Bending behavior of corrugated-core sandwich plates

ga b,*, he P he Pe e 15 S ior o d to wich ved i esent investigations of deflections and moments. Furthermore, the effects of several geometric parameters of a corrugated-core sandwich plate on its rigidity and state of stress are investigated. face sheets apart and stabilizes them by resisting vertical deformations, and also enables the whole structure to neering and other applications, where weight is an Plantema [2] and Allen [3]. In 1951, the elastic constants of corrugated-core sandwich panels were investigated by face are assumed to remain straight, but not necessarily normal to the middle surface after the plate distortion. Such a sandwich plate theory requires knowledge of cer- tain elastic constants for the type of corrugation. These constants provide an equivalence between the given 3D . * Corresponding author. Tel./fax: +1 814 863 7967. E-mail address: esvesm@engr.psu.edu (E. Ventsel). Composite Structures 70 (2 0263-8223/$ - see front matter � 2004 Elsevier Ltd. All rights reserved act as a single thick plate as a virtue of its shearing strength. This second feature imparts outstanding strength to the sandwich structures. In addition, unlike soft honeycomb-shaped cores, a corrugated-core resists bending and twisting in addition to vertical shear. Therefore, corrugated-core sandwich panels, due to their exceptionally high flexural stiffness-to-weight ratio are commonly used in aviation, aerospace, civil engi- Libove and Hubka [4]. The sandwich-plate theory used here is based on the Mindlin–Reissner plate theory and a structural idealization of a given 3D corrugated- core sandwich construction as a 2D homogeneous, structurally orthotropic thick plate. The modulus of elasticity in the z-, or the thickness, direction is assumed to be infinite; i.e. local buckling of the face sheets is not considered here. Straight lines normal to the middle sur- � 2004 Elsevier Ltd. All rights reserved. Keywords: Corrugated core; Mindlin–Reissner; Analytical solution; Geometric parameters; Various boundaries; Structurally orthotropic 1. Introduction A corrugated-core sandwich plate is comprised of a corrugation sheet between two thin face sheets. A corru- gated panel and its cross-sectional views are shown in Fig. 1. The significant feature of this structure is its high strength-to-weight ratio. The corrugated core keeps the important design issue. This construction approach to form a sandwich plate might be defined as �structurally composite�, since its behavior characteristics are defined by the composite action of its components. In 1948, Libove and Batdorf [1] developed a small deflection theory for flat sandwich plates. The details of the behavior of sandwich plates are described by Bending behavior of corru Wan-Shu Chang a, Edward Ventsel a Department of Civil and Environmental Engineering, T b Department of Engineering Science and Mechanics, T Available onlin Abstract In this paper, the comprehensive analysis for the linear behav solution based on the Mindlin–Reissner plate theory is presente ing with various boundary conditions. A three-dimensional sand orthotropic thick plate continuum. Some new phenomena obser ical analyses have been confirmed and reported, herein. The pr doi:10.1016/j.compstruct.2004.08.014 ted-core sandwich plates Ted Krauthammer a, Joby John b enn State University, University Park, PA 16802, USA nn State University, University Park, PA 16802, USA eptember 2004 f a corrugated-core sandwich plate is presented. A closed-form describe the behavior of corrugated-core sandwich plate bend- panel is reduced to an equivalent two-dimensional structurally n experimental investigations but not found in previous numer- numerical analysis agrees well with the reported experimental 005) 81–89 www.elsevier.com/locate/compstruct ndwic 82 W.-S. Chang et al. / Composite Structures 70 (2005) 81–89 sandwich construction and its 2D thick-plate model in the framework of the introduced assumptions [4,5]. The elastic constants required for such a conversion are derived by Libove and Hubka [4]. The sign conven- tions, symbols and expressions used in this paper are the same as in [4]. These elastic constants relate to the geometric param- eters shown in Fig. 1. Therefore, the effects of these geo- metric parameters, such as corrugation angle a, the core to face thickness ratio tc/tf, the pitch to corrugation depth ratio p/hc, and corrugation depth to core thickness ratio hc/tc, are investigated in this paper. In this study, a closed-form solution based on the Mindlin–Reissner plate theory is presented for the static analysis of corrugated-core sandwich plates. The present analytical solution with simply supported edges is cor- roborated by experimental results, the analytical solu- tion, and 3D finite element analysis of corrugated-core sandwich plate reported in [6]. It should be noted that the moments, Mx and My, in the x- and y-direction, respectively, obtained by 3D FEM of the corrugated- core sandwich plates and analytical solution in [6], are Fig. 1. Corrugated-core sa both positive. However, experimental results have shown (see [6]) that the signs ofMx andMy are opposite in this case, and My compared to Mx is very small. The solution presented in this paper for deflections and mo- ments in x- and y-direction agrees with all the experi- mental investigations presented in [6]. 2. Governing equations The equilibrium equations in plate bending are as follows: oMx ox þ oMxy oy � Qx ¼ 0 ð1aÞ oMxy ox þ oMy oy � Qy ¼ 0 ð1bÞ oQx ox þ oQy oy þ q ¼ 0 ð1cÞ The constitutive equations for orthotropic sandwich plates are given as follows: Mx ¼ Dx 1� txty ohx ox þ ty ohyoy � � My ¼ Dy 1� txty tx ohx ox þ ohy oy � � Mxy ¼ Dxy 2 ohx oy þ ohy ox � � ð2aÞ Qx ¼ DQx hx þ ow ox � � ; Qy ¼ DQy hy þ ow oy � � ð2bÞ where w is the deflection in z-direction and hx and hy are the slopes of the normal to the middle plane of the sand- wich plate about the yz and xz planes. Mx,My,Mxy and Qx and Qy are the moments and the shear forces, respec- tively. The above slopes are assumed to be independent variables with respect to w. Dx, Dy, Dxy and DQx ;DQy are the flexural and shear stiffnesses, respectively. The corre- hc 2 h 2 h tc 2p tf α x z y f h panel and a panel unit. sponding expressions for these elastic constants are gi- ven in [4]. In Eq. (2a) tx and ty are bending Poisson�s ratios in the x- and y-directions, respectively. Substituting Eqs. (2) into Eqs. (1), the governing equations can be represented in the following operator form: L uf gð Þ ¼ Pf g ð3Þ where L ¼ L11 L12 L13 L21 L22 L23 L31 L32 L33 2 64 3 75; fug ¼ hx hy w 2 64 3 75; fPg ¼ 0 0 q 8>< >: 9>= >; ð4Þ plate with all four edges are hard type simply supported, in the form of the following double Fourier series: X1 X1 mpx� � npy� � site S and another plate with all four edges clamped. 3.1. Hard type simply supported edge Two boundary conditions for a simply supported edge, x = constant are w ¼ 0; Mx ¼ 0 ð6Þ and the third boundary condition for the hard type sim- ply supported edge is of the form Qy ¼ 0 ð7Þ Thus, the boundary conditions in this case can be for- mulated in terms of the deflection and slopes of the problem, as follows: w ¼ 0; ohx ox þ ty ohyoy ¼ 0; hy þ owoy ¼ 0 on the edge x ¼ constant ð8Þ Since w = 0 on the edge x = constant, ow/oy is equal to zero. Therefore, the third condition, Eq. (8), is of the form hy = 0 and thus ohy/oy = 0. Finally, we can repre- and L11 ¼ Dxx o 2ð Þ ox2 þ Dxy 2 o2ð Þ oy2 � DQx; L12 ¼ Dxy 2 þ Dxxty � � o2ð Þ oxoy ; L13 ¼ �DQx oð Þox ð5aÞ L21 ¼ L12; L22 ¼ Dyy o 2ð Þ oy2 þ Dxy 2 o2ð Þ ox2 � DQy ; L23 ¼ �DQy oð Þoy ð5bÞ L31 ¼ �L13; L32 ¼ �L23; L33 ¼ DQx o 2ð Þ ox2 þ DQy o 2ð Þ oy2 ð5cÞ Dxx ¼ Dx 1� txty ; Dyy ¼ Dy 1� txty and q is the lateral loading. 3. Boundary conditions Let us consider appropriate boundary conditions pre- scribed on one edge of a rectangular sandwich structur- ally orthotropic plate. Since the governing differential equations (3) have the sixth order, three boundary con- ditions should be prescribed at any point on the sand- wich plate boundary. In this paper we consider one W.-S. Chang et al. / Compo sent the above boundary conditions in the form: w ¼ m¼1 n¼1 wmn sin a sin b hx ¼ X1 m¼1 X1 n¼1 Amn cos mpx a � � sin npy b � � hy ¼ X1 m¼1 X1 n¼1 Bmn sin mpx a � � cos npy b � � ð10aÞ q ¼ X1 m¼1 X1 n¼1 qmn sin mpx a � � sin npy b � � ð10bÞ where wmn, Amn, Bmn, and qmn represent coefficients to be determined. It can be easily verified that the expressions for the deflection and two slopes automatically satisfy the prescribed boundary conditions (9). Substituting Eqs. (10) into Eq. (3), the above unknown coefficients can be evaluated. 3.2. Clamped edge The boundary conditions for a clamped edge at say x = constant are given by w ¼ hx ¼ hy ¼ 0 ð11Þ The solutions sought are in the form of the following double Fourier series, ensuring that they satisfy with the boundary conditions, Eq. (9) w ¼ X1 m¼1 X1 n¼1 wmn sin mpx a � � sin npy b � � hx ¼ X1 m¼1 X1 n¼1 Amn sin mpx a � � sin npy b � � hy ¼ X1 m¼1 X1 n¼1 Bmn sin mpx a � � sin npy b � � ð12Þ q ¼ X1 m¼1 X1 n¼1 qmn sin mpx a � � sin npy b � � 4. Numerical examples 4.1. Comparison with existing results from Tan et al. [6] The behavior of the 6m by 2.1m corrugated-core sandwich plate studied by Tan et al. [6] is analyzed. In w ¼ 0; ohx ox ¼ 0; hy ¼ 0 on the edge x ¼ constant ð9Þ Hence, for the rectangular plate with all simply-sup- ported edges, the solutions of the Eq. (3) can be sought tructures 70 (2005) 81–89 83 this case, the load is taken to be 5520N/m2. Table 1 shows the geometric parameters and elastic constants for this corrugated-core sandwich plate. The modulus of elasticity is 208GPa and Poisson�s ratio is assumed to be 0.3. Table 2 shows the comparisons of the maximum deflection at the central point with analytical solution, 3D corrugated-core sandwich plate FEM (3D FEM) re- ported in [6], and the present solution. As can be seen, the discrepancy between the 3D FEM and the presented solution is 0.37%, and between the solution by Tan and the 3D FEM is 1.19%. The deflection of the present solution agrees closely with that given by the 3D FEM, and provides adequate predictions of the experi- mental deflections [6]. Table 3 shows the moments in the x- and y-direction at the central point. It should be noted that in the pre- sented solution My is very small as compared to Mx, and the signs of Mx and My are opposite, coinciding with the experimental investigation conducted by Tan et al. [6]. In that paper, the moments in both x- and y- direction from the 3D FEM and the solution by Tan et al. [6] have the same positive sign, namely, the com- pressions are at the top and tension at the bottom, as would be expected. However, in the experimental inves- tigations [6], Mx is positive but My is negative (i.e. the compression in the bottom and tension in the top in the y-direction). One would expect the plate to hog in the y-direction because of this negative moment but it is seen that the plate has a net downward deflection (sag- ging) throughout the whole plate, as shown in Fig. 2. This is because the transverse shear deformation is much greater than the bending deformation, and the effect of the negative My is not readily visible. Subsequently, we analyzed the hard-type simply sup- ported and fully clamped rectangular corrugated-core sandwich plates (6 · 2.1m) under a static load of 10kN/m2. We varied the geometric parameters of the plate, and studied its effect on the deflection and the Table 1 Geometric parameters and elastic constants p (mm) hc (mm) f (mm) tf (mm) tc (mm) 265 107.5 82.5 2.5 2.5 Dx (Nm) Dy (Nm) Dxy (Nm) DQx (N/m) DQy (N/m) 4.1 · 106 3.22 · 106 2.31 · 106 2.83 · 107 1.59 · 105 Table 2 Table 3 Moments at the central point Mx (Nm/m) My (Nm/m) 5.99E+03 �613.9807 84 W.-S. Chang et al. / Composite Structures 70 (2005) 81–89 Comparison of central deflection Maximum deflection, wmax (mm) Tan et al. 1989 Present solution Solution By Tan [6] 3D FEM analysis [6] 6.86 6.779 6.754 Fig. 2. Deflection of the corrug Discrepancy (%) between the present solution and 3D FEM Discrepancy (%) between solution by Tan and 3D FEM 0.37 1.19 ated-core sandwich plate. moments induced in the x- and y-direction in the plate. The elastic constants were computed using the expres- sions and data from [4]. The primary geometric param- eters affecting the behavior of the plate are the corrugation angle (a), core sheet to face sheet thickness ratio (tc/tf) and the pitch to core depth ratio (p/hc). In our analysis we maintained the core depth to core sheet thickness ratio (hc/tc = 20) and the core depth were maintained constant (hc = 0.1m). The above geometric parameters are shown in Fig. 1. 5. Hard-type simply supported plate Tables 4 and 5 show the results of the conducted numerical investigations for the certain ratios hc/tc = 20 and 40 but various a, tc/tf, and p/hc for simply supported corrugated-core sandwich plates. It is seen from these tables that these geometric parameters have less effect on the bending stiffness, but they significantly affect the shear stiffness, especially, the effect of corrugation angle on the DQy. With an increasing corrugation angle, Table 4 Comparison of various geometric parameters for simply support, hc/tc = 20 hc/tc = 20 q = 10kN/m 2 a tc/tf p/hc Dx (Nm) Dy (Nm) Dxy (Nm) DQx (Nm) DQy (Nm) w (mm) Mx (Nm/m) My (Nm/m) 60 0.6 1 1.32E+07 1.13E+07 8.56E+06 3.48E+08 3.47E+07 3.38E�01 2.10E+03 5.03E+03 1.2 1.33E+07 1.13E+07 8.56E+06 2.63E+08 1.43E+07 5.36E�01 2.50E+03 4.72E+03 1.4 1.34E+07 1.13E+07 8.56E+06 2.06E+08 8.74E+06 7.32E�01 2.94E+03 4.38E+03 1 1 8.39E+06 6.44E+06 4.84E+06 3.28E+08 2.45E+07 5.47E�01 2.15E+03 5.07E+03 1.2 8.48E+06 6.44E+06 4.84E+06 2.48E+08 1.02E+07 8.23E�01 2.53E+03 4.82E+03 1.4 8.54E+06 6.44E+06 4.84E+06 1.94E+08 5.66E+06 1.18E+00 3.08E+03 4.45E+03 1.25 1 7.04E+06 5.08E+06 3.80E+06 3.22E+08 2.15E+07 6.70E�01 2.18E+03 5.08E+03 1.2 7.13E+06 5.08E+06 3.80E+06 2.43E+08 9.34E+06 9.64E�01 2.54E+03 4.88E+03 1.4 7.19E+06 5.09E+06 3.80E+06 1.91E+08 5.14E+06 1.36E+00 3.07E+03 4.55E+03 70 0.6 1 1.37E+07 1.13E+07 8.56E+06 3.32E+08 1.07E+07 6.40E�01 2.81E+03 4.52E+03 1.2 1.37E+07 1.13E+07 8.56E+06 2.52E+08 5.99E+06 9.40E�01 3.56E+03 3.93E+03 1.4 1.37E+07 1.13E+07 8.56E+06 1.98E+08 3.89E+06 1.26E+00 4.38E+03 3.29E+03 1 1 8.87E+06 6.46E+06 4.84E+06 3.13E+08 8.17E+06 9.35E�01 2.79E+03 4.69E+03 1.2 8.87E+06 6.46E+06 4.84E+06 2.37E+08 4.78E+06 1.31E+00 3.41E+03 4.28E+03 1.4 8.88E+06 6.46E+06 4.84E+06 1.87E+08 3.14E+06 1.74E+00 4.15E+03 3.78E+03 1.25 1 7.52E+06 5.10E+06 3.80E+06 3.07E+08 7.79E+06 1.06E+00 2.76E+03 4.79E+03 1.2 7.52E+06 5.10E+06 3.80E+06 2.33E+08 5.92E+06 1.25E+00 3.01E+03 4.64E+03 1.4 7.53E+06 5.10E+06 3.80E+06 1.83E+08 2.96E+06 1.95E+00 4.06E+03 3.99E+03 80 0.6 1 1.42E+07 1.14E+07 8.56E+06 3.13E+08 5.02E+06 1.06E+00 4.00E+03 3.66E+03 1.2 1.41E+07 1.14E+07 8.56E+06 2.39E+08 3.37E+06 1.38E+00 4.85E+03 2.99E+03 1.4 1.40E+07 1.13E+07 8.56E+06 1.89E+08 2.43E+06 1.70E+00 5.69E+03 2.33E+03 1 1 9.31E+06 6.48E+06 4.84E+06 2.95E+08 4.15E+06 1.44E+00 3.80E+03 4.11E+03 1.2 9.24E+06 6.48E+06 4.84E+06 2.25E+08 2.86E+06 1.84E+00 4.52E+03 3.62E+03 1.4 9.19E+06 6.48E+06 4.84E+06 1.78E+08 2.11E+06 2.25E+00 5.26E+03 3.12E+03 1.25 1 7.96E+06 5.12E+06 3.80E+06 2.90E+08 4.05E+06 1.58E+00 3.68E+03 4.31E+03 1.2 7.89E+06 5.12E+06 3.80E+06 2.21E+08 2.74E+06 2.04E+00 4.40E+03 3.87E+03 75E+ 92E+ 24E+ 78E+ 75E+ 11E+ 68E+ 70E+ 07E+ W.-S. Chang et al. / Composite Structures 70 (2005) 81–89 85 1.4 7.84E+06 5.11E+06 3.80E+06 1. 90 0.6 1 1.46E+07 1.14E+07 8.56E+06 2. 1.2 1.45E+07 1.14E+07 8.56E+06 2. 1.4 1.44E+07 1.14E+07 8.56E+06 1. 1 1 9.76E+06 6.50E+06 4.84E+06 2. 1.2 9.61E+06 6.49E+06 4.84E+06 2. 1.4 9.51E+06 6.49E+06 4.84E+06 1. 1.25 1 8.41E+06 5.13E+06 3.80E+06 2. 1.2 8.27E+06 5.13E+06 3.80E+06 2. 1.4 8.16E+06 5.12E+06 3.80E+06 1.65E+ 08 1.99E+06 2.52E+00 5.15E+03 3.40E+03 08 2.91E+06 1.50E+00 5.43E+03 2.67E+03 08 2.20E+06 1.79E+00 6.18E+03 2.08E+03 08 1.68E+06 2.09E+00 6.96E+03 1.45E+03 08 2.64E+06 1.93E+00 4.96E+03 3.46E+03 08 1.92E+06 2.38E+00 5.78E+03 2.91E+03 08 1.51E+06 2.76E+00 6.47E+03 2.43E+03 08 2.55E+06 2.13E+00 4.83E+03 3.75E+03 08 1.87E+06 2.61E+00 5.60E+03 3.26E+03 08 1.43E+06 3.10E+00 6.36E+03 2.78E+03 = 40 Qx (N 54E+ 17E+ 13E+ 49E+ 13E+ 85E+ 86 W.-S. Chang et al. / Composite Structures 70 (2005) 81–89 Table 5 Comparison of various geometric parameters for simply support, hc/tc a tc/tf p/hc Dx (Nm) Dy (Nm) Dxy (Nm) D 60 0.6 1 5.98E+06 5.01E+06 3.79E+06 1. 1.2 6.02E+06 5.01E+06 3.79E+06 1. 1.4 6.05E+06 5.01E+06 3.79E+06 9. 1 1 3.92E+06 2.94E+06 2.21E+06 1. 1.2 3.96E+06 2.94E+06 2.21E+06 1. 1.4 3.99E+06 2.94E+06 2.21E+06 8. the shear stiffness DQy decreases drastically. In Table 5 we note that for higher ratios hc/tc, p/hc and higher cor- rugation angles the moments in the y-direction might have negative signs. The negative moment is because the slope hy is negative. The coefficients of hy relate to the elastic constants of the plate; therefore, only in these cases that the ratio Dxx/DQy is greater than 16, and Dxy/DQy is greater than 9 the coefficients of hy are nega- tive and My is negative. Furthermore, only when the shear stiffness DQy is low enough the above conditions can occur. In these cases the stiffness in the x-direction 1.25 1 3.32E+06 2.34E+06 1.75E+06 1.48E+ 1.2 3.36E+06 2.34E+06 1.75E+06 1.12E+ 1.4 3.39E+06 2.34E+06 1.75E+06 8.77E+ 70 0.6 1 6.22E+06 5.02E+06 3.79E+06 1.47E+ 1.2 6.22E+06 5.02E+06 3.79E+06 1.12E+ 1.4 6.22E+06 5.02E+06 3.79E+06 8.78E+ 1 1 4.15E+06 2.95E+06 2.21E+06 1.43E+ 1.2 4.16E+06 2.95E+06 2.21E+06 1.08E+ 1.4 4.16E+06 2.95E+06 2.21E+06 8.51E+ 1.25 1 3.56E+06 2.35E+06 1.75E+06 1.41E+ 1.2 3.56E+06 2.35E+06 1.75E+06 1.07E+ 1.4 3.56E+06 2.35E+06 1.75E+06 8.43E+ 80 0.6 1 6.44E+06 5.04E+06 3.79E+06 1.39E+ 1.2 6.41E+06 5.04E+06 3.79E+06 1.06E+ 1.4 6.38E+06 5.03E+06 3.79E+06 8.36E+ 1 1 4.38E+06 2.96E+06 2.21E+06 1.34E+ 1.2 4.34E+06 2.96E+06 2.21E+06 1.03E+ 1.4 4.32E+06 2.96E+06 2.21E+06 8.10E+ 1.25 1 3.78E+06 2.36E+06 1.75E+06 1.33E+ 1.2 3.75E+06 2.36E+06 1.75E+06 1.02E+ 1.4 3.72E+06 2.35E+06 1.75E+06 8.02E+ 90 0.6 1 6.66E+06 5.05E+06 3.79E+06 1.29E+ 1.2 6.59E+06 5.05E+06 3.79E+06 9.92E+ 1.4 6.54E+06 5.04E+06 3.79E+06 7.88E+ 1 1 4.60E+06 2.97E+06 2.21E+06 1.25E+ 1.2 4.53E+06 2.96E+06 2.21E+06 9.61E+ 1.4 4.48E+06 2.96E+06 2.21E+06 7.63E+ 1.25 1 4.01E+06 2.36E+06 1.75E+06 1.24E+ 1.2 3.93E+06 2.36E+06 1.75E+06 9.52E+ 1.4 3.88E+06 2.36E+06 1.75E+06 7.56E+ hc/tc = 40 q = 10kN/m 2 m) DQy (Nm) w (mm) Mx (Nm/m) My (Nm/m) 08 3.54E+06 1.75E+00 3.11E+03 4.26E+03 08 1.45E+06 3.17E+00 4.79E+03 2.94E+03 07 8.00E+05 4.55E+00 6.46E+03 1.62E+03 08 2.51E+06 2.62E+00 3.15E+03 4.41E+03 08 1.09E+06 4.56E+00 4.72E+03 3.34E+03 07 6.00E+05 6.62E+00 6.45E+03 2.14E+03 or corrugation direction is much higher than that in y- direction due to the irrelevantly high DQx and low DQy. Therefore, the most bending resistance is taken in x-direction and that causes the moment in y-direction to be relatively small or even become negative. According to the above analysis, the parameters a, hc/tc, and p/hc are the key factors to affect the shear stiffness DQy. Fig. 3 shows the effect of variation of tc/tf, p/hc, and a on the deflections at central point of the plates. It can be seen that as the p/hc ratio increases the stiffness of the plate decreases. This is because as the p/hc ratio increases 08 2.17E+06 3.13E+00 3.21E+03 4.46E+03 08 9.70E+05 5.29E+00 4.73E+03 3.50E+03 07 5.60E+05 7.51E+00 6.35E+03 2.47E+03 08 1.14E+06 3.66E+00 5.60E+03 2.41E+03 08 6.36E+05 5.09E+00 7.40E+03