Abstract
This study developed to solve the problem of prediction of the natural frequencies of free vibration for laminated beams. The study presented the natural frequencies of composite beams with four layered and different boundary conditions. In each boundary condition, two cases are assumed: movable ends and immovable ends. Numerical results are obtained for the same material to demonstrate the effects of the aspect ratio, fiber orientation, and the beam endmovements on the nondimensional natural frequencies of beams. Two aspect ratios are given in the numerical results, one is for relatively shortthick beams, while the other is for slender beams. It was found that the results of the nondimensional frequencies obtained from the shortthick beams are generally much less than those obtained from the other slender beams for same fiber orientation and generally, the frequencies of longitudinal vibration increase as the aspect ratio increased. It was also found the values of the nondimensional frequencies of the transverse modes are not affected by the longitudinal movements of the ends since these modes are generated by lateral movements only. However, the values of the natural frequencies of longitudinal modes are found to be the same for all beams with movable ends since they are generated by longitudinal movements only.
Highlights
 The study presented the natural frequencies of composite laminated beams.
 First order shear deformation (FSDT) theory is used, and finite element method is employed.
 The aspect or slenderness ratio has a considerable effect on all modes of vibration.
 The natural frequencies of a laminated beam generally increase with the aspect ratio.
 All beams with movable ends have equal longitudinal frequencies of vibration, while those of beams with immovable ends are different.
1. Introduction
Composite have been used in engineering structures over the last four decades or so. They could be seen in a variety of applications as in craft wings and fuselage, satellites helicopter blades, wind turbines boats and vessels, tubes and tanks etc. Their advantages over traditional materials are widely recognized and these are high strength to weight ratio, and their properties which can tailored according to need. Other advantages include high stiffness, high fatigue and corrosion resistance, good friction characteristics, and ease of fabrication. They are made of fiber such as glass, carbon, boron, etc. embedded in matrix or suitable resin that act as binding material. The increasing use of composites has been required a good understanding of composite mechanics and their behavior. Many mathematical models for laminates subjected to static and dynamic loading have been developed. This paper addresses free vibration. The knowledge of the few lower natural frequencies of a structure is utmost importance in order to save it in service from being subjected to unnecessary large amplitude of motion which can cause immediate collapse or ultimate failure by fatigue.
Free vibration analysis of laminated composite beams is presented by P. Subramanian, R. A. JafariTalookolaei et al. and A. Pagani [13] reference [1] used two higher order displacement based shear deformation theories, while references and [2, 3] used the first order shear deformation theory. M. Rueppel et al. [4] studied the damping of carbon fibre and flax fibre angleply composite laminate. Torabi K. et al. [5] Investigated on the effects of delamination size and its thicknesswise and lengthwise location on the vibration characteristics of crossply laminated composite beams. Analytical solutions for free vibration and buckling of composite beams using a higher order beam theory presented by He G. et al. [6]. Vibration prediction of thinwalled composite Ibeams using scaled models analyzed by M. E. Asl et al. [7]. Within that study, which is an extension of Authors’ previous work on design of scaled composite models [810], similitude theory is applied to the governing equations of motion for vibration of a thin walled composite Ibeam. Algarray et al. [11] studied the effects of end conditions of CrossPly laminated composite beams on their dimensionless natural frequencies
2. Modeling analysis
Fig. 1. Showed a composite laminated beam made up of $n$ layers with different orientation, thickness, and properties. Where $L$ is the length, b is breadth and $h$ is depth.
Fig. 1Composite laminated beam
Treat the beam as a plane stress problem and employ firstorder shear deformation theory. The longitudinal displacement ($U$) and the lateral displacement ($W$) can be written as follows:
where $u$ and $w$ are the midplane longitudinal and lateral displacements, $\varphi $ is the rotation of the deformed section about the $y$axis, $z$ is the perpendicular distance from midplane to the layer plane, and $t$ is time.
The StrainDisplacement Relations:
where: ${\epsilon}_{1}$ is the longitudinal strain, and ${\epsilon}_{5}$ is the throughthickness shear strain.
Fig. 2Composite laminated beam with 3noded lineal element
By employing 3noded lineal element as shown in Fig. 2.
The displacements can be expressed in terms of shape function ${N}_{i}$ and nodal displacements:
The shape functions are: ${N}_{1}=\frac{r}{2}\left(1r\right)$, ${N}_{2}=1{r}^{2}$, ${N}_{3}=\frac{r}{2}\left(1+r\right)$.
From Eqs. (2), (3), the strains can be written as:
where:
And ${a}^{e}$ is the vector of nodal displacements ${a}^{e}={\left[\begin{array}{ccc}{u}_{i}& {w}_{i}& {\varnothing}_{i}\end{array}\right]}^{T}$, $i=$ 1, 2, 3.
The stressstrain relation:
where $\sigma ={\left[\begin{array}{cc}{\sigma}_{1}& {\sigma}_{5}\end{array}\right]}^{T}$, $\u03f5={\left[\begin{array}{cc}{\u03f5}_{1}& {\u03f5}_{5}\end{array}\right]}^{T}$ and the matrix containing the transformed elastic constants:
Substitute Eq. (4) in Eq. (5):
The strain energy:
where:
The kinetic energy:
where $\rho $is density and the dot denotes differentiation with time:
where:
In the above derivation it is assumed the motion is harmonic and $\omega $ is circular frequency.
In the absence of damping and external nodal load, the total energy is:
The principle of minimum energy requires that:
The condition yields the equation of motion:
where:
and $n$ is number of elements. To facilitate the solution of Eq. (10), we introduce the following quantities:
where ${K}_{f}$ is the shear correction factor.
The transformed elastic constants are:
In which:
${c}_{66}^{\text{'}}={G}_{12},{c}_{55}^{\text{'}}={G}_{13},{c}_{44}^{\text{'}}={G}_{23},S=\mathrm{s}\mathrm{i}\mathrm{n}\theta ,C=\mathrm{c}\mathrm{o}\mathrm{s}\theta .$
And $\theta $ is the angle of orientation of the ply with respect to the beam axis:
Nondimensional quantities used in the analysis are:
${\stackrel{}{D}}_{11}=\left(\frac{1}{{E}_{1}{h}^{3}}\right){D}_{11},{\stackrel{}{A}}_{55}=\left(\frac{1}{{E}_{1}h}\right){A}_{55},{\stackrel{}{I}}_{1}=\left(\frac{1}{\rho h}\right){I}_{1},$
${\stackrel{}{I}}_{2}=\left(\frac{1}{\rho {h}^{2}}\right){I}_{2},{\stackrel{}{I}}_{3}=\left(\frac{1}{\rho {h}^{3}}\right){I}_{3},\stackrel{}{\omega}=\omega \sqrt{\frac{\rho {L}^{4}}{{E}_{1}{h}^{2}}}.$
The element stiffness matrix:
The mass matrix is 9×9 symmetrical matrix:
${M}^{e}=\int \left[\begin{array}{ccc}{I}_{1}{N}_{i}{N}_{j}& 0& {I}_{2}{N}_{i}{N}_{j}\\ 0& {I}_{1}{N}_{i}{N}_{j}& 0\\ {I}_{2}{N}_{i}{N}_{j}& 0& {I}_{3}{N}_{i}{N}_{j}\end{array}\right]dx.$
3. Results and discussion
3.1. Effect of aspect ratios
Two aspect ratios are given in the numerical results, which are 10 and 50. The first one is for relatively shortthick beams, while the other is for slender beams. The results of the nondimensional frequencies obtained from the aspect ratio 10 are generally much less than those obtained from the other aspect ratio 50 for same fiber orientation. For example, the fundamental mode of the nondimensional natural frequencies for a symmetric [30/–30/–30/30] angleply hinged ^{_}hinged beam with immovable ends is 1.9918 for the aspect ratio 10, and 2.1947 for the aspect ratio 50 as can be seen in Table 1.
This observation can be seen in Fig. 3 to Fig. 5 for symmetric [45/–45/–45/45] angleply laminated beams. These figures show the variation of the nondimensional frequencies with the aspect ratio range from 5 to 40 for the first three modes of vibration for all beams with immovable ends. It is obvious from the figure that the frequency increases rapidly for the range of aspect ratio from 5 to 20, and slows down beyond this range. When the aspect ratio is greater than 20, the beam is slender and consequently shear deformation and rotary inertia have small noticeable effects on the natural frequencies.
Table 2 shows the effect of aspect ratio in nondimensional frequencies for symmetric [45/–45/–45/45] angleply beams. The percentage increase in the nondimensional frequencies, for the first range of the aspect ratio, increases sharply as the mode order increased for all boundary conditions. For the second range, the percentage increase in frequencies is independent on the mode order. The longitudinal modes of free vibration are also affected by the change of aspect ratio. Generally, the frequencies of longitudinal vibration increase as the aspect ratio increased.
Table 1Nondimensional natural frequencies ω¯=ωρL4/E1h2 [30/–30/–30/30] composite beams with different aspect ratio
Mode No.  Beam type aspect ratio ($L/h=$ 10)  
CF  HH  CC  HC  HF  FF  
1  0.7465  1.9918  3.4380  2.7113  3.0503  4.3728 
2  3.7279  6.4128  7.4386  6.9645  7.9206  9.5329 
3  8.4193  11.4744  12.0720  11.7816  12.1545^{*}  14.9573 
4  12.1545^{*}  16.5865  16.9085  16.7518  13.1707  20.2046 
5  13.4194  21.6353  21.8199  21.7278  18.3742  24.3090^{*} 
6  18.5147  24.3090^{*}  24.3090^{*}  24.3090^{*}  23.4730  25.3532 
7  23.5463  26.6166  26.7237  26.6709  28.4805  30.3492 
8  28.5288  31.5448  31.6114  31.5778  33.4158  35.3042 
9  33.4400  36.4344  36.4743  36.4548  36.4635^{*}  40.0877 
10  36.4635^{*}  41.2968  41.3227  41.3093  38.2900  44.9060 
11  38.3118  46.1413  46.1540  46.1482  43.0983  48.6181^{*} 
12  43.1022  48.6181*  48.6181*  48.6181*  47.7785  48.9418 
Mode No.  Beam type aspect ratio ($L/h=$ 50)  
CF  HH  CC  HC  HF  FF  
1  0.7837  2.1947  4.8908  3.4027  3.4251  4.9666 
2  4.8503  8.6633  13.1481  10.8187  10.9431  13.4813 
3  13.3186  19.0823  24.9877  21.9844  22.3328  25.8305 
4  25.4001  32.9800  39.8324  36.3924  37.1019  41.4535 
5  40.6240  49.8113  57.1574  53.4997  54.6965  59.7934 
6  58.4417  69.0255  76.4775  72.7861  60.7725^{*}  80.3045 
7  60.7725^{*}  90.1153  97.3759  93.7899  74.5680  102.4978 
8  78.3401  112.6436  119.5087  116.1227  96.2177  121.5450^{*} 
9  99.8689  121.5450^{*}  121.5450^{*}  121.5450^{*}  119.2209  125.9612 
10  122.6521  136.2519  142.5999  139.4697  143.2325  150.3635 
11  146.3874  160.6577  166.4325  163.5836  167.9831  175.4481 
12  170.8386  185.6456  190.8379  188.2740  182.3175^{*}  201.0230 
(*) Modes with predominance of longitudinal vibration 
Table 2The effect of aspect ratio in nondimensional frequencies for symmetric [45/–45/–45/45] angleply beams
Beam type  Approximate % increase in nondimensional frequencies  
Aspect ratios from 5 to 20  Aspect ratios from 20 to 40  
1st. mode  2nd. mode  3rd. mode  1st. mode  2st. mode  3rd. mode  
CF  25  50  100  10  6  9 
HH  25  62  100  10  6  9 
CC  68  120  150  10  10  15 
HC  45  85  125  10  9  14 
HF  22  70  220  8  6  10 
FF  85  115  145  12  8  12 
3.2. Effect of axial movements of the ends
From the results of Table 1, that the values of the nondimensional frequencies of the transverse modes are not affected by the longitudinal movements of the ends since these modes are generated by lateral movements only (at the yellow shaded). However, the values of the natural frequencies of longitudinal modes are found to be the same for all beams with movable ends since they are generated by longitudinal movements only. Table 3 shows this observation for symmetric [60/–60/–60/60] laminated beams with aspect ratio of 10.
Fig. 3Effect of aspect ratio on natural frequencies of a symmetric [45/–45/–45/45] crossplay clampedfree beam
Fig. 4Effect of aspect ratio on natural frequencies of a symmetric [45/–45/–45/45] crossplay clampedclamped beam
Fig. 5Effect of aspect ratio on natural frequencies of a symmetric [45/–45/–45/45] crossplay freefree beam
Table 1 also shows the fundamental modes of longitudinal vibration for various beams with immovable ends for the symmetric case [30/–30/–30/30] and for two aspect ratios, 10, and 50. It could be noticed that the values of nondimensional natural frequencies of the longitudinal vibration for the clampedfree and hingedfree beams are equal, and those of the other beams are also the same. This phenomenon occurs since both clampedfree and hingedfree beams with immovable ends are the same when restricted from executing longitudinal motion at the ends. Similarly, the rest of beams with immovable ends have the same longitudinal end conditions.
Table 3The first two nondimensional modes of longitudinal free vibration of [60/–60/–60/60] laminated beams with aspect ratio 10
Beam ends  Mode No.  Beam type  
CF  HH  CC  HC  HF  FF  
Immovable  1  5.6426  11.2852  11.2852  11.2852  5.6426  11.2852 
2  16.9279  22.5705  22.5705  22.5705  16.9279  22.5705  
Movable  1  11.2852  11.2852  11.2852  11.2852  11.2852  11.2852 
2  22.5705  22.5705  22.5705  22.5705  22.5705  22.5705 
3.3. Verification
The natural frequencies results which obtained by this study are closer with Abramovich [12] results, as show in Table 4, and difference between two results less than 0.6 % for cantilever and clampclamp beams.
A thirdorder shear deformation theory was used by Kant et al. [13] in the analysis of the free vibration of composite and sandwich simply supported beams. Two comparisons of nondimensional natural frequencies between the present method (using FSDT) and the results of this reference are presented in Table 5, which presented a comparison for symmetric [0/90/90/0] crossply laminated beams respectively, with aspect ratio of ($L/h=$ 5), where the shear effect is significant. The comparison shows a difference of less than 3.3 % associated with the fundamental frequency and less than 4.5 % for higher modes. These differences are due to the employment of different shear theories as stated bellow.
Table 4Nondimensional frequencies of [0/90/90/0] composite beams with immovable ends and aspect ratio 10
Mode No.  Cantilever  Clamp clamp  
Present  Ref. [11]  Present  Ref. [11]  
1  0.8866  0.8819  3.6855  3.7576 
2  4.1062  4.0259  7.7244  7.8718 
3  8.9536  9.1085  12.381  12.573 
4  11.504  12.193  17.192  17.373 
5  13.924  14.080  22.119  22.200 
6  18.980  18.980  23.007^{*}  23.007 
Table 5Nondimensional frequencies of [0/90/90/0] composite beams with simple support ends and aspect ratio 5
Mode No.  Present  Ref. [13] 
1  1.7619  1.820 
2  4.2749  4.528 
3  6.7214  7.201 
4  9.1414  9.814 
5  11.5783^{*}  – 
(*) Mode with predominance of longitudinal vibration 
It is clear, from the above comparisons, that the differences are very small even for higher modes. This confirms the accuracy of the method of analysis and the computer program.
4. Conclusions
In this paper, free vibration of four layered composite beams has been studied. Both secondary effects of transverse shear deformation and rotary inertia were included in the analysis. A firstorder shear deformation theory was applied in the analysis. A finite element model has been formulated to predict the nondimensional natural frequencies and to study the influence of aspect ratio and movable ends of fibers on the natural frequencies. Different end conditions were studied which are clampedfree, hingedhinged, clampedclamped, hingedclamped, hingedfree, and freefree beams with immovable and movable ends. The main conclusion is the natural frequencies of a laminated beam generally increase with the aspect ratio and all beams with movable ends have equal longitudinal frequencies of vibration, while those of beams with immovable ends are different. Namely, clampedfree and hingedfree beams with immovable ends have equal longitudinal frequencies, and the other beams have also equal longitudinal frequencies.
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