一枚小可爱(15)
where the matrix elements are
(16a)
(16b)
(16c)
and
(17a)
(17b)
The terms and F are, respectively, the dimensionless load and foundation stiffness parameters. Solving equa-tion (15) for the largest eigenvalue λ yields the lowest buckling load P cr .
Finite-difference method.An alternative method for examining the elastic stability of a sandwich debond is
the finite-difference method (ref.10). The governing
equations at the onset of buckling are
(18)
In this method,N + 4 discrete evenly spaced gridpoints
x ?1 to x N +2 are used with spacing ? as shown in figure 4.Derivatives of w are approximated by
(19)
with simply supported boundary conditions (20)
Substituting equation (19) into equation (18) for points x 1 to x N along with the boundary conditions in equa-tion (20) provides N + 4 simultaneous algebraic equa-tions for w ?1 to w N +2. The nontrivial solution for the buckling load is represented by the lowest value of P for which the determinant of the coefficients of the N alge-braic equations vanishes (ref.9).
Finite-Element Model
Plane-strain ?nite-element analysis.Due to sym-metry about the panel’s half-length (x =L /2) and thick-ness, only a quarter of the debonded sandwich panel cross section was modeled by 4-noded quadrilateral assumed natural-coordinate strain (ANS) finite elements,as shown in figure 5 (refs.12 and 13). The element is
φ?m
ψ?U ?b []F μ4
-----U ?f []+ 1–??
[]A {}λA {}=U ?b ij d 2φ?i d ξ2----------d 2φ?
j d ξ2----------d ξ0
1
∫
=U ?f ij ψ?φ?i φ?j d ξ
1
∫
=
??ij
d φ?i
d ξ-------d φ?j d ξ
--------d ξ0
1
∫
=λ1
P ?---E b I PL 2---------==F K l d
4
E b I
--------=P
?w
IV
P
EI
-----w ″+0=w IV P EI -----w ″k EI -----w ++0
=for debonded region ()for bonded region ()
w ″n w n 1+2w n w n 1
–+–?
2
--------------------------------------------------=w n IV w n 2+4w n 1+6w n 4w n 1–w n 2–+–+–?4
-------------------------------------------------------------------------------------------------=
w 00
=w 1–w 1–=w N 1+0=w N w N 2+–=
Figure 4. Finite-difference grid with uniform spacing.
5
based on a Mindlin/Reissner shell formulation. Eight layers of 4-noded quadrilateral finite elements are used to model the faceskin and sandwich foam regions.Duplicate nodes for the faceskin and core at their com-mon boundary along the debonded region created the debonded region between the faceskin and core. (See fig.6.) These duplicate nodes are allowed to translate independently. A linear stability analysis was performed using NASA Langley Research Center’s COmputational MEchanics Testbed (COMET) code (ref.12) to compute the buckling load of the debonded panel. The linear elas-tic stability analysis in COMET is formulated using the concept of adjacent equilibrium. Buckling occurs when the membrane strain energy is converted to bending strain energy in the adjacent equilibrium configuration.
Figure 6. Local buckled region of debonded faceskin of sandwich panel strip.
Faceskin
Core
P
Symmetric L 2
Faceskin
Core
Debonded region
6
The linear elastic stability analysis reduces to an eigen-value analysis to find the bifurcation from the membrane state to the bending state. The eigenvalue satisfies
(21)
where K assembled linear elastic stiffness matrix K g (σ)assembled geometric stiffness matrix φi i th eigenvector or mode shape λi
i th eigenvector or buckling load factor
.Results and Discussion
Three analytical methods were used to calculate the buckling load of the debonded sandwich panel strip,which consisted of an aluminum faceskin and a foam core. Parametric studies were performed to study how
changes in debond length, core moduli, and Poisson’s ratios affect the buckling load. Tables 1 and 2 summarize the data used in the analyses for the Rayleigh-Ritz and finite-difference methods and the plane-strain finite-element analyses. The subscripts (i ,j = 1, 2) for E bij and E fij are along the x - and z -directions, respectively. In gen-eral, the aluminum faceskin and foam core are treated as isotropic materials. However, to simulate the Winkler foundation theory using finite-element analysis, the Poisson’s ratios and the foam longitudinal stiffness (x -direction) were given reduced constitutive properties.The Poisson’s ratios of the faceskin and core,νb and νf ,were set to 0, and Young’s modulus of the sandwich foam,E f 11, was set to approximately 0 (10?4) psi. The reduction of constitutive properties allowed evaluation of the effects of the simplifying assumptions of the elastic foundation approach. In addition, the tests evaluated the effects of transverse shear.
Table 1. Geometric and Material Properties
Length of sandwich beam L , in. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30.0Young’s modulus of beam faceskin E b , psi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.0× 106Elastic foundation modulus K , psi. . . . . . . . . . . . . . . . . . . . . . . . . . .Variable from 0 to 12500Thickness of faceskin t b , in. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0.2Thickness of core t f , in.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0.8
Table 2. Constituent Properties of Finite-Element Model
(a) Set 1
Aluminum faceskin (isotropic):
E b 11 =E b 22 =E b , psi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.0× 106Poisson ratio,νb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0.30
Shear modulus,G b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Core (isotropic):
E f 11 =E f 22 =E f , psi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Variable from 0 to 10000Poisson ratio,νf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0.30
Shear modulus,G f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(b) Set 2
Aluminum faceskin (reduced):
Poisson ratio,νb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0Core (reduced):
E f 11, psi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10?4Poisson ratio,νf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0
K φi λi K g σ()φi +0=i 12…,,=()
E b
21νb +()
----------------------E f
21νf +()
-----------------------
7
Effect of Sandwich Foam Stiffness on Buckling Load
Rayleigh-Ritz method.The Rayleigh-Ritz method was used to perform a parametric study of variations in the foundation stiffness of the core. The dimensionless foundation stiffness F varied from 0 (no core stiffness) to a typical foam core stiffness of 30.0 (K = 12500psi) for a specific value of the debond ratio μ. The debond length was fixed at 2.0 in., or μ = 0.067 in. The results from this study are presented in terms of the relative buckling load P cr /P E where P E denotes the Euler buckling load (π2E b I /L 2) of the debonded faceskin.
A plot of P cr /P E versus F in figure 7 shows the con-vergence of the Rayleigh-Ritz method as the number of shape functions is increased. As expected, the buckling load of the debonded sandwich panel strip increases as the foundation stiffness increases. Twenty-five terms in the Fourier sine series were sufficient to obtain a con-verged solution.
Finite-difference method.The finite-difference method was also used to predict the buckling load. Fig-ure 8 illustrates the convergence of buckling load as the number of gridpoints is increased when using the finite-difference method. Because the finite-difference method discretely models the elastic foundation as a series of closely spaced springs, a large number of gridpoints are
required for a convergent solution. As a result, conver-gence is slower than the Rayleigh-Ritz method. In addi-tion, the solution procedure of this method was inefficient because the buckling load could not be solved for directly as in the Rayleigh-Ritz method. Instead, the buckling load was determined by using successive itera-tions of the bisection method until a root of the polyno-mial equation was found (ref.14). A change of sign of the determinant of the coefficients in the algebraic equa-tions indicates a root (buckling load) between two suc-cessive iterations. Due to the length of the computational time for this analysis, 360 gridpoints were used as the final solution despite the lack of full convergence.Finite-element method.The 2-D plane-strain finite-element model shown in figure 5 was analyzed using 4-noded quadrilateral elements. The finite-element model consisted of 1280 elements and 1385 nodes for a symmetric model. Duplicate nodes are used to model the debonded region. Two eigenvalue analyses were performed using this model. In the first analysis, the aluminum faceskin and foam core were assumed to be isotropic with the properties shown in table 2. The finite-element results from the isotropic case (at F = 30.0)are about 20 percent higher than those from the Rayleigh-Ritz and finite-difference analyses, as shown in figure 9. This difference increases as the foundation stiffness increases.
Figure 7. Convergence of relative buckling load using Rayleigh-Ritz method.
