Buckling Analysis of Debonded Sandwich Panel Under Compressi

NASA Technical Memorandum 4701

Available electronically at the following URL address:632ea37a31b765ce05081447/ltrs/ltrs Printed copies available from the following:

NASA Center for AeroSpace Information National Technical Information Service (NTIS) 800 Elkridge Landing Road5285 Port Royal Road

Linthicum Heights, MD 21090-2934Springfield, VA 22161-2171

(301) 621-0390(703) 487-4650

Introduction

Sandwich construction has been used in aircraft applications over the last 40 years because of its light weight, high bending rigidity, and good fatigue proper-ties (ref.1). However, debonding of the faceskin from the core, which can be by impact damage or defects in the manufacturing process, significantly degrades sand-wich construction performance. In particular, debonding can significantly affect the compression stability and strength of the sandwich structure. Moreover, the pres-ence of debonding is a major concern in sandwich struc-tures because the debonded region may grow under compression (ref.2).

There have been several studies on the buckling of debonded sandwich and delaminated composite struc-tures loaded under compression (refs.2–8). Webster investigated the buckling loads of composite laminates with circular debond defects using the Rayleigh-Ritz method(ref.3). Chai et al. (refs.4 and5) studied buck-ling and postbuckling of delaminated composite plates by modeling the failure in one and two dimensions. Hwu and Hu (ref.2) and Somers et al. (ref.6) derived closed-form solutions to study the buckling and postbuckling of delaminated sandwich beams. They applied elastic frac-ture mechanics (LEFM), using J-integral and Griffith cri-terion to compute strain energy release rates for predicting debond growth. Vizzini and Lagace (ref.7) studied sublaminate delamination in composite lami-nates. In their approach, a one-dimensional (1-D) model of a delaminated sublaminate was developed in which the sublaminate rested on an elastic foundation that mod-eled the resin layer. The buckling load of the sublaminate was predicted using the Rayleigh-Ritz energy method. Kim and Dharan (ref.8) used the approach developed by Vizzini and Lagace with LEFM to study faceskin deb-onding for composite sandwich panels.

This paper examines a sandwich panel with initial through-the-width debonds between the faceskin and sandwich foam core. (See fig. 1.) The buckling load of the panel is computed from a 1-D model, which models a sandwich panel strip as a beam (faceskin) on an elastic foundation (core material), and a two-dimensional (2-D) model, which models the cross section of the panel. The Rayleigh-Ritz and finite-difference methods are used to analyze the 1-D model, while the finite-element method using 4-node quadrilateral plane-strain elements is used to analyze the 2-D model.

Although many investigators used the 1-D beam-on-elastic-foundation model to study the sandwich panel debond problem, little research has been performed (to the authors’ knowledge) to assess the adequacy of mod-eling the foam core material as elastic springs by more accurate analysis methods, such as finite-element analy-sis. In the finite-element model, the foam core material is modeled as an isotropic material. Therefore, because the finite-element model properly includes the behavior of the faceskin and foam core, more accurate predictions of buckling will result.

This paper presents three different numerical meth-ods (Rayleigh-Ritz, finite-difference, and finite-element) for determining the buckling load of a sandwich panel with initial through-the-width debonds. Parametric stud-ies investigating the effects of sandwich foam stiffness and debond length on the buckling load of the debonded Figure 1. Debonded sandwich panel.

x

Abstract

A sandwich panel with initial through-the-width debonds is analyzed to study the buckling of its faceskin when subject to an in-plane compressive load. The debonded faceskin is modeled as a beam on a Winkler elastic foundation in which the springs of the elastic foundation represent the sandwich foam. The Rayleigh-Ritz and finite-difference methods are used to predict the critical buckling load for various debond lengths and stiffnesses of the sandwich foam. The accuracy of the methods is assessed with a plane-strain finite-element analysis. Results indicate that the elastic founda-tion approach underpredicts buckling loads for sandwich panels with isotropic foam cores.

2

faceskin are performed. Finally, the assumption of using the elastic foundation approach to model the foam core is evaluated by comparing the Rayleigh-Ritz and finite-difference results from the 1-D elastic beam foundation model to the results from the 2-D plane-strain finite-element model.

Numerical Methods for Determining Buckling Load

As previously mentioned, the Rayleigh-Ritz, finite-difference, and finite-element methods were used to ana-lyze the buckling of the debonded sandwich panel. In the Rayleigh-Ritz and finite-difference methods, a sandwich panel strip was modeled as a beam on an elastic founda-tion. In the finite-element method, 2-D plane-strain quad-rilateral elements were used to model the cross section of the debonded sandwich panel.

Beam-on-Elastic-Foundation Model

Figure 2(a) shows a strip of unit width b of the sand-wich panel loaded in compression with symmetric through-the-width debonds between the faceskins and core. The debonded region of length l d is located in the middle of the sandwich panel, and the applied compres-sive load is P , as shown in figure 2(b). The thickness of the faceskins is denoted by t b , and the thickness of the sandwich core is denoted by 2t f . Due to symmetry, only the upper half of the sandwich panel strip of length L is modeled, as shown in figure 3. The Young’s modulus and the moment of inertia of the faceskins are denoted by E b and , respectively. The bonded regions l f have foundation support. The faceskin, modeled as a beam, is partially supported by an elastic foundation with modulus k and has simple supported boundary conditions at its ends.

Rayleigh-Ritz method.The total potential energy of the beam-on-elastic-foundation system is expressed as

(1)

where U m and U b are membrane and bending energies,respectively,U f is the potential energy of the elastic foundation, and ? is the potential energy of the compres-sive load P . These energy terms are defined as

(2a)

(2b)

(2c)

(2d)

where A is the cross-sectional area of the faceskin and ,and are the longitudinal and out-of-plane deflections,respectively (ref.9). The extensional strain of the face-skin can be written as

(3)

Introduction into equation (1) gives

(4)

The elastic foundation modulus k (x ) in equation (2c) is

given by

(5)

(a) Sandwich beam with initial symmetric debonded regions between core and faceskin.

(b) Buckled shape of sandwich beam.

Figure 2. Buckling of debonded sandwich panel strip modeled as a beam on an elastic foundation.

I t b 3/12=ΠU m U b U f ?

+++=U m E b A 2

----------εb 2

x

d 0

L

∫

=U b E b I 2

--------w ″()2

dx

L

∫

=U f 1

2

--k x ()w 2dx

L

∫

=?P u ′dx

L

∫

=u w εb u ′1

2

--w ′()

2+=ΠE b A 2----------u ′12--w ′2+

2E b I 2--------w ″()21

2--kw 2Pu ′+++dx 0

L

∫

=

k x ()K ψx ()

=

3

where K is a constant and can be represented as E f b /t f and E f is the Young’s modulus of the sandwich foam core. The variable ψ(x ) is defined as

(6)

For stability of a system in equilibrium, the second variation of potential energy needs to be positive semidefinite,

(7)

When the second variation of potential energy is zero, a neutral equilibrium configuration exists and indicates the onset of buckling. This point is characterized as the bifurcation point (ref.10). The Trefftz criterion states that equation (7) is identically satisfied when the varia-tion of the second variation is zero (ref.11).

The expression for δ2Π is obtained by perturbing the system in equation (4) such that

(8)

where u 0 and w 0 represent the prebuckling configuration and the variations u (x ) and w (x ) are infinitesimally small increments. For the prebuckling configuration,

(9)Introducing equations (8) and (9) into equation (4),assuming inextensional buckling (i.e.,u = 0), and collect-ing the second-order terms gives the second variation of the potential energy:

(10)

In the Rayleigh-Ritz method, the out-of-plane defor-mation at the onset of buckling w of the panel faceskin is approximated as

(11)

where φm are selected functions that satisfy the geometric boundary conditions and A m represents their amplitudes.For simple support boundary conditions, a sinusoidal function is used:

(12)

The Trefftz criterion used with the Rayleigh-Ritz method to determine the buckling load can be written as

(13)

Dimensional parameters are nondimensionalized by

the length L as follows:

(14)

Figure 3. Elastic foundation model of debonded sandwich beam.

P

z , w

x, u

ψx ()1if 0x l f ≤≤0if l f x L l f

–≤≤1if L l f x L

≤≤–

=δ2

Πw []0

≥u u 0u +→w w 0w +→

u 0P

E b A ----------x –=w 00=

δ2

ΠE b I w ″()2K ψw 2P w ′()2

–+[]dx

L

∫

=

w x ()A m φm x ()

m 1

=M

∑

=

φm x ()m πx

L

----------sin =?δ2

Πw []

?A m

----------------------0=m 12…M ,,,=()ξx /L =ηl f /L =μl d /L 12η–==

4

The variable η is the foundation ratio, where a value of zero is equivalent to a faceskin (beam) with no foun-dation (unsupported) and a value of one-half is equiva-lent to a fully supported faceskin (beam). The variable μis the debond ratio and represents the fractional amount of the strip which is debonded. The functions φm in equa-tion (12) and ψ in equation (6) become and ,respectively, after the nondimensionalized parameters in equation (7) are used. Incorporating equation (13) gives the eigenvalue problem:

(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.

Dimensionless foundation stiffness, F

R e l a t i v e b u c k l i n g l o a d , P c r /P E

8This discrepancy may have been induced by model-

ing the core material as a Winkler foundation in which

the elastic springs represent the core. To verify that the

discrepancy is caused by the simplifying assumptions of the elastic foundation approach, a second finite-element analysis was performed with reduced constituent Figure 8. Convergence of relative buckling load using finite-difference method.

Figure 9. Comparisons of relative buckling load results from different methods for variations in foundation stiffness.

240 pisions

Dimensionless foundation stiffness, F

R e l a t i v e b u c k l i n g l o a d , P c r /P

E Dimensionless foundation stiffness,

F R e l a t i v e b u c k l i n g l o a d , P c r /P E 7

9

properties. The Poisson’s ratios of the aluminum face-skin and foam core,νb and νf , were set to 0, and Young’s modulus of the foam,E f 11, was set to approximately 0(10?4) psi. This reduction of the material properties elim-inates the effects of the Poisson’s ratios and the longitu-dinal stiffness of the core. Figure 9 shows a comparison of the second set of finite-element results with the Rayleigh-Ritz and finite-difference methods. Excellent agreement is observed between the finite-element analy-sis using the reduced material properties and the results from the Rayleigh-Ritz and finite-difference methods.

Effect of Debond Length on Buckling Load In a second parametric study, the debond ratio μ was varied from 0 (fully bonded) to 1 (fully debonded) for a fixed value of F = 30.0 (K = 12500psi). The limiting value of μ = 0 yields the buckling load for a sandwich panel strip with no debonds, and a value of μ = 1 yields the Euler buckling load P E for a fully debonded faceskin.The results are presented in figure 10 in terms of the nor-malized buckling load P cr /P FULL , which is the ratio of the buckling load pided by the buckling load of a beam fully supported by an elastic foundation (ref.10):(22)

where n is the number of half sine waves required to give the lowest buckling load, depending upon the parameters.

Fifty terms in the Rayleigh-Ritz method and

360gridpoints in the finite-difference method were used in this study. In the finite-element analysis, two sets of constituent properties were again used. Set 1 used isotro-pic properties and Set 2 used reduced constituent proper-ties (νf 12 =νb 12 = 0,E f 11≈ 0). The results in figure 10show that the normalized buckling load decreases as μincreases, indicating that the buckling load for the unsup-ported skin decreases as the debonded length increases.The results from the isotropic plane-strain finite-element analysis are about 10–20 percent higher (depending on the debond length) than those predicted by the Rayleigh-Ritz and finite-difference methods. This suggests that the simplifying assumptions on Poisson’s ratio and the longi-tudinal stiffness of the core can substantially affect buck-ling load predictions. However, excellent agreement is observed once these properties are reduced.

Effect of Transverse Shear on Buckling Load Transverse shear effects were found to be insignifi-cant for the studies performed in this paper. From equa-tion (4.76) in reference (ref.10), the buckled fully bonded faceskin is calculated to have a wavelength of 2.68 in. for the largest K considered in this study (12500 psi). This corresponds to a l d /t b ratio of 13.4,which is estimated to contribute less than 2-percent error for the buckling load of the debonded faceskin using ref-erence (ref.15). Therefore, transverse shear effects were not considered in the Rayleigh-Ritz and finite-difference analyses.

P FULL n 2

π2

E b I L 2--------------kL 2

n 2π

2-----------+=n 12…,,=()Figure 10. Comparisons of normalized buckling load results from different methods for various debond ratios and F =30.0.

Debond ratio, μ

N o r m a l i z e d b u c k l i n g l o a d , P c r /P F U L L

≈ 0)

Conclusions

Three numerical methods were used to analyze the buckling of a debonded aluminum faceskin on a sand-wich panel with a foam core. The Rayleigh-Ritz and finite-difference methods were used to study a 1-D beam-on-elastic-foundation model that represents a strip of a sandwich panel with a through-the-width debond. The results were compared with the buckling loads cal-culated from a 2-D finite-element analysis using 4-noded quadrilateral plane-strain elements. Two parametric stud-ies were performed to evaluate the effects of the elastic foundation stiffness and debond length. Results from the three analytical methods were compared and discussed. As expected, the results showed that the buckling load increases with increasing core foundation stiffness and decreases with increasing debond length.

The Rayleigh-Ritz method was shown to be superior to the finite-difference method in obtaining the buckling loads because it requires less computational time and exhibits fast convergence. However, both the Rayleigh-Ritz and finite-difference methods assumed a Winkler core foundation and, thus, ignored the Poisson’s ratio and longitudinal stiffness effects of the core material. Results from the Rayleigh-Ritz method and the finite-difference method, which are based on the elastic foundation approach, show lower buckling loads than those pre-dicted by the 2-D plane-strain finite-element analysis, which models the sandwich foam core as an isotropic material. When the core is forced to behave as a Winkler foundation so that the Poisson’s ratios of the core are set to 0 and its longitudinal stiffness is reduced to be infini-tesimally small, the finite-element analysis agrees with the results from the Rayleigh-Ritz and finite-difference methods. Therefore, to accurately predict the buckling load of a debonded sandwich panel with sandwich foam cores, a finite-element model that includes the Poisson’s ratio and longitudinal core stiffness should be used. NASA Langley Research Center

Hampton, V A 23681-0001

October 16, 1995

References

1.Plantema, Frederik J.:Sandwich Construction. The Bending

and Buckling of Sandwich Beams, Plates, and Shells. John Wiley & Sons, Inc., 1966.

2.Hwu, Chyanbin; and Hu, Jian S.: Buckling and Postbuckling

of Delaminated Composite Sandwich Beams.AIAA J., vol. 30, no. 7, July 1992, pp. 1901–1909.

3.Webster, John D.:Flaw Criticality of Circular Disbond

Defects in Compressive Laminates—1980–81 Interim Report.

NASA CR-164830, June 1981.

4.Chai, Herzl; Babcock, Charles D.; and Knauss, Wolfgang G.:

One Dimensional Modelling of Failure in Laminated Plates by Delamination Buckling.Int. J. Solids & Struct., vol. 17, no.11, 1981, pp. 1069–1083.

5.Chai, Herzl; and Babcock, Charles D.: Two-Dimensional

Modelling of Compressive Failure in Delaminated Laminates.

J. Compos. Mater., vol. 19, Jan. 1985, pp. 67–98.

6.Somers, M.; Weller, T.; and Abramovich, H.: In?uence of Pre-

determined Delaminations on Buckling and Postbuckling Behavior of Composite Sandwich 632ea37a31b765ce05081447pos. Struct., vol. 17, no. 4, 1991, pp. 295–329.

7.Vizzini, A. J.; and Lagace, P. A.: The Buckling of a Delami-

nated Sublaminate on an Elastic Foundation.J. Compos.

Mater., vol. 21, Dec. 1987, pp. 1106–1117.

8.Kim, W. C.; and Dharan, C. K. H.: Facesheet Debonding

Criteria for Composite Sandwich Panels Under In-Plane Compression.Eng. Fract. Mech., vol. 42, no. 4, July 1992, pp.643–652.

9.Brush, Don O.; and Almroth, Bo O.:Buckling of Bars, Plates,

and Shells. McGraw-Hill, Inc., 1975.

10.Chen, Wai-Fah; and Atsuta, Toshio:Theory of Beam Columns.

V olume 1—In-Plane Behavior and Design. McGraw-Hill, Inc., 1976.

632ea37a31b765ce05081447nghaar, Henry L.:Energy Methods in Applied Mechanics.

John Wiley & Sons, Inc., 1962.

12.Stewart, Caroline B.:The Computational Structural Mechan-

ics Testbed User’s Manual. NASA TM-100644, 1989.

13.Park, K. C.; and Stanley, G. M.: A Curved C0Shell Element

Based on Assumed Natural-Coordinate Strains.J. Appl.

Mech., vol. 53, no. 2, June 1986, pp. 278–290.

14.Barnard, David T.; and Skillicorn, David B.:Effective

Fortran77 for Engineers & Scientists. William C. Brown Publ., 1988.

15.Timoshenko, S.:Theory of Elastic Stability. McGraw-Hill,

Inc., 1936.

10

本文来源：https://www.bwwdw.com/article/xlwq.html