Calculating Effective Rigidities of a Laminated Composite Beam (Classical Laminate Theory)


The application of fibre-reinforced composite materials in the aerospace industry extends from commercial to military aircraft, such as the Boeing F18, B2 Stealth Bomber, AV-8B Harrier (Jones, 1998). The attractiveness of composites lies in their mechanical properties; such as weight, strength, stiffness, corrosion resistance, fatigue life. Composites are widely used for control surfaces such as ailerons, flaps, stabilizers, rudders, as well as rotary and fixed wings. That is why the analysis of composite structures is imperative for aerospace industry. The main advantage of composites is their flexibility in design. Mechanical properties of the laminate can be altered simply by changing the stacking sequence, fibre lay-up and thickness of each ply which leads to optimization in a design process.


The composite beam is modeled based on the chord-wise bending moment (about the z-axis) being small compared to the span-wise moment (about the y axis, see Figure 2). The chord-wise moment is then neglected. The composite material pertaining to this research is a unidirectional fibre reinforced composite material. The given information of any unidirectional composite material is the elastic modulus in both the longitudinal and transverse axis (see Figures 1 and 2), Poison’s ratio and the shear modulus in the principle directions.

 Effective rigidities for a solid cross-section

The reduced stiffness constants in the material principle directions are:

where T  is the transformation matrix which is used to transform the reduced stiffness constants from the principal material fibre directions to a global (x, y, z) beam coordinates.

Then, the resulting transformed reduced stiffness constants for a unidirectional or orthotropic composite from its principal directions is (Berthelot, 1999):

Both equations (above) can be merged into a single equation commonly known as the “Constitutive Equation”. The constitutive equation describes the stiffness matrix of a laminate plate. The resultant forces and moments are functions of the in-plane strains and curvatures (Berthelot, 1999).

where  is the distance from the mid-plane of the laminate (Figure 3).

For a bending-torsion coupling behaviour the chord wise moment Mx is assumed to be zero so that the kx curvature can be eliminated from (above) and then the matrix  equation (11) reduces to the following form:


The EI, GJ and K represent the effective rigidities of the beam in the global (x, y, z) coordinate system. EI, GJ, and Krepresent, respectively, the bending rigidity, torsion rigidity and bending-torsion coupled rigidity. The effective rigidities are functions of ply angle, thickness, and stacking sequence.



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