%, while the volume fraction of the Epoxy ES-67 matrix is 45% in a lamina. Of the carbon fibers, 84% are in longitudinal, while 16% of the fibers are in the transversal direction in a layer.
The material properties of the investigated laminate are the following: elasticity modulus of a CFRP layer in the longitudinal direction is E 1 = 54 GPa, elasticity modulus in transversal direction is E 2 = 40 GPa and the shear modulus is G 12 = 5.5 GPa. The CFRP Polymers 2021, 13, 4348 5 of 21 face sheet's density is ρ c = 1.5 · 10 3 kg/m 3 . The CFRP lamina's thickness is t* = 0.25 mm, the number of layers in the laminate is n = 8, the thickness of the laminate is t c (t c = nt*). The Poisson's ratio is ν 12 = ν f V f + ν m V m = 0.15 [41], where ν f and ν m mean Poisson's ratio relating to the fiber and the epoxy matrix components.
Furthermore, V f and V m mean the volume fractions relating to the fibers and the epoxy matrix. The density of the Al square hollow section (30 mm × 30 mm × 2 mm) stiffener is ρ Al = 2.7 · 10 3 kg/m 3 .

CFRP Face Sheets-Measured Elasticity Modulus and Flexural Modulus Determination of Elasticity Modulus Using Tensile Test
The Tensile tests of the CFRP test specimens (manufactured according to MSZ ISO 527:1993 standard) ( Figure 3) were performed by the MTS tensile testing machine which capacity is 250 kN and the accuracy is 0.4-0.5% ( Figure 4). number of layers in the laminate is n = 8, the thickness of the laminate Poisson's ratio is ν12 = νfVf + νmVm = 0.15 [41], where νf and νm mean Pois to the fiber and the epoxy matrix components.
Furthermore, Vf and Vm mean the volume fractions relating to the fi matrix. The density of the Al square hollow section (     The maximal tensile force (F max ) can be defined by the Tensile tests. Based on the test results the values of the tensile forces are near the same (9137 N) in the case of the 5 test specimens. The relevant parts of the Tensile tests' diagrams are those parts that show where the failures of the CFRP test specimens have occurred (the highest points of the diagrams). The further parts of the diagrams show the downfalls of the extensometer after the failures of the specimens. But these parts are not relevant from the aspect of the determination of the tensile strength (σ max ) ( Figure 5).
The tensile strength (σ max ) and the reduced modulus of elasticity (E xred ) of the CFRP laminate can be defined by the following equations: ymers 2021, 13, x FOR PEER REVIEW • The results of the tensile tests can be seen in Figure 5.  The maximal tensile force (Fmax) can be defined by the Tensile tests. Based on the te results the values of the tensile forces are near the same (9137 N) in the case of the 5 te specimens. The relevant parts of the Tensile tests' diagrams are those parts that sho where the failures of the CFRP test specimens have occurred (the highest points of th diagrams). The further parts of the diagrams show the downfalls of the extensometer aft the failures of the specimens. But these parts are not relevant from the aspect of the dete mination of the tensile strength (σmax) ( Figure 5).
• The results of the tensile tests can be seen in Figure 5. The tensile strength (σmax) and the reduced modulus of elasticity (Exred) of the CFR laminate can be defined by the following equations: • The results of the tensile tests can be seen in Figure 5. The result of the measurements is summarized in Table 1.
The result of the measurements is summarized in Table 1.

Determination of Flexural Modulus Using Three-Point Bending Test
The geometry of the bending test specimens (manufactured according to MSZ 892-78 standard) and the three-point bending test machine are shown in Figures 6 and 7.  • The results of the three-point bending tests can be seen in Figure 8.   Figure 6. Geometry of the bending test specimens.
The result of the measurements is summarized in Table 1.

Determination of Flexural Modulus Using Three-Point Bending Test
The geometry of the bending test specimens (manufactured according to M 78 standard) and the three-point bending test machine are shown in Figures 6 an  • The results of the three-point bending tests can be seen in Figure 8.  • The results of the three-point bending tests can be seen in Figure 8.
The result of the measurements is summarized in Table 1.

Determination of Flexural Modulus Using Three-Point Bending Test
The geometry of the bending test specimens (manufactured according to MSZ 892-78 standard) and the three-point bending test machine are shown in Figures 6 and 7.  • The results of the three-point bending tests can be seen in Figure 8.  The flexural strength (σ f max ) and the flexural modulus of the CFRP laminate (E f xred ) can be defined by the following equations: The result of the measurements is summarized in Table 1.

Comparison of the Calculated and Measured Data
The elasticity modulus and the flexural modulus of the CFRP laminate defined by the CLT theory were compared to the experimental results ( Table 1).
Comparison of the results shows good agreements between the results of the calculated values and the experimental measurements of the different elastic and flexural modulus of the investigated laminated face sheets. This agreement confirms the correctness of the applied calculation methods.

Aluminum Square Hollow Section Stiffeners as Structural Element
The geometry of the aluminum square hollow section stiffeners (SHS) applied in the investigated structure is the following: L = 1200 mm, h = 30 mm, t w = 2 mm. The material of the Al profile is AlMgSi05, the density is ρ Al = 2.7 × 10 −6 kg/mm 3 (Figure 9).
The flexural strength (σ f max) and the flexural modulus of the CFRP laminate (E f xred) can be defined by the following equations: The result of the measurements is summarized in Table 1.

Comparison of the Calculated and Measured Data
The elasticity modulus and the flexural modulus of the CFRP laminate defined by the CLT theory were compared to the experimental results (Table 1). Comparison of the results shows good agreements between the results of the calculated values and the experimental measurements of the different elastic and flexural modulus of the investigated laminated face sheets. This agreement confirms the correctness of the applied calculation methods.

Aluminum Square Hollow Section Stiffeners as Structural Element
The geometry of the aluminum square hollow section stiffeners (SHS) applied in the investigated structure is the following: L = 1200 mm, h = 30 mm, tw = 2 mm. The material of the Al profile is AlMgSi05, the density is ρAl = 2.7×10 −6 kg/mm 3 (Figure 9).

Rivets for Joining the CFRP Face Sheets and the Al Stiffeners-Shear Test of Rivets
The connection between the face sheets and the stiffeners is provided by riveting ( Figure 10).

Rivets for Joining the CFRP Face Sheets and the Al Stiffeners-Shear Test of Rivets
The connection between the face sheets and the stiffeners is provided by riveting ( Figure 10). Shear tests of the rivets were carried out to calculate the optimal number of rivets ( Figure 11). The geometry of a rivet is 4 mm × 10 mm. The required number of rivets can be defined from the shear strength of one rivet. A shear test was completed for more rivets (1, 2 and 3). The results of the tests showed that the shear strength is linearly increasing as a function of the rivets' number. Shear tests of the rivets were carried out to calculate the optimal number of rivets ( Figure 11). The geometry of a rivet is Ø4 mm × 10 mm. The required number of rivets can be defined from the shear strength of one rivet. A shear test was completed for more rivets (1, 2 and 3). The results of the tests showed that the shear strength is linearly increasing as a function of the rivets' number. Shear tests of the rivets were carried out to calculate the optimal number of rivets ( Figure 11). The geometry of a rivet is 4 mm × 10 mm. The required number of rivets can be defined from the shear strength of one rivet. A shear test was completed for more rivets (1, 2 and 3). The results of the tests showed that the shear strength is linearly increasing as a function of the rivets' number. Figure 11. Rivets' shear specimen.
The adequate distance of the rivets can be defined by the following formula: where: FS-shear force; h-flange width of the Al stiffener; τ-shear strength of one rivet (measured value: τ = 1.45 MPa); R  -safety factor (=1.5). Based on the calculation the minimal required distance between the rivets is 31 mm. The result of the calculation defines the required number of rivets that has to be applied in the case of the real manufactured sandwich-like structure.  (Figure 1.) is the sum of the deflection (w) calculated by Betti's Theorem and the deflection caused by the relative movement (Dw) between the structural elements (between the face sheet and the stiffener). So the total deflection can be calculated as follows:  Figure 11. Rivets' shear specimen.

Calculation Methods for the Investigated Sandwich-Like Structure
The adequate distance of the rivets can be defined by the following formula: where: F S -shear force; h-flange width of the Al stiffener; τ-shear strength of one rivet (measured value: τ = 1.45 MPa); γ R -safety factor (=1.5).
Based on the calculation the minimal required distance between the rivets is 31 mm. The result of the calculation defines the required number of rivets that has to be applied in the case of the real manufactured sandwich-like structure. The total middle deflection of the investigated simple supported sandwich-like structure (Figure 1.) is the sum of the deflection (w) calculated by Betti's Theorem and the deflection caused by the relative movement (∆w) between the structural elements (between the face sheet and the stiffener). So the total deflection can be calculated as follows:

Results-The Elaborated Calculation Methods for the Investigated Sandwich-like Structure and Validation by Experimental Measurements and Finite Element Analysis
where: M-maximal bending moment; E xred -reduced elasticity modulus of the face sheet in x (longitudinal) direction; E Al -elasticity modulus of the stiffener; I CFRP -inertia moment of the CFRP face sheet; I Al -inertia moment of the Al stiffener. The bending stiffness for the CFRP face sheet can be calculated by the following equation: The bending stiffness for the Al stiffener can be calculated by the following equation: where: b c -face sheets' width; t c -laminates' thickness; h-height of the Al square stiffener; t w -wall thickness of the Al stiffener. The measured stress data can be used to determine the relative movement' effect between the structural elements, and can be defined as a ratio of the differences in stresses in the center of the stiffener and the face sheet. The difference of stresses (∆σ = |σ Al − σ c |) has an effect on the equivalent applied moment (∆M). The relative movement caused by the sliding can be defined by the following equation: where the difference in stress ∆σ results in the difference in moment ∆M: Concluding, the calculated total middle deflection of the investigated simple supported sandwich-like structure in case of a load of 500 N-based on the following equationis 2.56 mm.

Calculation of Stresses Occurred in the Structural Components of the Analyzed Sandwich-like Structure
The applied load is distributed on the stiffeners and the face sheets. It has to be taken into consideration during the calculations of the moments and stresses that occurred in the structural elements.
This ratio can be defined by the ratio of the bending stiffness of the structural elements (B i ). It can be calculated by the following formula: where: B i -bending stiffness of the Al stiffeners or bending stiffness of the CFRP face sheets; and B total = E Al n s I Al + E xred I CFRP .
The ratios of the bending stiffness of the structural elements can be calculated by the following equations: X Al = E Al nI Al E Al n s I Al + E xred I CFRP , where: X C -ratio of the bending stiffness relating to the CFRP face sheet; X Al -ratio of the bending stiffness relating to the Al stiffener; n s -number of Al stiffeners (=2). Stresses occurred in the CFRP face sheet (σ c ) and the Al stiffener (σ Al ): The calculated stresses in the structural elements in case of a load of 500 N are the following: σ c = 8.434 MPa; σ Al = 11.503 MPa.

Experimental Tests Relating to the Investigated Sandwich-like Structure
Experimental tests were performed relating to the investigated new sandwich-like structure's middle deflection and relating to the stresses that occurred in the structural elements of the structure (CFRP face sheets and Al stiffeners).

Experimental Tests Relating to the Sandwich-like Structure's Deflection
The sandwich-like structure's middle deflection was measured applying a displacement meter in the Al SHS stiffeners' center lines and in the face sheet's center point ( Figure 12) in case of different loading conditions. Table 2 shows the measurements' results.

Experimental Tests Relating to the Investigated Sandwich-Like Structure
Experimental tests were performed relating to the investigated new sandwich-like structure's middle deflection and relating to the stresses that occurred in the structural elements of the structure (CFRP face sheets and Al stiffeners).

Experimental Tests Relating to the Sandwich-Like Structure's Deflection
The sandwich-like structure's middle deflection was measured applying a displacement meter in the Al SHS stiffeners' center lines and in the face sheet's center point ( Figure  12) in case of different loading conditions. Table 2 shows the measurements' results. The structure and its elements' initial imperfections result in a difference between the deflection data relating to the two Al profiles.    The structure and its elements' initial imperfections result in a difference between the deflection data relating to the two Al profiles.

Experimental Tests Relating to the Stresses Occurred in the Structural Elements of the Sandwich-like Structure
The stresses that occurred in the stiffeners and in the face sheets are measured applying strain gauges (accuracy is ±1%) at the investigated structure's 7 points using 7 strain gauges (7 channels, Table 3) (Figure 13). These 7 points are located in the stiffener's midpoint (1); in the midpoint of the face sheets' riveting line (2,4); in the face sheets' midpoints (3,6) and near to the face sheets' riveting line (5, 7) (in that line in which the longitudinal fibers are continuous, not cut). the Sandwich-Like Structure The stresses that occurred in the stiffeners and in the face sheets are measured applying strain gauges (accuracy is ± 1%) at the investigated structure's 7 points using 7 strain gauges (7 channels, Table 3) (Figure 13). These 7 points are located in the stiffener's midpoint (1); in the midpoint of the face sheets' riveting line (2,4); in the face sheets' midpoints (3,6) and near to the face sheets' riveting line (5,7) (in that line in which the longitudinal fibers are continuous, not cut).

Finite Element Analysis of the Examined Sandwich-Like Structure
The I-deas software was used to analyze the mechanical behavior of the investigated structure. The aim of the Finite Element (FE) analysis was the verification of the correctness of the elaborated calculation methods.
The I-deas FE software is suitable for 2 and 3 dimensional modeling, provides fast and reliable calculations and different evaluations of the gained results.
The structural elements of the investigated sandwich-like structure were defined by shell elements. The material properties were defined as an isotropic material, the laminated CFRP face sheets were defined as an orthotropic material. Eight nodes parabolic quadratic elements were applied during the FE grid generation both of Al and CFRP components. The joining of the CFRP face sheets and Al stiffeners were achieved in the same locations and modes (riveting) as in the case of the real experimental setup. In the riveting point translations in directions x, y and z are active, the rotations in directions x, y and z are inactive. Boundary conditions in the first edge (Tx,y,z = 0, Rx,y,z = active), in the other edge (Tx,y = 0, Tz = active, Rx,y,z = active).

Finite Element Analysis of the Examined Sandwich-like Structure
The I-deas software was used to analyze the mechanical behavior of the investigated structure. The aim of the Finite Element (FE) analysis was the verification of the correctness of the elaborated calculation methods.
The I-deas FE software is suitable for 2 and 3 dimensional modeling, provides fast and reliable calculations and different evaluations of the gained results.
The structural elements of the investigated sandwich-like structure were defined by shell elements. The material properties were defined as an isotropic material, the laminated CFRP face sheets were defined as an orthotropic material. Eight nodes parabolic quadratic elements were applied during the FE grid generation both of Al and CFRP components. The joining of the CFRP face sheets and Al stiffeners were achieved in the same locations and modes (riveting) as in the case of the real experimental setup. In the riveting point translations in directions x, y and z are active, the rotations in directions x, y and z are inactive. Boundary conditions in the first edge (T x,y,z = 0, R x,y,z = active), in the other edge (T x,y = 0, T z = active, R x,y,z = active). Figure 14 shows the FE grid of the examined sandwich-like structure, the joining (riveting) of structural components (laminated face sheets and stiffeners) and the boundary conditions.
Polymers 2021, 13, x FOR PEER REVIEW 13 o Figure 14 shows the FE grid of the examined sandwich-like structure, the join (riveting) of structural components (laminated face sheets and stiffeners) and the bou ary conditions.
Graphical and numerical evaluation of mechanical behavior of the structure in c of different loading conditions can be completed. In the case of the examined conditions (in case of the given loading case boundary conditions and geometry) the FE calculation includes more than 14 The program applied an iterative solution during the calculation. Figure 15 shows the print screen of 3D graphical evaluation of the def investigated sandwich-like structure in case of the test conditions. Graphical and numerical evaluation of mechanical behavior of the structure in case of different loading conditions can be completed.
In the case of the examined conditions (in case of the given loading case, loading and boundary conditions and geometry) the FE calculation includes more than 1400 unknown. The program applied an iterative solution during the calculation. Figure 15 shows the print screen of 3D graphical evaluation of the deflection of the investigated sandwich-like structure in case of the test conditions.  Different colors depict the different values of deflections that occurred in different parts of the structure. The maximal deflection of the structure occurred in the middle line of the span where the applied load acts. The scaling of the colors provides an accurate estimation of the deflection as can be seen in Figure 15. The numerical result of the FE analysis relating to the middle deflection can be seen in Table 4. The FE analysis was also completed for stress distribution of the structure (Figure 16). High level of stress occurred in the middle line and in the supporting points of the ends. The numerical results of the FE analysis relating to the stress in the CFRP face sheet and to the stress in the Al stiffener can be seen in Table 5. Different colors depict the different values of deflections that occurred in different parts of the structure. The maximal deflection of the structure occurred in the middle line of the span where the applied load acts. The scaling of the colors provides an accurate estimation of the deflection as can be seen in Figure 15. The numerical result of the FE analysis relating to the middle deflection can be seen in Table 4.
The FE analysis was also completed for stress distribution of the structure ( Figure  16). High level of stress occurred in the middle line and in the supporting points of the ends. The numerical results of the FE analysis relating to the stress in the CFRP face sheet and to the stress in the Al stiffener can be seen in Table 5.   The gained FE results are suitable for verification of calculated and measured data relating to deflection ( Table 4).
The small difference between the analytically calculated, experimentally measured and Finite Element deflections validates the analytical approximation and the elaborated calculation methods.
The calculated, measured and FE results relating to stress distribution were also compared ( Table 5).
It can be summarized that there is also a small difference between the three data. The conclusion of comparison and the differences of data confirm, that the elaborated calculation methods and the applied FE model are valid and relevant. It provides the possibility of application of the above-mentioned methods for more complex structures, e.g., multi-cellular sandwich-like structures.

Structural Optimization of the Investigated Sandwich-like Construction
The other main purpose of the study was to elaborate on the newly developed sandwich-like construction's optimization method. During the research, the cost and the mass objective functions; furthermore, seven design constraints were elaborated and applied. The design variables were the stiffeners' height (h) and the stiffeners' wall thickness (t w ) in case of the examined 4 different combinations of the CFRP layers in the laminate

Cost Objective Function
The total cost of the construction is the sum of the material costs of the structural elements (CFRP face sheets, Al stiffeners and rivets); the heat treatment costs of the CFRP face sheets and the manufacturing costs. The cost objective function can be formulated: The highest cost is the material costs of the CFRP laminates. In the case of the investigated laminate, the layer's specific material cost is C CFRP = 26 USD/m 2 . The heat treatment costs of CFRP face sheets is depending on the dimension of the face sheets and the characteristics of the Epoxy ES-67 resin. In the case of the investigated laminate, the heat treatment's total cost is USD 4. The specific material cost of the Al stiffener is c Al = 4.94 USD/kg. The rivet's specific material cost is c R = 0.01 USD/pcs. The n s is the number of stiffeners; the n R is the number of rivets. The specific manufacturing cost is c f = 0.6 USD/min. The construction's total manufacturing cost is the sum of the CFRP face sheets' manufacturing costs, the cutting costs of the Al stiffeners and the total cost of the assembly of the structural elements.
The manufacturing cost can be defined as the function of the lead times of the manufacturing activity (in minutes) joining to the manufacturing of the CFRP face sheets. This manufacturing time is including the procedure of the following manufacturing processes: press form preparation, cutting of the laminas, sequencing of the laminas and finishing processes. The other component of the total manufacturing time is the final assembly of the structural elements including the face sheets' and stiffeners' drilling; furthermore, the time consumption of the riveting.

Mass Objective Function
The most important design aim-in the case of the application of FRP composite materials-is the reduction of the total weight of the construction.
The analyzed sandwich-like structure's total weight is the sum of the structural elements' weights (face sheets, stiffeners and rivets) (Figure 1).

Design Constraints
During the structural optimization the following seven design constraints were considered: 1. Total deflection of the structure, which can be calculated by the following equation: Polymers 2021, 13, 4348

Results of the Structural Optimization for Different Face Sheets' Layer Sequences
The optimization was carried out for four different layer sequences of the face sheets. These four different face sheets are the following: the manufactured [0 • , +45 • , −45 • , 0 • ] s ; [0 • , 90 • ] s ; [0 • , 90 • , 0 • ] and [0 • , 0 • ] layer sequences. The design variables were the stiffeners' height (h) and the stiffeners' wall thickness (t w ) in the case of the examined 4 different combinations of the CFRP layers. The mass and the cost structural optimization were carried out for the before mentioned 4 different face sheets (Figures 17 and 18).  The results of the optimization show that the most light-weighted sandwich structure (2.498 kg) can be constructed by the application of a CFRP laminate which layers; furthermore, the layer sequence is the following [0°, +45°, −45°, 0°]s. The op geometry of the Al stiffeners is 25 mm × 25 mm × 1.5 mm (Table 6, Figure 18).
The optimization results also show that the most cost-effective sandwich-like s ture (USD 180.958) can be constructed by the application of a CFRP laminate which layers; furthermore, the layer sequence is the following [0°, 90°, 0°]. The optimal geom of the Al stiffeners is 80 mm × 80 mm × 4 mm (Table 6, Figure 18).

Conclusions
• The main findings and the results of the research are the following: (1) A new light-weight sandwich-like structure was constructed, which consis CFRP face sheets and Al stiffeners (Figure 1). The face sheets are riveted to the stiffe This new structure was manufactured in order to take experimental measurements ure 12).
The new sandwich-like structure is the combination of different materials (CFRP Al) and structural elements (elements of sandwich structures and cellular plates); bec the face sheets made of CFRP composite (as in the case of sandwich structures); fur more, the stiffeners made of metal (as in case of cellular plates).
(2) Before the elaboration of the calculation methods for the newly constructed s   The results of the optimization show that the most light-weighted sandwich-like structure (2.498 kg) can be constructed by the application of a CFRP laminate which has 8 layers; furthermore, the layer sequence is the following [0°, +45°, −45°, 0°]s. The optimal geometry of the Al stiffeners is 25 mm × 25 mm × 1.5 mm (Table 6, Figure 18).
The optimization results also show that the most cost-effective sandwich-like structure (USD 180.958) can be constructed by the application of a CFRP laminate which has 3 layers; furthermore, the layer sequence is the following [0°, 90°, 0°]. The optimal geometry of the Al stiffeners is 80 mm × 80 mm × 4 mm (Table 6, Figure 18).

Conclusions
• The main findings and the results of the research are the following: (1) A new light-weight sandwich-like structure was constructed, which consists of CFRP face sheets and Al stiffeners (Figure 1). The face sheets are riveted to the stiffeners. This new structure was manufactured in order to take experimental measurements (Figure 12).
The new sandwich-like structure is the combination of different materials (CFRP and Al) and structural elements (elements of sandwich structures and cellular plates); because the face sheets made of CFRP composite (as in the case of sandwich structures); furthermore, the stiffeners made of metal (as in case of cellular plates).
(2) Before the elaboration of the calculation methods for the newly constructed sand-  The optimization results are tabulated in Table 6 which are obtained by the application of the Flexible Tolerance optimization method. During the optimization the mass and the cost objective functions (Equations (19) and (20)); furthermore, the before mentioned 7 design constraints (Equations (21)-(29)) were considered. The same optimal heights and wall thicknesses are obtained for the optimal Al stiffeners in the case of both objective functions for each of the different face sheets' layer sequences. It means that the obtained optimal constructions in case of the different layer sequences represent the most lightweighted and the most cost-effective, economic structures.
The most often used fiber orientations are the 0 • , 45 • and 90 • in industrial applications. The results of the preliminary calculations and the optimization ( Table 6) showed that the [0 • , +45 • , −45 • , 0 • ] s layer sequence provides the most light-weighted sandwich-like structure. The primary aim of the optimization was weight saving. It was the reason that the above-mentioned layer combination was manufactured and investigated during the measurements. The results of the optimization show that the most light-weighted sandwich-like structure (2.498 kg) can be constructed by the application of a CFRP laminate which has 8 layers; furthermore, the layer sequence is the following [0 • , +45 • , −45 • , 0 • ] s . The optimal geometry of the Al stiffeners is 25 mm × 25 mm × 1.5 mm (Table 6, Figure 18).
The optimization results also show that the most cost-effective sandwich-like structure (USD 180.958) can be constructed by the application of a CFRP laminate which has 3 layers; furthermore, the layer sequence is the following [0 • , 90 • , 0 • ]. The optimal geometry of the Al stiffeners is 80 mm × 80 mm × 4 mm (Table 6, Figure 18).

Conclusions
• The main findings and the results of the research are the following: (1) A new light-weight sandwich-like structure was constructed, which consists of CFRP face sheets and Al stiffeners ( Figure 1). The face sheets are riveted to the stiffeners. This new structure was manufactured in order to take experimental measurements ( Figure 12).
The new sandwich-like structure is the combination of different materials (CFRP and Al) and structural elements (elements of sandwich structures and cellular plates); because the face sheets made of CFRP composite (as in the case of sandwich structures); furthermore, the stiffeners made of metal (as in case of cellular plates).
(2) Before the elaboration of the calculation methods for the newly constructed sandwich-like structure, preliminary calculations and experimental measurements ( Figures 5, 8 and 11) had to be carried out relating to the structural elements (face sheets, stiffeners and rivets).
Then calculation methods were elaborated for the middle deflection of the new sandwichlike structure; at the same time for the stresses that occurred in the structural elements.
After it experimental measurements (Tables 2 and 3) and Finite Element analysis (Figures 15 and 16) were achieved. The results of the calculations, measurements and Finite Element analysis were near the same which confirmed the correctness of the elaborated calculation methods (Tables 4 and 5).
(3) Mass and cost optimization methods were elaborated for the investigated construction. During the optimization the Flexible Tolerance optimization method was applied considering the following seven design constraints: total deflection; buckling of face sheet; web buckling in stiffeners; stress in CFRP face sheets; stress in Al tubes; construction's eigenfrequency; constraints relating to the design variables. The results of the mass and the cost optimization were summarized in Table 6.
• The main added values of the research are the following: (1) The novelty of the study is the elaboration of the calculation methods on the one hand for the middle deflection of the investigated new sandwich-like structure; on the other hand, for the stresses that occurred in the structural elements. The elaborated calculation methods were validated by experimental measurements and Finite Element analysis.
(2) The other main contribution of the research is the elaboration of the mass and cost optimization method considering seven design constraints applying the Flexible Tolerance optimization method. But the primary aim of the optimization was to construct the most light-weighted structure.
The newly constructed light-weight sandwich-like structure can be widely used in many industrial applications where the primary design aim is weight saving, e.g., elements of vehicles (ship floors, airplane base plate, etc.); transport containers; building constructions (building floors, bridge decks, etc.).
In future research-based on the elaborated calculation methods-more complex structures can be investigated and optimized for other engineering applications. Furthermore, additional design constraints and other structural elements can be used during the optimization.