### NASA-CR-2330

- Version
- 351 Downloads
- 2.33 MB File Size
- 1 File Count
- November 27, 2015 Create Date
- November 27, 2015 Last Updated

Elastic Stability Of Laminated Flat And Curved Plates

.0 SUMMARY

A method is presented to predict theoretical buckling loads of long, rectangular flat and curved

laminated plates with arbitrary orientation of orthotropic axes in each lamina. The plate is

subjected to combined inplane normal and shear loads. Arbitrary boundary conditions may be

stipulated along the longitudinal sides of the plate. In the absence of inplane shear loads and

“extensional-shear” coupling, the analysis is also applicable to finite length plates.

Numerical results are presented for curved laminated composite plates with various boundary

conditions and subjected to various loadings. These results indicate some of the complexities

involved in the numerical solution of the analysis for general laminates. The results also show that

the “reduced bending stiffness” approximation when applied to buckling problems could lead to

considerable error in some cases and therefore must be used with caution.

2.0 INTRODUCTION

Fiber-reinforced composite materials are finding increased applications in aerospace structures.

Capability to predict the buckling behavior of thin laminated plates made from these materials,

when subjected to combined inplane normal and shear loads, is of prime interest to the structural

analyst. A particular characteristic of these laminated plates, in contrast to homogeneous plates, is

the possible coupling between inplane extension and out-of-plane bending, references 1 and 2. Such

coupling can significantly affect the load response of these plates, reference 3.

Considerable literature exists on the buckling of isotropic and orthotropic plates with various

boundary conditions and subjected to various loadings. References 4 through 12 are examples of

analysis for isotropic plates. Reference 13 presents an extensive treatment of orthotropic plates,

where the axes of orthotropy do not coincide with the plate axes, resulting in “mixed-order

derivatives” in the stability equations. A summary of this and other similar work is given in

references 14 and 15.

Buckling of laminated composite plates has been receiving increased attention in the recent

past. References 16 through 21 are examples of the analysis for flat plates. The phenomenon of

possible bending-stretching coupling in these plates is known to have a detrimental effect and adds

to the complexity of the buckling analysis. The use of “reduced bending stiffness” as formulated in

references 1 and 22 has been used in conjunction with classical orthotropic plate buckling analysis,

to allow for coupling effects, references 18, 20, and 21. The inplane boundary conditions do not

enter into this type of analysis. These boundary conditions are known to significantly influence the

buckling of flat plates in the presence of bending-stretching coupling, references 19 and 23.

Therefore, caution has to be exercised in applying the “reduced bending stiffness” concept. As the

number of laminas increases, the exact solution for certain types of lamina layups approaches the

orthotropic plate solution, reference 3.

Few analytical results are available for laminated composite rectangular curved plates. The

analysis of reference 24 may be readily applied to curved plates which are subjected to biaxial

inplane normal loads and wherein no “shear-extensional” coupling is present. For such laminates,

reference 25 considers the effect of the stacking sequence on buckling.

The buckling analysis presented here considers rectangular flat or curved general laminates

subjected to combined inplane normal and shear loads. The analysis is applicable to (i) finite length

plates, when the plate is “specially orthotropic” (i.e., “16” and “26” elements in equations (A—6)

and (A-7) are zero) and the combined inplane loads do not include shear, and (ii) infinitely long

plates, for all other cases. Arbitrary boundary conditions may be specified along the sides (y =

constant). See figure 1. For finite length plates, simply supported boundary conditions are

stipulated along the ends (x = constant).

The method of analysis is such that it may be readily extended to longitudinally stiffened

structures, subjected to combined inplane normal and shear loads, in a manner analogous to that of

reference 24. A stiffness matrix is derived (from a solution of the “stability equations") relating the

buckling displacements and the corresponding forces along the sides (y = constant) of the plate. The

elements of this matrix are transcendental functions of the external loading and the half-wavelength

of buckling, A, in the x-direction. In general, the stiffness matrix is complex and Hermitian in form.

The buckling criterion is formulated in a deterrninantal form, after enforcing the desired boundary

conditions along the sides of the plate. For a chosen half-wavelength, La buckling load is evaluated

by an iteration procedure using the algorithm discussed in reference 26. Therefore a series of

half-wavelengths must be investigated to determine the minimum buckling load.

File | Action |
---|---|

NASA-CR-2330-Elastic-Stability-of-Laminated-Flat-and-Curved-Plates.pdf | Download |

## Comment On This Post