A Beam Theory for Laminated Composites and Application to Torsion Problems Dr. BhavaniV. Sankar Presented By: Sameer Luthra EAS 6939 Aerospace Structural Composites 1
Introduction Composite beams have become very common in applications like Automobile Suspensions, Hip Prosthesis etc. Unlike beams of Isotropic materials, Composite beams may exhibit strong coupling between: Extensional Flexural & Twisting modes of Deformation. There is a need for simple and efficient analysis procedures for Composite beam like structures. 2
Beam Theories EULER-BERNOULLI BEAM THEORY Assumptions: 1. Cross-sections which are plane & normal to the longitudinal axis remain plane and normal to it after deformation. 2. Shear Deformations are neglected. 3. Beam Deflections are small. Euler-Bernoulli eq. for bending of Isotropic beams of constant cross-section: where: w(x): deflection of the neutral axis q(x): the applied transverse load 3
Beam Theories TIMOSHENKO BEAM THEORY Basic difference from Euler-Bernoulli beam theory is that Timoshenko beam theory considers the effects of Shear and also of Rotational Inertia in the Beam Equation. So physically, Timoshenko s s theory effectively lowers the stiffness of beam and the result is a larger deflection. Timoshenko s eq. for bending of Isotropic beams of constant cross-section: where: A: Area of Cross-section G: Shear Modulus : Shear Correction Factor 4
Beam Theories TIMOSHENKO BEAM THEORY(Contd.) Shear Correction Factor Timoshenko Defined it as: Significance of Shear Correction Factor : Multilayered plate and Shell finite elements have a constant shear distribution across thickness. This causes a decrease in accuracy especially for sandwich structures. This problem is overcome using shear correction factors. 5
Objective Derivation of a Beam Theory for Laminated Composites and Application to Torsion problems The solution procedure is indicated for the case of a Cantilever Beam subjected to end loads. A closed form solution is derived for the problem of Torsion of a Specially Orthotropic laminated beam (Coupling Matrix [B] = 0, A 16 = A 26 = D 16 = D 26 = 0). 6
Derivation of a Composite Beam Theory A Beam Theory for Laminated Composite Beams is derived from the shear deformable laminated plate theory. The equilibrium equations are assumed to be satisfied in an average sense over the width of the beam. The Principle of Minimum Potential Energy is applied to derive the Equilibrium equations and Boundary conditions. i.e Beam cross sections normal to the x-axis do not undergo any in-plane deformations. 7
Derivation of a Composite Beam Theory 8
Steps Followed for the Derivation The displacement field in the Beam is derived by retaining the First order terms in the Taylor Series expansion for the plate mid-plane deformations in the width coordinate. E.g. where U(x) is the displacement of points on the longitudinal axis of the beam The laminate constitutive relation is expressed in simple terms as: {F} = [C] {E} where: {F} : Vector of Force and Moment Resultants [C] : Laminate Stiffness Matrix {E} : Vector of Mid-Plane Deformations A new set of Force and Moment Resultants for the Beam are defined as: 9
Steps Followed for the Derivation The strain energy per unit area of the laminate : The strain energy per unit length of the beam : The Strain Energy in the Beam : The Principle of Minimum Potential Energy is applied to derive the Equilibrium equations and Boundary conditions. Force and Moment resultants t are substituted t in the differential equations of Equilibrium in terms of displacement variables to obtain differential equations of equilibrium. These differential equations are then solved for the particular case of a Cantilevered beam of Rectangular cross section subjected to end loads only. 10
Steps Followed for the Derivation Principle of Minimum Potential Energy According to the Principle of Minimum Potential energy, a structure or body shall deform or displace to a position that minimizes i i the total potential energy. The total potential energy,, is the sum of the elastic strain energy,, stored in the deformed body and the potential energy,,of the applied forces. 11
Torsion of Specially Orthotropic Laminated Beams Specially Orthotropic Laminated Beams: The property of Specially Orthotropic Laminated Beams used for this derivation is that they have no coupling effects. i.e. Coupling Matrix [B] = 0, A 16 = A 26 = D 16 = D 26 = 0 Actually Specially orthotropic Laminates is another name given to Symmetric Balanced Laminates. For a Specially Orthotropic Cantilever Beam Subjected to an End Torque T, Angle of Twist ( ) is derived as: where: ; ; 12
Results For the purpose of comparison with available results we introduce a Non-dimensional tip Rotation defined as: So our solution for the tip rotation takes the form: The first term on the right corresponds to classical theory solution for isotropic beams. The shear deformations effects are reflected in the second and third term The third term represents the effect of restrained end at x=0, where warping is prevented. In Figure 2, is plotted as a function of. It shows that restrained end effects are only felt for. Further, the restrained effects are less pronounced as the ratio increases. 13
Figure 2: Results The results obtained can be compared with 2 available results: 1. If we ignore shear deformation i.e. let : This result is identical for an isotropic beam (Boresi, Sidebottom, Seely and Smith, 1978) 2. If we ignore the restrained end effects by letting : This result can be compared with that of (Tsai, Daniel and Yaniv, 1990) for a 0 0 Unidirectional Composite Beam. The Maximum difference between the results is about 11% 14
Conclusions A beam theory for Laminated Composites has been derived. A closed form solution is derived for the problem of Torsion of a Specially Laminated Orthotropic Laminate. The result for Angle of Twist compare well with available soultions. 15
References B.V. Sankar (1993) "A Beam Theory for Laminated Composites and Application to Torsion Problems", Journal of Applied Mechanics, 60(1):246-249. 16