Flexural Vibrations of Beams with Delamination

Shripad P. Thakur Under the guidance of

(110010008) Prof. Mira Mitra

AE 493 BTP - I

Abstract

• The effect of delamination on the natural vibration characteristics of laminated beam type structures is studied.

• An analytical model is presented for beams with through-width delaminations parallel to the beam surface located arbitrarily in both the spanwise and thicknesswise directions.

• The beam is modelled as four separate component segments, each analyzed as an Euler beam.

• An analysis is presented of the case of vibration when the two segments of the beam in the delamination region are constrained to have identical transverse deformation.

Delamination • It is a mode of failure for composite materials.

• In laminated materials, repeated cyclic stresses, impact, and so on can cause layers to separate, forming a mica-like structure of separate layers, with significant loss of mechanical toughness.

• It developes inside the material without much effect on the surface.

Why study Delamination

This all signifies the importance of studying delamination.

• Composites forming integral part in construction of aircrafts

• Reduction in Material properties

• Severity of damage

• Shifts & Create some extra frequencies which may cause resonance if close to resonance frequency

• E.g. Impact loading

Ref: Google images

Use of Composites

Ref: google images

Literature Review

• Shu and Fan presented constrained model based on bimaterial inhomogeneous beam.• Effect of delamination, because of its size and location, on natural

frequency has been studied.

• Shu et. al. have analyzed free vibration of delaminated multilayered beams. They have used both free and constrained mode of analysis. • Two parameter, namely the normalized axial stiffness and the

normalized bending stiffness, are introduced which give better insight on the vibration behavior of delaminated beam.

• Effect of relative slenderness ratio on the vibration of beam is studied. Further, double delamination is also studied.

Literature Review

• Wang et al. have examined the free vibrations of an isotropic beam with a through-width delamination by using four Euler Bernoulli beams connected at the delamination boundaries. In their formulation he considered the coupling effect of longitudinal and flexural motions in delaminated layers. He assumed that the delaminated layers deformed `freely' without touching each other (`free mode'), which was shown to be physically inadmissible by Mujumdar and Suryanarayan.

• Mujumdar and Suryanarayan then proposed a `constrained mode' where the delaminated layers are assumed to be in touch along their whole length all the time, but are allowed to slide over each other. This model was physically admissible and the results was also in the vicinity of the experimental results.

Timoshenko beam and Bernoulli beam

• In the Euler-Bernoulli the cross section is perpendicular to the bending line.

• In a Timoshenko beam you allow a rotation between the cross section and the bending line. This rotation comes from a shear deformation, which is not included in a Bernoulli beam.

• Therefore, the Bernoulli beam is stiffer. However, if the relation between length and thickness is large enough the error between both models is small. You need the Timoshenko beam which works for shorter beam structures.

Problem Statement

• An analytical model is reproduced then proposed for the full cycle analysis of vibration of beams with through-width delaminations.

• Assumption made is that the delaminated layers of the beam are constrained to have identical transverse displacements.

• Formulation is based on a one-dimensional Euler beam analysis in a constrained mode.

• The formulation in this report is done for isotropic beam.

• This will be extended to anisotropic beams in BTP stage 2

Modelling of Delamination

Ref: Flexural vibrations of bemas with delaminations, Mujumdar and Suryanarayan

Analytical Modelling

Free Mode Model Constrained Mode Model

Laminates allowed to slide over each other.

Relative motion between two layers is restricted.

Analytical modelling

• The tendency of one of the delaminated layers to overlap on the other will be resisted by the development of a contact pressure distribution between the adjacent layers.

• Such a pressure distribution would constrain the transverse deformation of these adjacent layers to be identical and thus ensure compatibility.

Constrained mode model

Formulation

Ref: Mechanical Vibrations, S. S. Rao

Formulation

• For integral segments:

• For delaminated segments:

Pd is the magnitude of axial load

• For harmonic motion:

Where ,

Formulation

Boundary conditions• Depending upon the end support:

Continuity conditions• Transevrse displacement:

• Normal slopes:

• Continuity of beniding moments:

• Continuity of axial displacements:

Formulation

Continuity conditions• Total axial contraction/extension:

• Final expression for axial compatibility:

• Substituting this value makes equations homogeneous which can be solved to get eigen vectors and eigen values.

Conclusion And Future Work

• This study includes an analytical formulation for the full cycle linear analysis of the vibration behavior of beams with delaminations. This formulation is applicable to the general case of an arbitrary through-width delamination parallel to the beam surface located anywhere within the beam.

• Effect of the delamination on the frequencies depends not only on its size but is also very sensitive to its location, and on the boundary conditions and the vibration mode.

• Weakening produced by the delaminations are the magnitude of the shear force distribution and the average curvature of the beam over the delamination zone.

• Knowing the theory for isotropic beam very well, it will be easy to extend the formulation for anisotropic laminated composite beams in the stage two of BTech Project.