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A study on composite honeycomb sandwich panel structure
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Materials & Design
Materials and Design 29 (2008) 709–713 www.elsevier.com/locate/matdes
Short Communication
A study on composite honeycomb sandwich panel structure Meifeng He, Wenbin Hu
*
State Key Laboratory of Metal Matrix Composites, Shanghai Jiao Tong University, Shanghai 200030, PR China Received 23 May 2006; accepted 2 March 2007 Available online 19 March 2007
Abstract Honeycomb sandwich structure combines high flexural rigidity and bending strength with low weight. Sandwich construction plays an increasing role in industry, and sandwich structural designing is an available method for sandwich structures. However, the absence of the design variable is the principal problem of composite sandwich construction. In this paper, the structure and mechanical properties of honeycomb sandwich panels are introduced. The weight ratio range of honeycomb core that is deduced on the basis of optimum mechanical properties offer a principle foundation for designing the structure of honeycomb sandwich panels. The satisfying weight condition of the honeycomb core weight is 50–66.7% of the weight of the whole honeycomb sandwich panels by theoretical analysis. Based on that conclusion, the honeycomb sandwich panels were designed and the results were verified by further experiments. Agreement between the theoretical values of the sample and experimental results is good. 2007 Elsevier Ltd. All rights reserved.
1. Introduction Honeycomb sandwich materials are being used widely in weight sensitive and damping structures where high flexural rigidity is required, in many fields especially in the automobile industry [1–3]. Honeycomb core sandwich panel is formed by adhering two high-rigidity thin-face sheets with a low-density honeycomb core possessing less strength and stiffness. By varying the core and the thickness and material of the face sheet, it is possible to obtain various properties and desired performance, particularly high strength-to-weight ratio [4]. Several types of core shapes and core material have been applied to the construction of sandwich structures [5,6]. In this paper, the honeycomb core that consists of very thin mild-steel panel in the form of regular hexagonal cells perpendicular to the facings was applied. Whereas its honeycomb core must be stiff enough to prevent one face panel of the honeycomb sandwich panel from sliding over the other, when a honeycomb sandwich panel subjected to flexural loading. Such rigidities are called the transverse shear stiffness of the honeycomb core. *
Corresponding author. Tel.: +86 21 6293 3585; fax: +86 21 6282 2012. E-mail address: [email protected] (W. Hu).
0261-3069/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2007.03.003
When the core depth is much larger than the thickness of the face panels (most honeycomb sandwich panels belong to this category), the transverse shear stiffness of the sandwich plate is almost entirely contributed by its core. For simplicity and efficiency, the cellular honeycomb core is idealized as a homogeneous material and its equivalent mechanical properties are used in analysis and design. Therefore, the knowledge of the equivalent transverse shear stiffness of honeycomb is very important for the analysis and design of sandwich plates [7]. Even if the concept of sandwich construction is not very new, it has primarily been adopted for non-strength part of structures in the last decade. This is because there are a variety of problem areas to be overcome when the sandwich construction is applied to design of dynamically loaded structures. To enhance the attractiveness of sandwich construction, it is thus essential to better understand the local strength characteristics of individual sandwich panel members. To optimize further the structural design, the present trend is to apply the principle of flexural rigidity and bending stiffness of sandwich plates. The weight ratio range of honeycomb core that is deduced on the basis of optimum mechanical properties offer a principle foundation for designing the structure of honeycomb sandwich
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M. He, W. Hu / Materials and Design 29 (2008) 709–713
Nomenclature r1, r2 t1, t2 E1, E2 Ef e1, e2 b h hf hc
yield stress of two facing material thickness of two facing skins (assuming that both skins are of same each other) elastic (Young’s) modulus of two facing material elastic (Young’s) modulus of facing material strain of two facing skins breadth of the sandwich panel height of sandwich panel including facing skins thickness of facing skins height of honeycomb core
Z Ieq D M rmax Dmax Mmax y qf qc
distance from facing skin to centroidal axis equivalent sectional inertia moment flexural rigidity flexural torque maximum tensile strength of facing skin maximum flexural rigidity maximum flexural torque half height of sandwich panel including two facing skins density of facing material density of honeycomb core material
panels. Noteworthy, the experimental data agree satisfactorily with the theoretical predictions. The useful results are extended to give a available design tool for sandwich panels manufacturers. 2. The mechanical properties of honeycomb sandwich panels Honeycomb sandwich structure possesses high flexural rigidity and bending strength with low weight. Flexural rigidity and bending strength are very important mechanical properties of honeycomb sandwich panels. The underlying assumptions of the theory presented in the deducing procedure are as follows [8]: (1) The faceplates are very thin compared with the total thickness of the sandwich. (2) The shear stress over the depth of the core is considered constant. This assumption may be made if the core is too weak to provide a significant contribution to the flexural rigidity of the sandwich. (3) The honeycomb construction is anisotropy. (4) The rotary inertia of the face plates about their own centroidal axes is negligible. (5) The sandwich has equal thickness face plates (symmetric sandwich) and both face plates are of the same material with identical properties.
Fig. 1. Calculated sectional area and equivalent sectional area.
r1 t1 ¼ r2 t2 : Then, r1 t1 ¼ : r2 t2
ð1Þ
ð2Þ
The Eq. (3) can be achieved by applying Hooke’s law. r1 r2 ð3Þ e1 ¼ ; e2 ¼ ; E1 E2 so,
2.1. Flexural rigidity Honeycomb core was used to apart one face sheet from another in honeycomb sandwich structure. Because of this special structure the sectional inertia moment of honeycomb sandwich panel was increased and induced an obvious increase of the flexural rigidity of panels [9,10]. The flexural rigidity of panels is one of the most important parameters in structural design. The calculated sectional area, as shown in Fig. 1, are assumed that the sectional area of two facing sheets are both effective. According to the mechanical equilibrium condition, the equality can be obtained as follow:
e1 r1 E2 t2 E2 ¼ ¼ : e2 E1 r2 t1 E1
ð4Þ
The distance from facing skin to centroidal axis Z of sandwich plate has been noted. There also have several equalities. c hþhc e2 hþh e2 e1 2 ; Z¼ ð5Þ ¼ hþhc ¼ e1 2 : e1 þ e2 þ1 Z Z e2 2 When Eq. (4) is used in Eq. (5), so that Eq. (5) reduces to c E1 t1 hþh 2 Z¼ : ð6Þ E2 t2 þ E1 t1
M. He, W. Hu / Materials and Design 29 (2008) 709–713
As shown in Fig. 1.The equivalent sectional inertia moment Ieq is given by 2 h þ hc E2 t3 1 E2 3 Z þ t2 Z 2 þ 1 þ t I eq ¼ t1 2 E1 12 12 E1 2 " # 2 h þ hc h þ hc E2 2 ¼ t1 Z Z þ t2 Z 2 2 2 E1 1 3 E2 3 t þ t : þ ð7Þ 12 1 E1 2 When Eq. (6) is used in Eq. (7), so that Eq. (7) reduces to 2 h þ hc E2 t2 t1 1 3 E2 3 t1 þ t2 : I eq ¼ þ ð8Þ 12 2 E1 t1 þ E2 t2 E1 The flexural rigidity D of the sandwich panels is also given by ð9Þ
D ¼ Ef I eq :
Assuming that both faces are of the same each other, the facing material and the facing thickness are same. So we can obtain E1 = E2 = Ef and t1 = t2 = hf. In many practical sandwiches application, hf/hc lies in the range from 0.02 to 0.1. The limiting conditions of t1 = t2 hc and h hf hc always exist. When above several conditions are satisfied in these cases, so that Eq. (9) reduces to 2 2 h þ hc ð E f hf Þ D ¼ E1 I eq ¼ 2 2Ef hf 2 1 h þ h 2hf 1 2 ¼ E f hf ¼ Ef hf ðh hf Þ 2 2 2 1 Ef hf h2c : ð10Þ 2 The expression of the flexural rigidity D of honeycomb sandwich panels is 1 D ¼ Ef hf h2c : 2
ð11Þ
2.2. Bending strength The honeycomb sandwich structure does not get shear fracture when one face sheet is tensioned and the other is compressed, which is the optimal flexural failure of the sandwich panels. This makes the honeycomb sandwich structure can get better performed under the same mechanical condition. The fracture of the face sheet is the optimal flexural failure. According to the theory of mechanics of materials, the equality can be obtained as follow: rmax
M max ¼ y; I eq
rmax
Ef M max Ef M max y: ¼ y¼ Ef I eq D
ð12Þ
Then, M max ¼
rmax D : Ef y
ð13Þ
711
c h2c always exists The limiting condition of y ¼ hf þh 2 because the thickness of facing skin is very thin. When above condition is satisfied in this case, so that Eq. (13) reduces to
M max ¼
rmax 12 Ef hf bh2c ¼ rmax hf hc b: Ef h2c
ð14Þ
3. Deducing the weight range of honeycomb core on the basis of optimum mechanical properties 3.1. Deducing from the flexural rigidity From Eq. (11) we have already gotten the expression of the flexural rigidity D of honeycomb sandwich panels. 1 D ¼ Ef hf h2c : 2 The mass area ratio W of the honeycomb sandwich panels is associated with the density of facing material qf and the density of honeycomb core material qc, as given by W ¼ 2qf hf þ qc hc ;
ð15Þ
where for convenience we write x ¼ qc hc =W ;
ð16Þ
where x is the weight ratio of honeycomb core, then we can obtain the follow equality from Eqs. (15) and (16) hf ¼
W ð1 xÞ: 2qf
ð17Þ
Eqs. (16) and (17) provide two homogeneous equalities for the unknowns hf and hc in Eq. (11), the flexural rigidity D can then be determined 2 E f W W D ¼ x2 ð1 xÞ: ð18Þ 2 qc 2qf When the flexural rigidity D and the weight ratio of honeycomb core x satisfy the condition of dD ¼ 0, the maximum dx flexural rigidity Dmax will exist. The weight ratio of honeycomb core x is now determined by the above limiting condition dD ¼ 2x 3x2 ¼ 0: dx
ð19Þ
The solutions of Eq. (19) are x ¼ 0 or
x ¼ 2=3:
ð20Þ
x = 0 does not satisfy the practical application and only x = 2/3 is the useful root. According to the definition of x, when the weight of honeycomb core is 2/3 of the weight of the whole honeycomb sandwich panels, the maximum flexural rigidity can be obtained. In other words, the weight of honeycomb core is two times of the weight of face sheets.
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M. He, W. Hu / Materials and Design 29 (2008) 709–713
3.2. Deducing from the bending strength From Eq. (14) we have already obtained the expression of the flexural torque M of honeycomb sandwich panels. M¼
rmax 12 Ef hf bh2c ¼ rmax hf hc b: Ef h2c
Similarly, the above expression also includes the unknowns hf and hc which are shown in Eqs. (16) and (17), the flexural torque M can be derived as follow: W W M ¼ rmax b ½xð1 xÞ: ð21Þ q 2q c
f
In like manner, when flexural torque M and the weight ratio of honeycomb core x satisfy the condition of dM ¼ 0, dx the maximum flexural torque Mmax will exist. Where for convenience we write W W dM ð22Þ ¼ rmax b ð1 2xÞ ¼ 0: dx q 2q c
f
The solution of Eq. (22) is 1 x¼ : 2 According to the above root of Eq. (22), when the weight of honeycomb core is the same as the weight of face sheets, the maximum bending strength can be obtained. By above theoretical analysis, the weight ratio of honeycomb core xwould meet a condition of 0.5 6 x 6 0.667, so that the optimum mechanical properties can be obtained. 4. Experiments Based on the range of the weight ratio of honeycomb core x, the honeycomb sandwich panels were designed and the results were verified by further experiments. The mild-steel sheets Q235 were used as the honeycomb core and face sheets materials in the present study. The sample, 130mm · 100 mm · 4.2 mm, was composed of two face sheets, 130 mm · 100 mm · 0.6 mm, and a honeycomb core, 130 mm · 100 mm · 3 mm. The cell shape of honeycomb core is regular hexagon. The length of side of regular hexagon is 5 mm. The wall thickness of core is 2.65 mm. The core thickness is 3 mm. The weight of honeycomb core is 133.85 g and whole honeycomb sandwich g panels are 255.53 g. For x ¼ 133:85 ¼ 0:524, the designed 255:53 g sample was in the range of x. A suitable assistant bonding coating was prepared between face sheets and honeycomb core. The sample had been performed a heat treatment at 830 C for 10 min, and then cooled down until room temperature in the furnace. To avoid oxidation of sample, pure argon gas (purity 99.995%) was flowed through the furnace during the heat treatment. Then the sample was taken out to the air. After the processing, the 3-point bending test was performed to obtain the flexural rigidity and bending
Fig. 2. Schematic diagram of 3-point bending test.
Table 1 Contrastive values of honeycomb sandwich sample
Testing value Theoretical value
Weight (g)
Bending strength (MPa)
Flexural rigidity (N m)
253.02 255.53
327.0 370
462.7 518.4
strength of the honeycomb sandwich beam with a dimension of 130 mm · 15 mm · 4.2 mm that had been subjected to the 3-point loading. The sample was tested by 3-point bending using Shimadzu UTM machine, as shown in Fig. 2 (Schematic diagram of 3-point bending test), applying loading through a roller of diameter d of 10 mm in accordance with the ASTM standard C393-62. The crosshead speed 0.05 mm/s is kept constant, so that the maximum load occurs between 3 and 6 min after the start of the test. During the bending test, a foil sheet of thickness 0.2 mm was placed in between the sample and the indentation roller to prevent failure due to local indentation. By 3-point bending test, the experimental results of the flexural rigidity and the bending strength were obtained. The theoretical values of the flexural rigidity and bending strength can be calculated using Eqs. (11) and (14). The contrastive values are shown in Table 1. The theoretical values of the model agree well with the experimental results. 5. Conclusion According to the theoretical analysis, we have a good understanding of the effects on designing the sandwich structure. To achieve the maximum flexural rigidity and bending strength, the satisfying weight condition of the honeycomb core weight is 50–66.7% of the weight of the whole honeycomb sandwich panels. To verify the conclusion experimentally, the flexural rigidity and the bending strength of honeycomb sandwich sample are measured. The measured flexural rigidity and bending strength are in relatively good agreement with the calculated values. By comparing our experimental results with the results calculated theoretically, we conclude that the weight condition of the honeycomb sandwich core is appropriate. That can help the researchers to design the honeycomb sandwich structure.
M. He, W. Hu / Materials and Design 29 (2008) 709–713
Acknowledgments The authors are grateful for the university Science and Technology Beijing. References [1] Yu SD, Cleghorn WL. Free flexural vibration analysis of symmetric honeycomb panels. J Sound Vib 2005;284:189–204. [2] Wang B, Yang M. Damping of honeycomb sandwich beams. J Mater Process Technol 2000;105:67–72. [3] Kim Hyeung-Yun, Hwang Woonbong. Effect of debonding on natural frequencies and frequency response functions of honeycomb sandwich beams. Compos Struct 2002;55:51–62. [4] Petras A, Sutcliffe MPF. Failure mode maps for honeycomb sandwich panels. Compos Struct 1999;44:237–52.
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[5] Renji K, Nair PS, Narayanan S. Modal density of composite honeycomb sandwich panels. J Sound Vib 1996;195(5):687–99. [6] Paik Jeom Kee, Thayamballi Anil K, Kim Gyu Sung. The strength characteristics of aluminum honeycomb sandwich panels. Thin Wall Struct 1999;35:205–31. [7] Cunningham PR, White RG, Aglietti GS. The effects of various design parameters on the free vibration of doubly curved composite sandwich panels. J Sound Vib 2000;230:617–48. [8] Cunningham PR, White RG. A new measurement technique for the estimation of core shear strain in closed sandwich structures. Compos Struct 2001;51:319–34. [9] Khatua TP, Cheng YK. Bending and vibration of multiplayer sandwich beams and plates. Int J Numer Meth Eng 1973;6: 11–24. [10] Yu L, Yang S, Gibson RF, Chen WH. Modal parameter evaluation of degraded adhesively bonded composite beams. Compos Struct 1988;43:79–91.
Materials & Design
Materials and Design 29 (2008) 709–713 www.elsevier.com/locate/matdes
Short Communication
A study on composite honeycomb sandwich panel structure Meifeng He, Wenbin Hu
*
State Key Laboratory of Metal Matrix Composites, Shanghai Jiao Tong University, Shanghai 200030, PR China Received 23 May 2006; accepted 2 March 2007 Available online 19 March 2007
Abstract Honeycomb sandwich structure combines high flexural rigidity and bending strength with low weight. Sandwich construction plays an increasing role in industry, and sandwich structural designing is an available method for sandwich structures. However, the absence of the design variable is the principal problem of composite sandwich construction. In this paper, the structure and mechanical properties of honeycomb sandwich panels are introduced. The weight ratio range of honeycomb core that is deduced on the basis of optimum mechanical properties offer a principle foundation for designing the structure of honeycomb sandwich panels. The satisfying weight condition of the honeycomb core weight is 50–66.7% of the weight of the whole honeycomb sandwich panels by theoretical analysis. Based on that conclusion, the honeycomb sandwich panels were designed and the results were verified by further experiments. Agreement between the theoretical values of the sample and experimental results is good. 2007 Elsevier Ltd. All rights reserved.
1. Introduction Honeycomb sandwich materials are being used widely in weight sensitive and damping structures where high flexural rigidity is required, in many fields especially in the automobile industry [1–3]. Honeycomb core sandwich panel is formed by adhering two high-rigidity thin-face sheets with a low-density honeycomb core possessing less strength and stiffness. By varying the core and the thickness and material of the face sheet, it is possible to obtain various properties and desired performance, particularly high strength-to-weight ratio [4]. Several types of core shapes and core material have been applied to the construction of sandwich structures [5,6]. In this paper, the honeycomb core that consists of very thin mild-steel panel in the form of regular hexagonal cells perpendicular to the facings was applied. Whereas its honeycomb core must be stiff enough to prevent one face panel of the honeycomb sandwich panel from sliding over the other, when a honeycomb sandwich panel subjected to flexural loading. Such rigidities are called the transverse shear stiffness of the honeycomb core. *
Corresponding author. Tel.: +86 21 6293 3585; fax: +86 21 6282 2012. E-mail address: [email protected] (W. Hu).
0261-3069/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2007.03.003
When the core depth is much larger than the thickness of the face panels (most honeycomb sandwich panels belong to this category), the transverse shear stiffness of the sandwich plate is almost entirely contributed by its core. For simplicity and efficiency, the cellular honeycomb core is idealized as a homogeneous material and its equivalent mechanical properties are used in analysis and design. Therefore, the knowledge of the equivalent transverse shear stiffness of honeycomb is very important for the analysis and design of sandwich plates [7]. Even if the concept of sandwich construction is not very new, it has primarily been adopted for non-strength part of structures in the last decade. This is because there are a variety of problem areas to be overcome when the sandwich construction is applied to design of dynamically loaded structures. To enhance the attractiveness of sandwich construction, it is thus essential to better understand the local strength characteristics of individual sandwich panel members. To optimize further the structural design, the present trend is to apply the principle of flexural rigidity and bending stiffness of sandwich plates. The weight ratio range of honeycomb core that is deduced on the basis of optimum mechanical properties offer a principle foundation for designing the structure of honeycomb sandwich
710
M. He, W. Hu / Materials and Design 29 (2008) 709–713
Nomenclature r1, r2 t1, t2 E1, E2 Ef e1, e2 b h hf hc
yield stress of two facing material thickness of two facing skins (assuming that both skins are of same each other) elastic (Young’s) modulus of two facing material elastic (Young’s) modulus of facing material strain of two facing skins breadth of the sandwich panel height of sandwich panel including facing skins thickness of facing skins height of honeycomb core
Z Ieq D M rmax Dmax Mmax y qf qc
distance from facing skin to centroidal axis equivalent sectional inertia moment flexural rigidity flexural torque maximum tensile strength of facing skin maximum flexural rigidity maximum flexural torque half height of sandwich panel including two facing skins density of facing material density of honeycomb core material
panels. Noteworthy, the experimental data agree satisfactorily with the theoretical predictions. The useful results are extended to give a available design tool for sandwich panels manufacturers. 2. The mechanical properties of honeycomb sandwich panels Honeycomb sandwich structure possesses high flexural rigidity and bending strength with low weight. Flexural rigidity and bending strength are very important mechanical properties of honeycomb sandwich panels. The underlying assumptions of the theory presented in the deducing procedure are as follows [8]: (1) The faceplates are very thin compared with the total thickness of the sandwich. (2) The shear stress over the depth of the core is considered constant. This assumption may be made if the core is too weak to provide a significant contribution to the flexural rigidity of the sandwich. (3) The honeycomb construction is anisotropy. (4) The rotary inertia of the face plates about their own centroidal axes is negligible. (5) The sandwich has equal thickness face plates (symmetric sandwich) and both face plates are of the same material with identical properties.
Fig. 1. Calculated sectional area and equivalent sectional area.
r1 t1 ¼ r2 t2 : Then, r1 t1 ¼ : r2 t2
ð1Þ
ð2Þ
The Eq. (3) can be achieved by applying Hooke’s law. r1 r2 ð3Þ e1 ¼ ; e2 ¼ ; E1 E2 so,
2.1. Flexural rigidity Honeycomb core was used to apart one face sheet from another in honeycomb sandwich structure. Because of this special structure the sectional inertia moment of honeycomb sandwich panel was increased and induced an obvious increase of the flexural rigidity of panels [9,10]. The flexural rigidity of panels is one of the most important parameters in structural design. The calculated sectional area, as shown in Fig. 1, are assumed that the sectional area of two facing sheets are both effective. According to the mechanical equilibrium condition, the equality can be obtained as follow:
e1 r1 E2 t2 E2 ¼ ¼ : e2 E1 r2 t1 E1
ð4Þ
The distance from facing skin to centroidal axis Z of sandwich plate has been noted. There also have several equalities. c hþhc e2 hþh e2 e1 2 ; Z¼ ð5Þ ¼ hþhc ¼ e1 2 : e1 þ e2 þ1 Z Z e2 2 When Eq. (4) is used in Eq. (5), so that Eq. (5) reduces to c E1 t1 hþh 2 Z¼ : ð6Þ E2 t2 þ E1 t1
M. He, W. Hu / Materials and Design 29 (2008) 709–713
As shown in Fig. 1.The equivalent sectional inertia moment Ieq is given by 2 h þ hc E2 t3 1 E2 3 Z þ t2 Z 2 þ 1 þ t I eq ¼ t1 2 E1 12 12 E1 2 " # 2 h þ hc h þ hc E2 2 ¼ t1 Z Z þ t2 Z 2 2 2 E1 1 3 E2 3 t þ t : þ ð7Þ 12 1 E1 2 When Eq. (6) is used in Eq. (7), so that Eq. (7) reduces to 2 h þ hc E2 t2 t1 1 3 E2 3 t1 þ t2 : I eq ¼ þ ð8Þ 12 2 E1 t1 þ E2 t2 E1 The flexural rigidity D of the sandwich panels is also given by ð9Þ
D ¼ Ef I eq :
Assuming that both faces are of the same each other, the facing material and the facing thickness are same. So we can obtain E1 = E2 = Ef and t1 = t2 = hf. In many practical sandwiches application, hf/hc lies in the range from 0.02 to 0.1. The limiting conditions of t1 = t2 hc and h hf hc always exist. When above several conditions are satisfied in these cases, so that Eq. (9) reduces to 2 2 h þ hc ð E f hf Þ D ¼ E1 I eq ¼ 2 2Ef hf 2 1 h þ h 2hf 1 2 ¼ E f hf ¼ Ef hf ðh hf Þ 2 2 2 1 Ef hf h2c : ð10Þ 2 The expression of the flexural rigidity D of honeycomb sandwich panels is 1 D ¼ Ef hf h2c : 2
ð11Þ
2.2. Bending strength The honeycomb sandwich structure does not get shear fracture when one face sheet is tensioned and the other is compressed, which is the optimal flexural failure of the sandwich panels. This makes the honeycomb sandwich structure can get better performed under the same mechanical condition. The fracture of the face sheet is the optimal flexural failure. According to the theory of mechanics of materials, the equality can be obtained as follow: rmax
M max ¼ y; I eq
rmax
Ef M max Ef M max y: ¼ y¼ Ef I eq D
ð12Þ
Then, M max ¼
rmax D : Ef y
ð13Þ
711
c h2c always exists The limiting condition of y ¼ hf þh 2 because the thickness of facing skin is very thin. When above condition is satisfied in this case, so that Eq. (13) reduces to
M max ¼
rmax 12 Ef hf bh2c ¼ rmax hf hc b: Ef h2c
ð14Þ
3. Deducing the weight range of honeycomb core on the basis of optimum mechanical properties 3.1. Deducing from the flexural rigidity From Eq. (11) we have already gotten the expression of the flexural rigidity D of honeycomb sandwich panels. 1 D ¼ Ef hf h2c : 2 The mass area ratio W of the honeycomb sandwich panels is associated with the density of facing material qf and the density of honeycomb core material qc, as given by W ¼ 2qf hf þ qc hc ;
ð15Þ
where for convenience we write x ¼ qc hc =W ;
ð16Þ
where x is the weight ratio of honeycomb core, then we can obtain the follow equality from Eqs. (15) and (16) hf ¼
W ð1 xÞ: 2qf
ð17Þ
Eqs. (16) and (17) provide two homogeneous equalities for the unknowns hf and hc in Eq. (11), the flexural rigidity D can then be determined 2 E f W W D ¼ x2 ð1 xÞ: ð18Þ 2 qc 2qf When the flexural rigidity D and the weight ratio of honeycomb core x satisfy the condition of dD ¼ 0, the maximum dx flexural rigidity Dmax will exist. The weight ratio of honeycomb core x is now determined by the above limiting condition dD ¼ 2x 3x2 ¼ 0: dx
ð19Þ
The solutions of Eq. (19) are x ¼ 0 or
x ¼ 2=3:
ð20Þ
x = 0 does not satisfy the practical application and only x = 2/3 is the useful root. According to the definition of x, when the weight of honeycomb core is 2/3 of the weight of the whole honeycomb sandwich panels, the maximum flexural rigidity can be obtained. In other words, the weight of honeycomb core is two times of the weight of face sheets.
712
M. He, W. Hu / Materials and Design 29 (2008) 709–713
3.2. Deducing from the bending strength From Eq. (14) we have already obtained the expression of the flexural torque M of honeycomb sandwich panels. M¼
rmax 12 Ef hf bh2c ¼ rmax hf hc b: Ef h2c
Similarly, the above expression also includes the unknowns hf and hc which are shown in Eqs. (16) and (17), the flexural torque M can be derived as follow: W W M ¼ rmax b ½xð1 xÞ: ð21Þ q 2q c
f
In like manner, when flexural torque M and the weight ratio of honeycomb core x satisfy the condition of dM ¼ 0, dx the maximum flexural torque Mmax will exist. Where for convenience we write W W dM ð22Þ ¼ rmax b ð1 2xÞ ¼ 0: dx q 2q c
f
The solution of Eq. (22) is 1 x¼ : 2 According to the above root of Eq. (22), when the weight of honeycomb core is the same as the weight of face sheets, the maximum bending strength can be obtained. By above theoretical analysis, the weight ratio of honeycomb core xwould meet a condition of 0.5 6 x 6 0.667, so that the optimum mechanical properties can be obtained. 4. Experiments Based on the range of the weight ratio of honeycomb core x, the honeycomb sandwich panels were designed and the results were verified by further experiments. The mild-steel sheets Q235 were used as the honeycomb core and face sheets materials in the present study. The sample, 130mm · 100 mm · 4.2 mm, was composed of two face sheets, 130 mm · 100 mm · 0.6 mm, and a honeycomb core, 130 mm · 100 mm · 3 mm. The cell shape of honeycomb core is regular hexagon. The length of side of regular hexagon is 5 mm. The wall thickness of core is 2.65 mm. The core thickness is 3 mm. The weight of honeycomb core is 133.85 g and whole honeycomb sandwich g panels are 255.53 g. For x ¼ 133:85 ¼ 0:524, the designed 255:53 g sample was in the range of x. A suitable assistant bonding coating was prepared between face sheets and honeycomb core. The sample had been performed a heat treatment at 830 C for 10 min, and then cooled down until room temperature in the furnace. To avoid oxidation of sample, pure argon gas (purity 99.995%) was flowed through the furnace during the heat treatment. Then the sample was taken out to the air. After the processing, the 3-point bending test was performed to obtain the flexural rigidity and bending
Fig. 2. Schematic diagram of 3-point bending test.
Table 1 Contrastive values of honeycomb sandwich sample
Testing value Theoretical value
Weight (g)
Bending strength (MPa)
Flexural rigidity (N m)
253.02 255.53
327.0 370
462.7 518.4
strength of the honeycomb sandwich beam with a dimension of 130 mm · 15 mm · 4.2 mm that had been subjected to the 3-point loading. The sample was tested by 3-point bending using Shimadzu UTM machine, as shown in Fig. 2 (Schematic diagram of 3-point bending test), applying loading through a roller of diameter d of 10 mm in accordance with the ASTM standard C393-62. The crosshead speed 0.05 mm/s is kept constant, so that the maximum load occurs between 3 and 6 min after the start of the test. During the bending test, a foil sheet of thickness 0.2 mm was placed in between the sample and the indentation roller to prevent failure due to local indentation. By 3-point bending test, the experimental results of the flexural rigidity and the bending strength were obtained. The theoretical values of the flexural rigidity and bending strength can be calculated using Eqs. (11) and (14). The contrastive values are shown in Table 1. The theoretical values of the model agree well with the experimental results. 5. Conclusion According to the theoretical analysis, we have a good understanding of the effects on designing the sandwich structure. To achieve the maximum flexural rigidity and bending strength, the satisfying weight condition of the honeycomb core weight is 50–66.7% of the weight of the whole honeycomb sandwich panels. To verify the conclusion experimentally, the flexural rigidity and the bending strength of honeycomb sandwich sample are measured. The measured flexural rigidity and bending strength are in relatively good agreement with the calculated values. By comparing our experimental results with the results calculated theoretically, we conclude that the weight condition of the honeycomb sandwich core is appropriate. That can help the researchers to design the honeycomb sandwich structure.
M. He, W. Hu / Materials and Design 29 (2008) 709–713
Acknowledgments The authors are grateful for the university Science and Technology Beijing. References [1] Yu SD, Cleghorn WL. Free flexural vibration analysis of symmetric honeycomb panels. J Sound Vib 2005;284:189–204. [2] Wang B, Yang M. Damping of honeycomb sandwich beams. J Mater Process Technol 2000;105:67–72. [3] Kim Hyeung-Yun, Hwang Woonbong. Effect of debonding on natural frequencies and frequency response functions of honeycomb sandwich beams. Compos Struct 2002;55:51–62. [4] Petras A, Sutcliffe MPF. Failure mode maps for honeycomb sandwich panels. Compos Struct 1999;44:237–52.
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[5] Renji K, Nair PS, Narayanan S. Modal density of composite honeycomb sandwich panels. J Sound Vib 1996;195(5):687–99. [6] Paik Jeom Kee, Thayamballi Anil K, Kim Gyu Sung. The strength characteristics of aluminum honeycomb sandwich panels. Thin Wall Struct 1999;35:205–31. [7] Cunningham PR, White RG, Aglietti GS. The effects of various design parameters on the free vibration of doubly curved composite sandwich panels. J Sound Vib 2000;230:617–48. [8] Cunningham PR, White RG. A new measurement technique for the estimation of core shear strain in closed sandwich structures. Compos Struct 2001;51:319–34. [9] Khatua TP, Cheng YK. Bending and vibration of multiplayer sandwich beams and plates. Int J Numer Meth Eng 1973;6: 11–24. [10] Yu L, Yang S, Gibson RF, Chen WH. Modal parameter evaluation of degraded adhesively bonded composite beams. Compos Struct 1988;43:79–91.