Calculating Effective Rigidities of a
Laminated Composite Beam (Classical Laminate Theory)
Introduction
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.
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:
where,
The EI, GJ
and K represent the effective rigidities of the beam in the global (x,
y, z) coordinate system. EI, GJ, and K represent,
respectively, the bending rigidity, torsion rigidity and bending-torsion
coupled rigidity. The effective rigidities are functions of ply angle,
thickness, and stacking sequence.
The calculation of
composite rigidities for a box-beam is presented by Armanios and Badir
(1995) and Berdichevsky et al (1992). The Circumferentially
Asymmetric Stiffness (CAS) configuration would produce a bend-twist
coupling. The reduced axial stiffness A(s), coupling stiffness
B(s)
and shear stiffness C(s) can then be developed from the constitutive
equation (9) as:
The resulting effective rigidities are then obtained as:
