Research Article  Open Access
Naihui Yu, Jianzhong Shang, Yujun Cao, Dongxi Ma, Qiming Liu, "Comparative Analysis of AlLi Alloy and Aluminum Honeycomb Panel for Aerospace Application by Structural Optimization", Mathematical Problems in Engineering, vol. 2015, Article ID 815257, 12 pages, 2015. https://doi.org/10.1155/2015/815257
Comparative Analysis of AlLi Alloy and Aluminum Honeycomb Panel for Aerospace Application by Structural Optimization
Abstract
AlLi alloy and aluminum honeycomb panel (AHP) are both excellent materials for aeronautical structures. In this paper, a platetype aeronautical structure (PAS), which is a base mounting structure for 172 kg functional devices, is selected for comparative analysis with different materials. To compare systemlevel performance under multidisciplinary constraints, mathematical models for optimization are established and then structural optimization is carried out using Altair OptiStruct. For AHP, its honeycomb core is regarded as orthotropic material and its mechanical properties are calculated by Allen’s model in order to establish finite element model (FEM). The heights of facing sheet and honeycomb core are selected as design variables for size optimization. For AlLi alloy plate, topology optimization is carried out to obtain its most efficient load path; and then a reconstruction process is executed for practical manufacturing consideration; to obtain its final configuration, accurate size optimization is also used for reconstructed model of AlLi alloy plate. Finally, the optimized mass and performance of two PASs are compared. Results show that AHP is slightly superior to AlLi alloy.
1. Introduction
With the development of aerospace technology, the demand of highstrengthlowdensity materials is becoming more and more urgent. Severe mechanical environment and aerodynamic coupling are inevitable because of high launch acceleration and high frequency vibration, so the requirements for strength and stiffness of aeronautical structures are extremely high. Moreover, launch costs have strong restrictions on the overall mass of spacecraft, so the mass of aeronautical structures must be minimized as far as possible. Severe contradiction between strength and mass spurs extensive utilization of advanced alloy material and composite material in aerospace applications, such as aluminum lithium (AlLi) alloys [1], titanium alloy [2, 3], carbon fiber/epoxy composites, and aramid fiber/epoxy composites [4, 5]. The AlLi products offer opportunities for significant improvements in aerostructural performance through density reduction, stiffness increase, increases in fracture toughness and fatigue crack growth resistance, and enhanced corrosion resistance [6]. It has been found that addition of 1% of lithium to aluminum reduces alloy density up to 3% and increases modulus by 6% [7]. Besides, a basic trend towards increased utilization and integration of laminated anisotropic composites into the construction of aeronautical and aerospace vehicles has manifested in the last decade [8]. And sandwichtype honeycomb panel [9] which is a typical laminated anisotropic composite is an important alternative aeronautical material. Due to their high strengthtoweight ratio and stiffnesstoweight ratio, the use of honeycomb panels is particularly attractive in various aeronautical structures [10, 11].
Since highstrength alloy material and honeycomb panels are both excellent options for aeronautical structures, it is necessary to conduct a comparative analysis under same boundary conditions and load case. Currently, most researches concentrate on the performance of separate material [12, 13], but there are very few systematic studies on the performance comparison of AHP and AlLi alloy. Aeronautical structures usually have multidisciplinary requirements like minimum mass, high strength and stiffness, eased manufacturing and assembly techniques, and functional needs [14]. In the design of aeronautical structures, choosing what kind of materials is one of the key problems which must be considered firstly. However, it may be a very challenging task even for sophisticated designers only based on their professional knowledge and experience. The choice of materials directly determines the dimension design of structures, its processing methods, and even configurations of structures, especially when the honeycomb panels are in the scope of selection. As a laminated cellular structure, honeycomb panel is a type of structural material considering lightweight design and is a typical anisotropic material, so it cannot be directly compared with alloy material. Performance comparison is meaningful only when the optimal structures of two materials are obtained under the same conditions. Therefore, after the functional loads and performance constraints are almost confirmed, structural optimization technology is needed to obtain the eventual configurations.
In structural optimization, design variables can be categorized into three groups as topology, shape, and sizing variables. Topological design variables determine an initial structural layout whereas shape and sizing parameters give the shape and dimensions of structures, respectively [15]. The three levels of structure optimization are shown in Figure 1. Topology optimization is used in the conceptual design stage to obtain an initial structural configuration and to optimize material layout within a given design space for a given set of loads and boundary conditions such that the resulting layout meets a prescribed set of performance targets. Shape optimization belongs to preliminary design stage. By modifying the structural boundaries, for example, detailed designs for notches, holes, and fillets, concerned mechanical performances are improved during the optimization procedure [16]. Size optimization is used in a detailed design stage and to determine the ideal thickness of a material based on the performance goals and the forces expected to be placed on the component. In the field of aerospace or aeronautical industry, structural optimization technology is widely adopted for many years and is utilized in the design of most aeronautical structures, such as space station [17], aircraft [18], rockets [19], spacecrafts [20]. Maute and Allen presented a topology optimization methodology for the conceptual design of aeroelastic structures accounting for the fluidstructure interaction and the geometrical layout of the internal structure is optimized by material topology optimization [21]. A topology optimization method is proposed to minimize the resonant response of plates with constrained layer damping treatment under specified broadband harmonic excitations [22]. For honeycomb panels, Ermolaeva et al. [23] presented the application of a structural optimization system to the optimal choice of foams as a core material for sandwiches with aluminum alloy faces. Hansel and Becker [24] present a simple heuristic optimization algorithm implemented by ANSYSmacros to determine weightminimal laminate structures. However, there are no papers about systemlevel performance comparison of honeycomb panel and alloy material using structural optimization.
In the current paper, a PAS, which is a base mounting structure for 13 different functional devices, is selected for comparative analysis. Highstrength AlLi alloy and aluminum honeycomb panel are both ideal material for simple PAS, so it is very important for structure designers to determine which material is more superior. The commercial finite element package HyperMesh and OptiStruct 8.0 [25] are used for structural optimization. Final size optimization of facing sheet and honeycomb core heights could be directly executed for comparative analysis, as the configuration of honeycomb core is almost definite. The pure AlLi alloy plate needs topology and size optimization to obtain the optimized configuration. Mathematical models for optimization are established and the mass of structure is chosen as objective function. The rest of the paper is organized as follows. Section 2 briefly introduces the platetype aeronautical structure and its performance requirements considering the launch conditions. Section 3 introduces the structure of AHP and presents its calculation method of orthotropy mechanical properties based on Allen’s model, which is important to established its finite element model; the mathematical models for structural optimization are also presented for AHP and AlLi alloy. Section 4 presents the results of structural optimization and comparative analysis.
2. PlateType Aeronautical Structure and Its Performance Requirements
Figure 2 shows a typical PAS which is often used in a spacecraft. direction is rocket’s flight direction. Assume that 13 different function devices which form an independent subsystem are installed on this PAS and their total weight is 172 kg. After the layout of these function devices is determinate, the first thing which must be decided by the structural designers is to select materials for this PAS. According to the requirement of installing space, the dimension of PAS is confirmed and the width and the length are 980 mm and 1075 mm, respectively. Two lengths of this rectangle are fixed when the rocket launches. According to the rocket launching environment, performance constraints of the system are given and are shown in Table 1, where is 9.81 m/s^{2}.

(a) Front
(b) Back
Because the frequency range of harmonic response is 0 Hz~100 Hz, mechanical resonance will not happen as the fundamental frequency of subsystem is not less than 115 Hz. Therefore, harmonic response constraints need not be considered in the procedure of structural optimization and performance verifications should be carried out after optimized analysis. Lightweight design is one of the most important tasks of structure design for aerospace applications, so the mass of PAS should be selected as optimization objective and structural stress, deformation, and the first order modal as design constraints.
AHP and AlLi alloy are very different aeronautical materials. AHP is a kind of typical structured material and its configuration is almost determinate, so only size optimization is needed to obtain its optimized structure. Different optimization methods for AHP and AlLi alloy are shown in Figure 3. Software types used in the paper are SolidWorks for geometry creation, HyperMesh for meshing, OptiStruct for optimization, and HyperView for postprocessing. CAD models of function devices and PAS created in SolidWorks are imported into the HyperMesh for preprocessing. Preprocessing of models includes creating FE model, selection of material properties, creation of load, and applying boundary conditions on model. For AHP, HyperLaminate module is used to model this orthotropy material. For the optimization purpose, optimization criteria are selected. According to criteria, the designable and nondesignable portions are generated. The optimization process is executed in OptiStruct and the result can be viewed in HyperView.
3. Finite Element Models and Mathematical Models for Structure Optimization
3.1. Honeycomb Panels
Honeycomb panel, which is shown in Figure 4, is a sort of sandwich panel consisting of two alloy or composite plates as face sheet and hexagonal honeycomb cell as core materials, bonded together by bonding adhesive or macromolecule lamination film. In the sandwich construction, the facing sheets are spaced to provide most of the bending rigidity. They also resist all or nearly all the applied edgewise loads and flatwise bending moments. The core material spaces the facing sheets and transmits shear between them so that they are effective about a common neutral axis. The core also provides most of the shear rigidity of the construction. The coretofacing bonding adhesive must be adequate to transfer the stresses from the facing sheets to the core materials so that the full properties of the two are utilized [10].
As shown in Figure 4, is the height of hexagonal honeycomb core and is the height of facing sheet. For hexagonal cell, is cell wall thickness, is cell wall width, and is the diameter of inscribed circle of hexagon. In the process of optimization analysis, 2A12T4 aluminum alloy is selected as the material of facing sheet. Honeycomb core is aluminum hexagonal honeycomb 1/820240.003 of Hexcel Corporation, in which 1/8 (inch) represents the diameter of inscribed circle of hexagon, 2024 is its material grade, and 0.003 (inch) represents cell wall thickness. So the cell wall width could be simply calculated by the equation and its value is 1.83 mm after unit conversion. The mechanical properties of 2A12T4 and 2024 are shown in Table 2.

In order to establish finite element model of honeycomb panels, honeycomb core can be equivalent to orthotropic material and its mechanical properties be calculated by Allen’s model [26] which is widely used in mechanical engineering [27]:where is Young’s modulus of honeycomb core, is shear modulus, is a correction factor, and its value depending on the manufacturing process is generally 0.4~0.6. By the equivalent calculation, mechanical parameters of aluminum hexagonal honeycomb are shown in Table 3.

A finite element model of this subsystem is established using Altair OptiStruct. Honeycomb panel is meshed in HyperMesh using fournoded linear quad elements (CQUAD4) and composite properties are applied, where the core is orthotropic material and the sheets are isotropic material. All function devices are simplified by shell element and their mass is equivalent to the thickness of shells. Bolt connections are simplified as multipoint constraints and modeled by the flexible unit (RBE3). Two length of honeycomb panel are fixed in all six degrees of freedom. The finite element model is shown in Figure 5.
For structure optimization, performance constraints shown in Table 1 are used and the height of hexagonal honeycomb core and the height of facing sheet are selected as the design variables. The mass of subsystem is objective function. And mathematical models for honeycomb panel optimization could be described as where is the total system mass, is performance constraint, is the actual stress value while is yield strength of facing sheet material, is safety factor and the value is selected as 1.5 in this paper, is maximum deformation of PAS, is fundamental frequency of this system, is design variable and , and and are, respectively, lower and upper limit values of design variables and are shown in Table 4.

3.2. PAS of AlLi Alloy
The performance of frame structure largely depends on the properties of material. In this paper, AlLi alloy 2090 is selected for comparative analysis with honeycomb panel while all boundary conditions, input loads, and layout of function devices are identical. 2090 is a kind of high performance material and its elastic modulus increased by about 10% while the density decreased by about 10% compared with the conventional aluminum alloy. So its stiffnesstoweight ratio and strengthtoweight ratio are improved remarkably. The mechanical parameters of 2090 versus conventional aluminum alloy 2Al12 are shown in Table 5.

For a planeframe structure of pure AlLi alloys, topology optimization is firstly used for a conceptual design proposal and then finetuned for manufacturability. Initial thickness of the planeframe structure is a critical factor for topology optimization and the value of initial thickness directly determined the optimization result. Therefore, four values of initial thickness (40 mm, 50 mm, 60 mm, and 70 mm) are used, respectively, and the best thickness is determined by comparing the optimum results. At last, size optimization is executed to obtain a desired thickness of planeframe structure. Similarly with honeycomb panel, AlLi alloys plate is meshed with fournoded linear quad elements (CQUAD4) and the function devices are meshed in the same way. The finite element model is shown in Figure 6. In order to retain installation points of function devices, the meshed plate will be divided into designable and nondesignable portions. The geometry for the design space is defined by inspection of the bounds. As shown in Figure 5, green areas are selected as nondesignable portions and insure that all fixing points (yellow points) are included.
(a) Finite element model
(b) Designable and nondesignable portions
In the process of topology optimization of AlLi alloys plate, the initial thickness is defined and the design variable is element density of meshed plate . And mathematical model could be described as
After topology optimization, size optimization should be carried out to obtain a further optimized thickness of AlLi alloys plate, so the thickness is selected as design variable and its initial value would be the selected result of topology optimization. The mathematical model of size optimization is shown in
4. Optimization Results and Comparative Analysis
4.1. Optimization Results of Honeycomb Panel
After seven times of iteration, the results of structure optimization are obtained and shown in Figure 7. The ultimate dimensions of design variables and are 1 mm and 99.4 mm, respectively. The ultimate mass of subsystem is 213.40 kg and the mass of honeycomb panel is 41.40 kg. However, bolt sockets embedded in honeycomb panel are inevitable and their mass must be taken into account. The number of bolt sockets used to fix function devices by M5 bolt is 103 and the mass of each bolt socket is 0.005 kg. Similarly, the number of bolt sockets used to fix honeycomb panel by M10 bolt is 32 and the mass of each bolt socket is 0.025 kg. Therefore, the total mass of bolt sockets is 1.315 kg and the mass of honeycomb panel including bolt sockets is 42.72 kg, which is 24.84% of total mass of all function devices. The maximum von Mises stress and deformation of honeycomb panel under input loads of three directions are shown in Table 6.

(a) Iterations of core’s height
(b) Iterations of facing sheet’s height
(c) Iterations of subsystem’s mass
Modal analysis is carried out after structure optimization, and first two order modes of subsystem are shown in Figure 8. Natural frequencies of first eight order modes are shown in Table 7. Harmonic response analysis is also conducted using performance constraints shown in Table 1. Modal damping is a critical parameter for harmonic response. The damping coefficient of composite material is commonly 5%~8% and 6% is taken for honeycomb panel in this paper. 13 codes of each function device are selected as output nodes during harmonic analysis. The maximum acceleration response of all nodes in , , and directions is , , and , respectively. So the acceleration response cannot exceed the requirement of subsystem, namely, .

(a) First order mode
(b) Second order mode
4.2. Optimization Results of AlLi Alloy Structure
The topology optimization result using material distribution method is a density distribution of the finite elements in the design domain [16]. Density contours of AlLi alloy structure are shown in Figure 9. Different thicknesses, 40 mm, 50 mm, 60 mm, and 70 mm, are chosen to implement topology optimization.
(a) 40 mm
(b) 50 mm
(c) 60 mm
(d) 70 mm
The iterative process curves of AlLi alloy structure of different thickness are shown in Figure 10. The ultimate mass of 40 mm plate is 54.71 kg, 50 mm plate is 48.30 kg, 60 mm plate is 47.43 kg, and 70 mm plate is 48.54 kg. Therefore, the best thickness is 60 mm considering the light weight design. And a successive size optimization is necessary to obtain a more optimized thickness. Since boundaries of topology optimization result are discontinuous, the optimized structure cannot satisfy practical manufacturing requirements. So model reconstruction is needed according to density contour. The structure of topology optimization is most efficient load path for various constraints, so reconstructed model would inevitably lead to degrading the performance or augmenting the mass, or even degrading the performance while augmenting its mass. The reconstructed model of AlLi alloy plate considering manufacturing requirements is shown in Figure 11. All inclined beams or irregular structures are modified to straight ones. In order to enhance the strength of fixed lengths, all of the other color parts of two lengths are changed to red parts. Other details are also adjusted to meet the requirements of processing. The mass of reconstructed plate is 47.23 kg and slightly lower than original 47.43 kg. Fundamental frequency is 110.7 Hz, so the performance of reconstructed plate degraded and further optimization is needed.
Size optimization of reconstructed model is carried out based on the mathematical model presented in Section 3.2. After two times of iteration, the optimized thickness of AlLi alloy plate is got and the value is 62.54 mm. The mass of AlLi alloy plate increases to 48.39 kg and the fundamental frequency increases to 115.4 Hz, simultaneously. The maximum von Mises stress and deformation of AlLi alloy plate under input loads of three directions are shown in Table 8. Stress nephogram of subsystem in the direction is shown in Figure 12. The damping coefficient of AlLi alloy plate is supposed to be 3% and its maximum acceleration response of all function devices in , , and directions is , , and , respectively.

Modal analysis is carried out after size optimization, and first two order modes of subsystem are with AlLi alloy plate shown in Figure 13. Natural frequencies of first eight order modes are shown in Table 9.

(a) First order mode
(b) Second order mode
4.3. Comparative Analysis of Two Materials
After structural optimization, the optimized performance of two materials is obtained and shown in Table 10. All performance indices met the requirements and the eventual mass of APH is slightly lighter than AlLi alloy. The first order modals of two structures are both the vibration perpendicular to the plate face ( direction) and the deformation of direction is maximum. The maximum von Mises stress of AlLi alloy structure is much larger than that of honeycomb panel. Fundamental frequency and harmonic response are almost same. As can be seen from the comparison results, the performance of honeycomb panel is slightly superior to AlLi alloy.

5. Conclusion
In order to compare the performance of AlLi alloy and honeycomb panel which are both excellent materials for aeronautical structures, the mathematical models of structural optimization are established for these two distinctively different materials. Altair OptiStruct is used to carry out the optimized analysis. Honeycomb panel is seen as an integral structure and only size optimization is used to obtain its optimized heights. Topology and size optimization are applied to obtain the optimized configuration and thickness of the AlLi alloy plate. The mass of optimized AHP including bolt sockets is 42.72 kg and that of AlLi alloy plate is 48.39 kg, so AHP is superior to AlLi alloy for weight reduction consideration. The optimization procedure shows that structural optimization is an excellent method for material selection in the conceptual design phase.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors would like to express their appreciation to National Natural Science Foundation of China (Grants nos. 51175505 and 51305455) for providing the financial support to this work.
References
 C. Giummarra, B. Thomas, and R. Rioja, “New aluminum lithium alloys for aerospace applications,” in Proceedings of the 3rd International Light Metals Technology Conference, Bombardier Aerospace and Alcoa Trade Study, Québec, Canada, September 2007. View at: Google Scholar
 R. R. Boyer, “An overview on the use of titanium in the aerospace industry,” Materials Science and Engineering A, vol. 213, no. 12, pp. 103–114, 1996. View at: Publisher Site  Google Scholar
 M. Peters, J. Kumpfert, C. H. Ward, and C. Leyens, “Titanium alloys for aerospace applications,” Advanced Engineering Materials, vol. 5, no. 6, pp. 419–427, 2003. View at: Publisher Site  Google Scholar
 E. C. Botelho, R. A. Silva, L. C. Pardini, and M. C. Rezende, “A review on the development and properties of continuous fiber/epoxy/aluminum hybrid composites for aircraft structures,” Materials Research, vol. 9, no. 3, pp. 247–256, 2006. View at: Google Scholar
 P. D. Mangalgiri, “Composite materials for aerospace applications,” Bulletin of Materials Science, vol. 22, no. 3, pp. 657–664, 1999. View at: Publisher Site  Google Scholar
 R. J. Rioja and J. Liu, “The evolution of AlLi base products for aerospace and space applications,” Metallurgical and Materials Transactions A: Physical Metallurgy and Materials Science, vol. 43, no. 9, pp. 3325–3337, 2012. View at: Publisher Site  Google Scholar
 R. K. Gupta, N. Nayan, G. Nagasireesha, and S. C. Sharma, “Development and characterization of AlLi alloys,” Materials Science and Engineering A, vol. 420, no. 12, pp. 228–234, 2006. View at: Publisher Site  Google Scholar
 G. G. Galletti, C. Vinquist, and O. S. EsSaid, “Theoretical design and analysis of a honeycomb panel sandwich structure loaded in pure bending,” Engineering Failure Analysis, vol. 15, no. 5, pp. 555–562, 2008. View at: Publisher Site  Google Scholar
 J. K. Paik, A. K. Thayamballi, and G. S. Kim, “Strength characteristics of aluminum honeycomb sandwich panels,” ThinWalled Structures, vol. 35, no. 3, pp. 205–231, 1999. View at: Publisher Site  Google Scholar
 B. J. Kim and D. G. Lee, “Development of a satellite structure with the sandwich Tjoint,” Composite Structures, vol. 92, no. 2, pp. 460–468, 2010. View at: Publisher Site  Google Scholar
 G. Bianchi, G. S. Aglietti, and G. Richardson, “Development of efficient and costeffective spacecraft structures based on honeycomb panel assemblies,” in Proceedings of the IEEE Aerospace Conference, pp. 1–10, Big Sky, Mont, USA, March 2010. View at: Publisher Site  Google Scholar
 L. Librescu and T. Hause, “Recent developments in the modeling and behavior of advanced sandwich constructions: a survey,” Composite Structures, vol. 48, no. 1, pp. 1–17, 2000. View at: Publisher Site  Google Scholar
 Y. Lin, Z. Zheng, S. Li, X. Kong, and Y. Han, “Microstructures and properties of 2099 AlLi alloy,” Materials Characterization, vol. 84, pp. 88–99, 2013. View at: Publisher Site  Google Scholar
 A. R. Srinivas, Robust Design and Realization of Spacecraft Payload Elements Using Optistruct as an Optimization Tool, Space Application Centre ISRO, 2008.
 N. Noilublao and S. Bureerat, “Simultaneous topology, shape and sizing optimisation of a threedimensional slender truss tower using multiobjective evolutionary algorithms,” Computers & Structures, vol. 89, no. 2324, pp. 2531–2538, 2011. View at: Publisher Site  Google Scholar
 J. Zhu, W. Zhang, P. Beckers, Y. Chen, and Z. Guo, “Simultaneous design of components layout and supporting structures using coupled shape and topology optimization technique,” Structural and Multidisciplinary Optimization, vol. 36, no. 1, pp. 29–41, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 A. S. Gendy, S. N. Patnaik, D. A. Hopkins, and L. Berke, “Optimization of Space Station components using design code CometBoards,” Computer Methods in Applied Mechanics and Engineering, vol. 129, no. 12, pp. 133–149, 1996. View at: Publisher Site  Google Scholar
 S. Buchanan, “Development of a wingbox rib for a passenger jet aircraft using design optimization and constrained to traditional design and manufacture requirements,” in Proceedings of the CAE Technology Conference, April 2007. View at: Google Scholar
 P. L. Schoonover, W. A. Crossley, and S. D. Heister, “Application of a genetic algorithm to the optimization of hybrid rockets,” Journal of Spacecraft and Rockets, vol. 37, no. 5, pp. 622–629, 2000. View at: Publisher Site  Google Scholar
 E. R. Taylor, “Evaluation of multidisciplinary design optimization techniques as applied to spacecraft design,” in Proceedings of the IEEE Aerospace Conference Proceedings, vol. 1, pp. 371–384, IEEE, Big Sky, Mont, USA, March 2000. View at: Publisher Site  Google Scholar
 K. Maute and M. Allen, “Conceptual design of aeroelastic structures by topology optimization,” Structural and Multidisciplinary Optimization, vol. 27, no. 12, pp. 27–42, 2004. View at: Publisher Site  Google Scholar
 Z. Fang and L. Zheng, “Topology optimization for minimizing the resonant response of plates with constrained layer damping treatment,” Shock and Vibration, vol. 2015, Article ID 376854, 11 pages, 2015. View at: Publisher Site  Google Scholar
 N. S. Ermolaeva, M. B. G. Castro, and P. V. Kandachar, “Materials selection for an automotive structure by integrating structural optimization with environmental impact assessment,” Materials & Design, vol. 25, no. 8, pp. 689–698, 2004. View at: Publisher Site  Google Scholar
 W. Hansel and W. Becker, “Layerwise adaptive topology optimization of laminate structures,” Engineering Computations (Swansea, Wales), vol. 16, no. 7, pp. 841–851, 1999. View at: Publisher Site  Google Scholar
 Altair Engineering Inc, Optistruct 8.0 User Manual, Altair Engineering Inc, 2007.
 H. G. Allen, Analysis and Design of Structural Sandwich Panels, Pergamon Press, Oxford, UK, 1969.
 A. K. Noor, W. S. Burton, and C. W. Bert, “Computational models for sandwich panels and shells,” Applied Mechanics Reviews, vol. 49, no. 3, pp. 155–199, 1996. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2015 Naihui Yu 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.