eFunda: Classical Lamination Theory
engineering fundamentals Classical Lamination Theory
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Strength Needed in More Than One Direction

Considering its light weight, a lamina (ply) of fiber reinforced composite is remarkably strong along the fiber direction. However, the same lamina is considerably weaker in all off-fiber directions. To address this issue and withstand loadings from multiple angles, one would use a lamination constructed by a number of laminae oriented at different directions.

Basic Assumptions of Classical Lamination Theory

Similar to the Euler-Bernoulli beam theory and the plate theory, the classical lamination theory is only valid for thin laminates (span a and b > 10×thinckness t) with small displacement w in the transverse direction (w << t). It shares the same classical plate theory assumptions:

Kirchhoff Hypothesis
1. Normals remain straight (they do not bend)
2. Normals remain unstretched (they keep the same length)
3. Normals remain normal (they always make a right angle to the neutral plane)

 
In addition, perfect bonding between layers is assumed.

Perfect Bonding
1. The bonding itself is infinitesimally small (there is no flaw or gap between layers).
2. The bonding is non-shear-deformable (no lamina can slip relative to another).
3. The strength of bonding is as strong as it needs to be (the laminate acts as a single lamina with special integrated properties).

Classical Lamination Theory From Classical Plate Theory

The classical lamination theory is almost identical to the classical plate theory, the only difference is in the material properties (stress-strain relations). The classical plate theory usually assumes that the material is isotropic, while a fiber reinforced composite laminate with multiple layers (plies) may have more complicated stress-strain relations.

The four cornerstones of the lamination theory are the kinematic, constitutive, force resultant, and equilibrium equations. The outcome of each of these segments is summarized as follows:

Kinematics:
where u0, v0, and w0 are the displacements of the middle plane in the x, y, and z directions, respectively. Please note that some literature may define kxy as the total skew curvature which eliminates the factor of 2. Also note that Kirchhoff's assumptions are introducted to simplify the displacement fields.
Constitutive:
alternatively,

where the subscript k indicates the kth layer counting from the top of the laminate.
Resultants:

Again, the subscript k indicates the kth layer from the top of the laminate and N is the total number of layers. Note that perfect bonding is assumed so we can move the integration inside the summation.
Equilibrium:

Forming Stiffness Matrices: A, B, and D

The plate is assumed to be constructed by a homogeneous but not necessarily isotropic material and subjected to both transverse and in-plan loadings. Also, the Cartesian coordinate system is used. The goal is to develop the relations between the external loadings and the displacements. However, the relations between the resultants (forces N and moments M) and the strains (strains e and curvatures k) are of most interest in practice.

Replace the stresses in the force and moment resultants with strains via the constitutive equations, we have

By applying the summation and integration operations to their respective components, the force and moment resultants can be further simplified to

Combine the above equations we can write:

where A is called the extensional stiffness, B is called the coupling stiffness, and D is called the bending stiffness of the laminate. The components of these three stiffness matrices are defined as follows:

where tk is the thickness of the kth layer and is the distance from the mid-plan to the centroid of the kth layer. Forming these three stiffness matrices A, B, and D, is probably the most crucial step in the analysis of composite laminates.

In some situations, strains expressed in terms of resultants are more handy. The strain-resultant relations can be derived with appropriate matrix operations:

where

Note that A, B, D and A*, B*, D* are all symmetric matrices. Among them, A, B, and D are considered universal notations in the field of composites, i.e., the same notations appear in almost all literature of composite materials. A*, B*, and D*, on the other hand, are not.

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