18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS BUCKLING OF SANDWICH BEAMS USING THE EXTENDED HIGH-ORDER SANDWICH PANEL THEORY AND COMPARISON WITH ELASTICITY C. Phan 1* , G. Kardomateas 1 , Y. Frostig 2 1 Department of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA, USA, 2 Civil and Environmental Engineering, Technion Israel Inst of Technology, Haifa 32000, Israel * Corresponding author ( phanc@gatech.edu ) Keywords : sandwich composites, buckling, wrinkling, high- order theory core. Global buckling is studied, followed by 1 Introduction wrinkling. Sandwich composites are a unique composite lay-up 2 Characteristics of EHSAPT that consists of two stiff metallic or composite thin face sheets separated by a thick core of low density. The EHSAPT was recently formulated in [4] based This configuration gives the sandwich material on variational principles. The main characteristics system high stiffness and strength with little of the EHSAPT are the following: resultant weight penalty. Classical structural • 7 generalized coordinates are used to model theories neglect the transverse and shear the displacement field of the sandwich deformation of the core while experimental results t t c c c b b φ u , w , u , , w , u , and w composite: 0 0 0 0 0 0 0 [1] have shown non-neglible core compression, and where the superscripts { t, b, c } indicate the shear failure modes in the core to occur under blast lay-up as either top, bottom, or core, loading. Several sandwich panel theories exist that respectively; u and w indicate the axial and make various assumptions to better model the core. transverse displacement of the given lay-up, The differences in these theories is that they either respectively; φ indicates rotation; and the take into account or neglect the axial, transverse subscript 0 indicates that the location of normal, or shear stiffness of the core. With regard to the generalized coordinate occurs at the buckling, Allen’s thick formulation takes into account the core's shear stiffness only [5]. The High- midsection of the respective lay-up. Order Sandwich Panel theory (HSAPT) [2] takes • Face sheets are Euler-Bernoulli type into account the core's transverse and shear beams. The core has polynomial stiffnesses, while the Extended Higher Order displacement fields; up to O(z 3 ) and Sandwich Panel Theory (EHSAPT) takes into O(z 2 ) for the axial and transverse account the axial, transverse, and shear stiffnesses in displacement fields, respectively, where the core [4]. In this paper, the characteristics of the ‘z’ is the through the thickness EHSAPT are presented and the equations that coordinate. Displacement fields satisfy all determine the critical load for a general asymmetric interface conditions geometry and different face sheet material are • 7 coupled differential equations govern presented. The case study of a simply supported the behavior of the sandwich of total (S-S) sandwich beam undergoing uniform order 18. In order to predict global strain/edge beam loading with symmetric geometry buckling phenomenon, nonlinear axial and same face sheet materials is used to compare the predicted critical load given by Allen, HSAPT, and strains in the face sheets were EHSAPT to Elasticity [3]. Three solution considered. Nonlinear axial strains in approaches using EHSAPT were conducted to the core were also considered but were explore simplifying the loading condition to found to not significantly change the concentrated loads applied to the face sheet, and accuracy of predicting the critical load in including/excluding nonlinear axial strains in the
the global buckling case study. 3.2 EHSAPT (3 cases) Nonlinear axial strains cause two of the EHSAPT takes into account the axial, transverse differential equations to be nonlinear. normal, and shear rigidity of the core. We have • 18 boundary conditions result from the used the EHSAPT to solve three cases: (a) axial variational principle, 9 at each end. load applied exclusively to the top and bottom face sheets and linear axial strain in the core; (b) 3 Buckling of a S-S sandwich beam uniform axial strain applied through the entire An Elasticity solutions exists to predict the thickness and, again, linear axial strain in the critical global buckling [3] and wrinkling [6] core; and (c) uniform axial strain applied loads of a S-S sandwich beam undergoing through the entire thickness but with nonlinear compressive loading through an edge beam. In axial strain in the core, whereas HSAPT was the following sections, we list the formulas for solved using just Case (a). The three cases were global buckling of sandwich columns from chosen to investigate the effect of the different Allen, HSAPT, and EHSAPT (3 cases), give loading conditions and excluding and including global bucking results that compare these nonlinear axial strain in the core. A theories to the Elasticity solution given in [3], perturbation approach was used to solve for and then give wrinkling results that compare each case. Buckling using EHSAPT was also HSAPT and EHSAPT (1 case) to the Elasticity presented in [7] where a numerical solution was solution given in [6]. given, but is now solved using the perturbation 3.1 Allen’s formulation and HSAPT approach and compared to Elasticity in this paper. Allen’s thick face sheet formulation accounts for the shear rigidity of the core and is given as Case (a): The critical load for concentrated equation (12) in [3]. HSAPT accounts for loading just on the face sheets can be transverse normal and shear stiffness in the core. determined by finding the value of P for which The critical buckling load for a S-S beam with compressive loads applied to the face sheets was the perturbed system has a nontrivial solution, solved using HSAPT in [2]. Some algebraic or finding P by zeroing the determinant: manipulation of equation (82) in [2] leads to 2 π [ ] � � [ ] − = det{ K G I } 0 HSAPT’s asymmetric global buckling load, written LC a 2 a (3) in a form similar to Allen: K LC contains stiffness and material constants for � � � � a sandwich with linear axial strains in the core. 2 P P � � � � Ef Ef + µ − P 1 � � � � K LC also appears in Case (b) and (c). G a E 2 P P P � � � � c c E 2 = contains the loading per unit width parameter P . P cr _ HSAPT � � P − P � � E 2 Ef K LC and G a are given in the Appendix. + µ 1 � � � P � (1) c where, Case (b): The critical load for Uniform strain ( ) 2 c 2 xz π 2 c G loading with linear axial strains in the core can µ = + 1 c 2 12 E a be determined by finding P from the following (2) z perturbed characteristic equation: c → ∞ As E , HSAPT reduces to Allen. For z 2 π [ ] � � [ ] long beams (i.e. large length-to-the-total-core- − = det{ K G I } 0 LC b 2 (a/(2c)) 2 ), a thickness ratio squared the (4) G b contains the loading per unit width which c contribution from the E term is small. The z only has contributions from the face sheets (see definition of P E2 , P Ef , and P c are given in [3]. Appendix). This is because without nonlinear For wrinkling, equation (76) in [2] was used.
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