1) The document presents an analytical model for studying the natural vibration characteristics of beams with through-width delaminations located arbitrarily in the spanwise and thicknesswise directions. 2) The beam is modeled as four separate Euler beam segments, and the model considers the case where the two segments in the delamination region are constrained to have identical transverse deformation. 3) Formulations are developed based on the constrained mode where delaminated layers are assumed to touch and slide over each other, with their transverse displacements constrained to be identical. Boundary conditions and continuity equations are applied to solve for natural frequencies and mode shapes.

1. Flexural Vibrations of Beams with
Delamination
Shripad P. Thakur Under the guidance of
(110010008) Prof. Mira Mitra
AE 493 BTP - I

2. 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.

3. 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.

4. 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

5. Use of Composites
Ref: google images

6. 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.

7. 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.

8. 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.

9. 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

10. Modelling of Delamination
Ref: Flexural vibrations of bemas with delaminations, Mujumdar and Suryanarayan

11. Analytical Modelling
Free Mode Model Constrained Mode Model
Laminates allowed to slide over
each other.
Relative motion between
two layers is restricted.

12. 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

13. Formulation
Ref: Mechanical Vibrations, S. S. Rao

14. Formulation
• For integral segments:
• For delaminated segments:
Pd is the magnitude of axial load
• For harmonic motion:
Where ,

15. Formulation
Boundary conditions
• Depending upon the end support:
Continuity conditions
• Transevrse displacement:
• Normal slopes:
• Continuity of beniding moments:
• Continuity of axial displacements:

16. 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.

17. 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.