Vibration Analysis of Multilayered beam of Graphite Epoxy, Epoxy E- Glass Composites based on layup sequence, fibre orientation and boundary conditions

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 10 Issue: 09 | Sep 2023 www.irjet.net p-ISSN: 2395-0072
© 2023, IRJET | Impact Factor value: 8.226 | ISO 9001:2008 Certified Journal | Page 410
Vibration Analysis of Multilayered beam of Graphite Epoxy, Epoxy E-
Glass Composites based on layup sequence, fibre orientation and
boundary conditions
Pothuraju Tharun1, Dr Kalapala Prasad2
1PG student, Department of Mechanical Engineering, UCEK, JNTU, Kakinada, AP, India
2Assistant Professor, Department of Mechanical Engineering, UCEK, JNTU, Kakinada, AP, India
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Abstract - Beam is a basic structural element that primarily resists loads applied laterallytoitsaxis. Beamshaveawiderangeof
engineering applications such as airplane wings, Helicopter blades, robot arm, medical instruments, turbine blades, automotive
industries, sports equipment etc. When these beams are made up of laminated composites their strength to weightratioincreases
and these can be used in different applications by varying the stacking sequenceinthelaminatewithsame weightand dimensions.
So, this requires a complete analysis of laminated composite beams. It is importanttostudymodalanalysisofcompositestructures
as they operate in complex environmental conditions and are frequently exposed to a variety of dynamic excitations. So,thiswork
aims to study the natural frequencies and mode shapes of four layered composite beam under various stacking sequences,
materials and boundary conditions. In this research work Classicalbeamtheory isusedforvibrationanalysisandnon-dimensional
natural frequencies are calculated by the FEM modal prepared by using 281 shell elementwhich ishaving8 nodeswith sixdegrees
of freedom at each node in Ansys APDL. Finally, a Mat lab code is developed to validate the results obtained in Ansys. It is observed
that as the angle of orientation increases the natural frequency decreases, as the stiffness increases the natural frequency
increases.
Key Words: Laminated composite beam, free vibration, Natural frequencies, Finite element analysis
1.INTRODUCTION
Composite materials have gained many applications in recent decades like in aerospace, automotive, and civil engineering
structures due to their many advantages, Such as Highstrength/rigiditytoweightratio,highstiffnesstoweightratio,Corrosion
resistant and tuning of fibre angles in different layers to obtain the required properties. The complex structures in various
fields of engineering such as Aerospace, Mechanical, Civil,Naval andautomobilearemadeupofsimplestructural memberslike
beams. So, Evaluation of free Vibration behaviour of a structure is an essential Consideration in the design of a structure.
Understanding the fundamental frequency parameter of beams will be helpful inthedesignofstructural members intheinitial
stages of design. Numerous methods and materials have been developed by various researchestoknowdynamic behaviourin
past decades. Rudy Lukez [1] described about the various applications of Graphite epoxy composites because of their high
strength to weight ratio, high stiffness to weight ratio and near-zero coefficient of thermal expansion. Graphite epoxy
composite solves the problem with the space environment like radiation, physical demands based on size and weight etc.
Ganesh kumar Tirumalasetty [2] discussed about the applicationofglass-fiberreinforcedepoxycompositeinmanufacturingof
train Components with its greater strength and impact resistance. Nitesh Talekara et al [3]describedmathematical procedure
for the free Vibration analysis of four layered composite cantilever beam using first order shear deformation beam theory by
Varying layup sequence and Thickness ratios. They observed natural frequencies of all the modes are highly sensitive to a
smaller layup angle than the higher lay up for all the boundary conditions.
Channabasavaradhy Suragimath [4] studiedtheVibrationanalysisofcomposite beam usingmatlab.Thestudyinvolvedfinding
the natural frequency and mode shapes of structure made up of Glass-epoxy, Carbon epoxy and Graphite fibre reinforced
polyamide materials. Euler's Bernoulli beam theory is used for analytical Solution and to construct Mat lab codes. The natural
frequency is maximum at the fixed-fixed Condition when compared with all other boundary conditions and higher natural
frequencies were found in Carbon-epoxy composites due to higher flexural rigidity (EI) when comparedwithothercomposite
materials. Graphite - fibre reinforced polyamide Composite has shown higher natural frequencies when comparedwithglass-
epoxy composites. Nitesh Talekar [5] described mathematical procedure for the free Vibration analysis of four layered
composite beam using first order shear deformation beam theory by various boundary conditions and effectof poissonsratio.
Mahmoud yassin osman et.al [6] discussed about the study of free vibration of rectangle laminated Composite beamsbyusing
first order Shear deformation theory. In this paper, formulated the mathematical equations and verified with fem method for
Graphite epoxy composite structure for various boundary conditions. The results obtained were compared with previous
papers and found in good agreements. Priyadarshini Das and shishir kumar Sahuta [7] discussed about experimental and

numerical free Vibrationanalysisofindustry-driven wovenfibrelaminatedglass/epoxycompositebeams.Theresultsconclude
that the free vibration finite element predictions for glass/epoxy are Sensitive to effects of different boundary conditions and
Span-to-thickness ratio. The nature of the supportsattheedgesinfluencesthefreeVibrationfrequencies.Dueto rigidcondition
at both the ends, fixed-fixed beam shows higher frequencies than other types of beams. Rajesh Kumar and v. Hariharan [8]
discussed about free Vibration of Hybrid Composite beams by varying aspect ratio using Ansys 12.0. The natural frequencies
are maximum for smaller aspect ratios and it decreases once the aspect ratio increases. Also,thetwistingoccursatlowermode
number for smaller aspect ratio and occurs at higher mode number for higher aspect ratios. chandrasekhara and Bangera [9]
presents the equation of motion for laminated Composite beams based on a higher order platetheory.Thenatural frequencies
for symmetric and unsymmetric laminated beams under various boundary conditions are discussed and suggested that the
mode shapes for cross ply clamped-clamped beams indicate thatthe effectsofsheardeformationaregreaterforhighermodes.
Jun-et al. [10] introduced a dynamic finite element method by first order shear deformation theory for free VibrationAnalysis
of generally layered Composite beam. Hamilton'sprincipleisusedtodesirethecoupleddifferential equationswhichgovernthe
free Vibration of generally layered composite beam. Shi and Lam [11] used third order beam theory for a new finite element
formulation for the free vibration analysis of thelayered compositebeams.Thecouplingmassmatricesandhigher orderhave a
negligible effect on the laminated fundamental frequencies,buttheyhavea significant effectonthe highermodal frequenciesof
flexural vibration. Bhimaraddi and Chandrasekhar [12] found the basic equations of the beam theory based on the parabolic
Shear de formation theory for the laminated beams by a systemic reductionoftheconstituterelationsof thethree-dimensional
anisotropic body. Banerjee and williams [13] presented anexactdynamicstiffnessmatrixforcomposite beam with theimpacts
of Shear deformation, rotary inertia and coupling between the bending and torsional deformations.
Table -1: Nomenclature
, , In Plane Forces Bending stiffness
, Bending moment Transformed elasticity constants
Twisting moment Bending stiffness about y-axis
, , Mid- plane strains L Length of the Beam
, Bending curvatures b Width of the Beam
twisting curvature h Height of the Beam
Extensional stiffness ρ Density of the Beam
Coupling stiffness ө Angle of orientation of the Fiber
2. MATERIALS AND MECHANICAL PROPERTIES
Table -2: Properties of Composite Materials
Material Young’s Modulus Shear Modulus Poisson’s Ratio
E1 E2 E3 G12 G23 G13 µ12 µ23 µ13
Graphite-
Epoxy
144.8e9 pa 9.65e9 pa 9.65e9 pa 4.14e9 pa 3.45e9 pa 4.14e9 pa
0.3 0 0
EpoxyE-glass 45e9 pa 10e9 pa 10e9 pa 5e9 pa 5e9 pa 3.84e9 pa 0.3 0.3 0.4

3. METHODOLOGY
3.1. Analytical Method –
Consider a beam of length L, breadth b and total thickness h, which is laminated of a finite number of orthotropic layers of
thickness hi with the principal material axes of each layer being oriented with respect to the beam mid-plane.
.
Fig, 1 Geometry of Laminated Composite Beam
By the classical lamination theory, the constitutive equations of the laminate can be obtained as:
Where,
, and are the in-plane forces,
, and are the bending and twisting moments,
, , are the mid- plane strains,
, , are the bending and twisting curvatures.
, , and , are the extensional stiffness, coupling stiffness, bending stiffness respectively.
For the case of laminated composite beam:
, and =0, , and the curvature assumed to be non-zero.
Then, equation (1) can be written as:
Where,
The transformed reduced stiffness constants (i, j = 1, 2, and 6) are given as:

)
,
3.2. Bernoulli-Navier Hypothesis –
The Euler Bernoulli beam theory (or classical beam theory - CBT) assumes that straight lines perpendicular to the mid-plane
before bending remain straight and perpendicular to the mid-plane after bending. As a result of this assumption, transverse
shear strain is neglected.
Vibration in x-z plane is given by:
=
Where
is the bending stiffness about y axis in N.
L is the length of the beam, ρ is the mass per unit length, and for different boundary conditions is given in table 3.
The subscript i= 1, 2…., indicates the first, second and so forth modes.
For symmetric orthotropic laminated beam:
The bending stiffness about y axis can be obtained by using the relation
= in N.
Where is the element 1-1 of the laminate bending compliance matrix (1/Nm.)
Table 3 The constants for different boundary conditions are:
Boundary Condition Mode 1 Mode 2 Mode 3
Fixed-free 1.87 4.694 7.85
Fixed-fixed 4.73 7.85 10.99
Fixed-pinned 3.92 7.068 10.2102
4. RESULTS
The theoretical formulation in the previous section is appliedtocomputethenatural frequenciesandmodeshapes ofgenerally
layered composite beam. The validation of numerical results is done by using ANSYS APDL software.

The following are the three boundary conditions:
1. Clamped- Clamped (C-C)
2. Clamped- Simply supported (C-S)
3. Clamped- Free (C-F)
Table4.Represents the Fundamental natural frequency of [θ/-θ/-θ/θ] orientation beam for varying boundary conditions.
Graphite epoxy has the following material properties: E1= 144.8 * Pa, E2= E3= 9.65 * Pa,G12=G13=4.14* Pa,G23=
3.45 * Pa. µ12=0.3, ρ= 1389.23 kg/m3, L=0.381m, b=0.0254m, h=0.0254m
Table -4: Fundamental natural frequency (Hz) of [θ/-θ/-θ/θ] orientation of Graphite Epoxy Composite beam
BC’s 0 15 30 45 60 75 90
Ansys Ref Ansys Ref Ansys Ref Ansys Ref Ansys Ref Ansys Ref Ansys Ref
C-C 1376.7 1384.3 1130.4 1125.7 824.63 818.3 564.06 548.4 468.74 475.1 457.04 460.9 459.19 463.7
C-S 1058.6 1090.5 838.72 856.1 593 592.4 390.19 386.3 325.21 323.2 318.84 321 320.65 320.6
C-F 278.23 278.4 209.8 207.2 140.09 137.9 90.903 89.3 75.285 74.7 73.781 73.7 74 74.2
Table-5 Represents the Fundamental natural frequency of [θ/-θ/-θ/θ] orientation beam for varying boundary conditions.
Epoxy E-Glass has the following material properties: E1= 45* Pa, E2= E3= 10 * Pa, G12=5 * Pa, G13= 3.84 * Pa,
G23= 5 * Pa. µ12=0.3, ρ= 2000 kg/m3, L=0.381m, b=0.0254m, h=0.0254m
Table-5. Fundamental natural frequency (Hz) of [θ/-θ/-θ/θ] orientation of Epoxy E-Glass Composite beam
BCs 0 15 30 45 60 75 90
Clamped-clamped(C-C) present 778.65 702.79 571.64 464.75 410.22 393.32 390.5
Clamped-SS(C-S) present 557.51 499.89 401.86 324.07 285.74 274.26 272.4
Clamped-Free(C-F) Present 132.8 117.98 93.814 75.177 66.166 63.426 62.996
Table6. Represents the natural frequencyof[θ/-θ/θ/-θ] orientationbeamforvarying boundaryconditions.Graphiteepoxyhas
the following material properties: E1= 144.8 * Pa, E2= E3= 9.65 * Pa, G12=G13= 4.14 * Pa, G23= 3.45 * Pa.
µ12=0.3, ρ= 1389.23 kg/m3, L=0.381m, b=0.0254m, h=0.0254m
Table-6. Natural frequencies (Hz) of [θ/-θ/θ/-θ] orientation of Graphite Epoxy Composite beam
BCs 0 15 30 45 60 75 90
Clamped-clamped(C-C) ω1 1376.7 1135 855.54 572.93 469.74 457.17 459.19
ω2 1403.5 1462.1 1020.2 630.58 532.12 520.33 524.28
ω3 2083.3 2239.4 2014.3 1475 1243.5 1213.4 1217.9
ω4 3075 2503.5 2272.5 1720 1451.0 1410.6 1399
Clamped-SS(C-S) ω1 1058 879.29 625.27 398.06 325.74 318.89 320.65
ω2 1397.6 1452.9 1016.3 628.42 529.57 517.33 519.50
ω3 2078.8 2175.1 1774.7 1223.9 1022.9 1002.9 1007.9
ω4 2826.1 2319.1 2193 1714.1 1444.2 1402.3 1385.1
Clamped-Free(C-F) ω1 278.47 233.66 154.86 93.326 75.443 73.792 74

ω2 281.34 248.47 161.58 98.737 83.769 82.552 84.331
ω3 1039.2 1080.2 874.88 563.29 462.29 452.69 455.16
ω4 1468.4 1216.9 997.56 614.24 521.11 512.04 518.72
Table7. Represents the natural frequency of [θ/-θ/θ/-θ] orientationbeamforvarying boundaryconditions.Epoxy E-Glasshas
the following material properties: E1= 45* Pa, E2= E3= 10 * Pa, G12=5 * Pa, G13= 3.84 * Pa, G23= 5 * Pa.
µ12=0.3, ρ= 2000 kg/m3, L=0.381m, b=0.0254m, h=0.0254m
Table-7. Natural frequencies (Hz) of [θ/-θ/θ/-θ] orientation of Epoxy E-Glass Composite beam
BCs 0 15 30 45 60 75 90
Clamped-clamped(C-C) ω1 759.84 701.86 586.80 473.65 413.35 395.62 392.96
ω2 809.92 774.98 655.53 525.22 471.79 457.82 454.56
ω3 1776.2 1731.2 1485.4 1239.4 1099.5 1057.4 1051.3
ω4 1869.7 1814 1749.7 1418.8 1275.8 1232 1217.2
Clamped-SS(C-S) ω1 549.41 506.21 416.62 330.81 287.47 275.19 273.37
ω2 806.10 771.07 652.19 522.60 468.53 453.76 449.97
ω3 1620.5 1497 1261.1 1029.3 905.81 870.27 865.09
ω4 1772.7 1804 1739 1411.2 1266.7 1220.5 1204
Clamped-Free(C-F) ω1 132.43 121.66 98.615 77.030 66.447 63.480 63.035
ω2 137.69 128.16 105.19 83.389 74.970 73.090 72.8941
ω3 773.50 712.70 588.40 468.69 408.15 390.91 388.35
ω4 817.41 773.87 646.92 516.26 464.23 451.62 449.21
Table8. Represents the natural frequency of [θ/θ/θ/θ] orientation beam for varyingboundaryconditions.Graphiteepoxyhas
the following material properties: E1= 144.8 * Pa, E2= E3= 9.65 * Pa, G12=G13= 4.14 * Pa, G23= 3.45 * Pa.
µ12=0.3, ρ= 1389.23 kg/m3, L=0.381m, b=0.0254m, h=0.0254m
Table-8. Natural frequencies (Hz) of [θ/θ/θ/θ] orientation of Graphite Epoxy Composite beam
BCs 0 15 30 45 60 75 90
ω2 1403.5 1047.0 709.62 574.01 530.22 522.78 524.28
ω3 2083.3 2329.8 1654.5 1342.9 1230.3 1212.5 1217.9
ω4 3075 2568.7 1904.9 1553.5 1430.1 1399.7 1399
Clamped-SS(C-S) ω1 1058 697.42 449.02 354.03 322.69 318.75 320.65
ω2 1397.6 1043.4 706.29 570.89 526.46 517.99 519.50
ω3 2078.8 1951.3 1369.5 1106.2 1014.7 1002.5 1007.9
ω4 2826.1 2542.3 1896.0 1545.3 1420.3 1386.9 1385.1
Clamped-Free(C-F) ω1 278.47 166.58 103.94 81.875 74.66 73.757 74

ω2 281.34 169.32 105.35 82.801 75.515 74.630 84.331
ω3 1039.2 952.35 627.15 500.60 458 452.47 455.16
ω4 1468.4 1004.8 648.52 513.08 467.36 460.02 518.72
Table9. Represents the natural frequency of [θ/θ/θ/θ] orientation beam for varying boundary conditions. Epoxy E-Glass has
the following material properties: E1= 45* Pa, E2= E3= 10 * Pa, G12=5 * Pa, G13= 3.84 * Pa, G23= 5 * Pa.
µ12=0.3, ρ= 2000 kg/m3, L=0.381m, b=0.0254m, h=0.0254m
Table-9. Natural frequencies (Hz) of [θ/θ/θ/θ] orientation of Epoxy E-Glass Composite beam
BCs 0 15 30 45 60 75 90
ω2 809.92 774.98 655.53 525.22 471.79 457.82 454.56
ω3 1776.2 1731.2 1485.4 1239.4 1099.5 1057.4 1051.3
ω4 1869.7 1814 1749.7 1418.8 1275.8 1232 1217.2
Clamped-SS(C-S) ω1 549.41 479.98 373.65 311.64 283.64 274.15 273.37
ω2 806.10 771.07 652.19 522.60 468.53 453.76 449.97
ω3 1620.5 1497 1261.1 1029.3 905.81 870.27 865.09
ω4 1772.7 1804 1739 1411.2 1266.7 1220.5 1204
Clamped-Free(C-F) ω1 132.43 112.59 86.525 72.036 65.581 63.397 63.035
ω2 137.69 128.16 105.19 83.389 74.970 73.090 72.8941
ω3 773.50 712.70 588.40 468.69 408.15 390.91 388.35
ω4 817.41 773.87 646.92 516.26 464.23 451.62 449.21
Table10. Represents the natural frequency obtained from Euler Bernoulli’s Equation and Ansys APDL for Graphite Epoxy
Composite Beam
Boundary Conditions Numerical Ansys APDL
Clamped-Clamped (C-C) 1513.12 1376.7
Clamped-Simply supported (C-S) 829.231 1058.6
Clamped-Free (C-F) 295.735 278.23
Table11. Represents the natural frequency obtained from Euler Bernoulli’s Equation and Ansys APDL for E-Glass Epoxy
Composite Beam
Boundary Conditions Numerical Ansys APDL
Clamped-Clamped (C-C) 985.38 778.65
Clamped-Simply supported (C-S) 538.5 557.51
Clamped-Free (C-F) 165.091 132.8

Chart -1: Natural frequency Vs Orientations of Chart-2: Natural frequency Vs Orientations of
Clamped- Clamped Boundary Condition for Graphite Clamped-ClampedBoundaryConditionforE-GlassEpoxy
Composite Beam Epoxy Composite beam
3.
CONCLUSIONS
The finite element model for the free vibration characteristicofthecomposite beamisanalysedinthispaper.Theimpactof lay-
up angle and boundary conditions on the natural frequencies of laminated composite beams are investigated for Graphite
Epoxy and E-Glass Epoxy composites. It is shown that the natural frequenciesobtainedwithAnsysandnumerical aresimilarto
each other. Natural frequency of all the modes is observed to be highly sensitive to a smaller layup angle than a higher lay up
for all the boundary conditions.
The following observations have been made:
1. From the graphs it is clear that as angle of orientation increases, the natural frequency of the beam decreases.
2. From the graphs it is clear that as stiffness of the plate increases, the natural frequency increases.
3. The stiffness is more for Clamped- Clamped(C-C) and less for Clamped-Free(C-F) Boundary Conditions.
Chart -3: Bar chart representing variation between Chart -4: Bar chart representing variation numerical
and Ansys APDL results of Graphite epoxy between numerical and Ansys APDL
beam results of E-Glass epoxy beam

4. The increasing beam stiffness order and natural frequency values for the boundary conditions considered in the
analysis is:
Clamped-Free < Clamped-pinned < Clamped-Clamped
5. From the chart 1 and 2 it is clear that [θ/-θ/θ/-θ] has the highest natural frequency in between 0 to 90 degrees
compared to [θ/-θ/-θ/θ] and [θ/θ/θ/θ] orientations.
6. From the chart 1 and 2 it is clear that [θ/-θ/-θ/θ] has the more natural frequency in between 0 to 90 degrees than
[θ/θ/θ/θ] orientations.
7. From the bar chart 3 & 4 it is clear that the numerical and Ansys results hold well with fewer variations.
REFERENCES
[1] Rudy Lukez, “The use of Graphite/Epoxy composite structures in space applications.
[2] Tirumalasetty, G. K. (2005). APPLICATION OF GLASS FIBER REINFORCED EPOXY COMPOSITEIN MANUFACTUREOF
TRAIN COMPARTMENTS.
[3] Nitesh Talekara (2019). Modal Analysis of Four Layered Composite Cantilever Beam with lay-upsequenceandlength
to thickness ratio. Elsevier.
[4] Chandrashekhara, K., & Bangera, K. M. (n.d.). Free vibration of composite beams using a refined shear flexible beam
element. Computers & Structures (ISSN 0045-7949), vol. 43, no. 4, May 17, 1992, p. 719-727.
[5] Nitesh Talekara (2019). free Vibration analysis of four layered composite beam using first order shear deformation
beam theory by various boundary conditions and effect of poisons ratio. Elsevier.
[6] Osman, M. Y. (2017). Free Vibration Analysis of Laminated Composite Beams using Finite Element Method.
International Journal of Engineering Research and Advanced Technology.
[7] Priyadarshi, & S. K. (2022). Experimental and numerical free vibration. CURRENT SCIENCE, VOL. 122, NO. 9.
[8] R. Hariharan (2012). Free Vibration Analysis of Hybrid-Composite. International Conference on Advances In
Engineering, Science And Management.
[9] J. Bannerjee, F. W. (1996). EXACT DYNAMIC STIFFNESS MATRIX FOR COMPOSITE TIMOSHENKO BEAMS WITH
APPLICATIONS. Journal of Sound and Vibration.
[10] JUN LI, C. S. (2012). Free Vibration Analysis of Generally Layered Composite. Taylor & Francis
[11] Shi G, Lam. (1999). Finite element vibration analysis of composite beams based on higher order beam theory. Journal
of Sound and Vibration.
[12] Chandrashekhara, K., & Bangera, K. M. (n.d.). Free vibration of composite beams using a refined shear flexible beam
element. Computers & Structures (ISSN 0045-7949), vol. 43, no. 4, May 17, 1992, p. 719-727.
[13] F.W.J. Bannerjee, “Exact Dynamic Stiffness Matrix for Composite Timoshenko beams with Applications” , Journal of
Sound and Vibration, 1996.

Vibration Analysis of Multilayered beam of Graphite Epoxy, Epoxy E- Glass Composites based on layup sequence, fibre orientation and boundary conditions

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