Forced axisymmetric response of polar orthotropic circular plates of linearly varying thickness resting on Winkler type of elastic foundation has been studied on the basis of classical plate theory. An approximate solution of problem has been obtained by Rayleigh Ritz method, which employs functions based upon the static deflection of polar orthotropic circular plates. The effect of transverse loadings has been studied for orthotropic circular plate resting on elastic foundation. The transverse deflections and bending moments are presented for various values of taper parameter, rigidity ratio, foundation parameter, and flexibility parameter under different types of loadings. A comparison of results with those available in literature shows an excellent agreement.
1. Introduction
The study of vibrational characteristics of plates has been of great interest due to their increasing use in various engineering applications. The development of fibre-reinforced composite materials and their extensive use in fabrication of plate type structural components in aerospace, ocean engineering, and electronic and mechanical components has led to the study of dynamic response of anisotropic plates. The consideration of thickness variation together with anisotropy not only reduces the size and weight of components but also meets the desirability of high strength, corrosion resistance, high temperature performance, and economy. One can fabricate the orthotropic nature of plate in different ways, that is, rectangular and polar orthotropy. The consideration of polar orthotropy in circular and annular plates provides the best approximation to the results, because polar coordinate axes are also the axes for the materials symmetry. An excellent review of vibration of plates has been given by Leissa in his monograph [1] and a series of review articles [2–7]. Reddy [8] and Lal and Gupta [9] have presented an up-to-date survey analysing transverse vibrations of nonuniform rectangular orthotropic plates. In a recent study, Sayyad and Ghugal [10] presented a review of recent literatures on free vibration analysis of Laminated Composite and Sandwich Plates.
A considerable amount of work dealing with vibration of polar orthotropic circular and annular plates of uniform/nonuniform thickness in presence/absence of elastic foundation has been done by a number of workers and is reported in [11–31] to mention a few. The orthotropic plates resting on an elastic foundation find their application in foundation engineering, such as reinforced concrete pavements of high runways, foundation of deep wells and storage tanks, and slabs of buildings (Lekhnitskii [11], p. 136). In this connection, various models such as Winkler [12–17], Pasternak [18], and Vlasov [19] have been proposed in the literature. Laura et al. [13] studied the effect of Winkler Foundation on vibrations of solid circular plate of linearly varying thickness. Gupta et al. [14] analysed the effect of elastic foundation on axisymmetric vibrations of polar orthotropic circular plates of variable thickness. Further the work has been extended to asymmetric vibration of polar orthotropic plates resting on elastic foundation by Ansari and Gupta [15]. Gupta et al. [16] studied buckling and vibration of polar orthotropic circular plate attached to Winkler Foundation. Orthotropic circular/annular plates such as deck, diaphragm, and bulk heads are used as structural components in launch vehicles.
In some technological situations, a plate is exposed to transverse loads on the surfaces with a downward and/or upward thrust, that is, ship container and aeronautical structural components. The stability of these components increases with the support of elastic foundations. Thus the study of the combined effect of transverse loads and elastic foundation on vibrational characteristics of plates is of practical importance. Gupta et al. [16] analysed the complicating effect of elastic foundation on elastic properties of the plates. A number of research papers are available showing the study of transverse deflection and bending moments of polar orthotropic/isotropic plates of variable thickness resting on elastic foundation [20–31] with complicating effects.
The present work is concerned with the study of transverse loads on vibration of polar orthotropic circular plates of linearly varying thickness resting on Winkler type of elastic foundation. The support of elastic foundation provides greater stability to these structural components exposed to transverse loads. An approximate solution is obtained by Ritz method employing functions based on static deflection given by Lekhnitskii [11]. The present choice of functions has a faster rate of convergence as compared to polynomial coordinate functions employed by Laura et al. [13, 21, 26, 27].
2. Analysis
Consider a thin circular plate of radius a and variable thickness h=h(r), resting on elastic foundation of modulus kf, elastically restrained against rotation by springs of stiffness kϕ and subjected to Prcosωt type of excitation extending from r=r0 to r=r1. Let (r,θ) be the polar coordinates of any point on the neutral surface of the circular plate referred to as the centre of the plate as origin (shown in Figure 1(a)).
(a) Plate geometry with side and surface views. (b) Loading structure of the plate.
The maximum kinetic energy of the plate is given by(1)Tmax=12ρω2∫0a∫02πhw2rdθdr,where w is the transverse deflection, ρ the mass density, and ω the frequency in radians per second.
The maximum strain energy of the plate is given by(2)Umax=12∫0a∫02πDr∂2w∂r22+2υθ∂2w∂r21r∂w∂r+Dθ1r∂w∂r2+kfw2rdrdθ+12akϕ∫02π∂wa,θ∂r2dθ,where 1/kϕ is the rotational flexibility of the spring and flexural rigidities of the plate are(3)Dr=Erh3121-υrυθ,Dθ=Eθh3121-υrυθ.The work done by the external force P(r) acting on the plate in the direction parallel to z-axis is given by(4)Vmax=∫02πdθ∫r0r1wPrrdr.
3. Method of Solution: Ritz Method
Ritz method requires that the functional (5)Jw=Umax-Vmax-Tmax=12∫0a∫02πDr∂2w∂r22+2υθ∂2w∂r21r∂w∂r+Dθ1r∂w∂r2+kfw2rdrdθ+12akϕ∫02π∂wa,θ∂r2dθ-∫02πdθ∫r0r1wPrrdr-12ρω2∫0a∫02πhw2rdθdrbe minimised.
Uniformly distributed load P0 extending from r0 to r1 is given by (6)Pr=q0=P0πr12-r02Ur-r0-Ur-r1.Introducing the nondimensional variables R=r/a,R1=r1/a,R0=r0/a and assuming the deflection function as W(R)=w/(a4q0Dr0)(7)WR=∑i=0mAiFiR=∑i=0mAi1+αiR4+βiR1+pR2i,where Ai are undetermined coefficients, p2=Eθ/Er, and αi, βi are unknown constants to be determined from boundary conditions (Leissa [1], p. 14): (8)KϕdFi1dR=-1-α3d2FidR2+υθ1RdFidRR=1Fi1=0.Linearly thickness variation of the plate is assumed as h=h0(1-αR), where α and h0 are taper parameter and thickness of the plate at the centre, respectively.
The choice of the function approximating the deflection of the plate W(R) given in (7) is based upon the static deflection polar orthotropic plates (Lekhnitskii [11]), which has faster rate of convergence (Gupta et al. [16]) as compared to the polynomial coordinate functions used by earlier researchers [13, 21, 26, 27].
Introducing the nondimensional variables W and R in (5), the functional J(W) becomes (9)JW=πDr0∫011-αR3∂2W∂R22+2υθ∂2W∂R21R∂W∂R+p21R∂W∂R2+KfW2RdR+Kϕ∫02π∂W1∂R2dθ-∫R0R12WRRdR-Ω2∫01∫02π1-αRW2RdR,where(10)Dr0=Erh03121-υrυθ,Ω2=a4ω2ρh0Dr0,Kf=a4kfDr0,KΦ=akΦDr0.
The minimisation of the functional J(W) given by (9) requires(11)∂JW∂Ai=0,i=0,1,2,3,…,m.This leads to a system of nonhomogeneous equations in Aj, (12)aij-bijAj=Ci,i,j=0,1,…,m,where A=aij and B=[bij] are square matrices of order m+1 given by(13)aij=∫011-αR3Fi′′Fj′′+2υθFi′′Fj′R+p2Fi′RFj′R+KfFiFjRdR+KϕFi′1Fj′1,bij=∫011-αRFiFjRdR,Cij=∫012FiRdR.The solution of system of (12) gives the value of Ai and thus the transverse deflection W and the radial and transverse bending moments (14)Mra2q0=-1-αR3d2WdR2+υθ1RdWdR,Mθa2q0=-1-αR3υθd2WdR2+p21RdWdRare computed.
4. Numerical Results
Numerical results have been calculated for different values of taper parameter α(=±0.3), Eθ/Er(=5.0), and foundation modulus Kf(=0.01,0.02,0.03) for forced vibration of polar orthotropic circular plate with simply supported (SS-plate) and clamped (CL-plate) edges. The plate is subjected under different types of loadings such as uniform loading on entire plate, annular loading, and disk loading. The natural frequencies for free vibration are obtained by putting P(r)=0. In case of forced vibration the nondimensional frequency parameter is taken as Ω=ηΩ00 for Ω<Ω00 and Ω=Ω00+η(Ω01-Ω00) for Ω00<Ω<Ω01, where η=0.2. The normalised deflection and bending moments are obtained for fixed value of Poisson’s ratio as 0.3.
5. Discussion
Forced axisymmetric response of polar orthotropic circular plates of linearly varying thickness has been analysed for various values of plate parameters. Transverse deflection and bending moments are obtained for circumferentially stiffened plate, that is, Eθ>Er. Bending moments of the plate cannot be obtained for radially stiffened plate Eθ>Er, because infinite stress is developed at the centre in this case (Lekhnitskii [11], p. 372). Transverse deflection and bending moments are presented here under different types of loadings:
When load is distributed uniformly on the disk extending from R0=0.0 to R1=0.5 (disk loaded plate).
When load is distributed uniformly on the annular region extending from R0=0.3 to R1=0.7 (annular loaded plate).
When load is distributed uniformly on the entire region extending from R0=0.0 to R1=1.0 (uniformly loaded plate).
The total load on the plate is the same in all three cases (shown in Figure 1(b)).
The whole analysis of numerical results is presented in Figures 2, 3, 4, 5, 6(a), and 6(b) for Ω<Ω00 and Eθ/Er=5.0 for different values of plate parameters of simply supported (SS) and clamped (CL) edges. The transverse deflection for all three cases of loading for SS- and CL-plate is presented in Figures 2, 3(a), and 3(b). Figure 2(a) presents the transverse deflection of polar orthotropic simply supported plates along the radius vector R for taper parameter α=±0.3 and foundation parameter Kf=0.01,0.02, and 0.03 under disk loading. The transverse deflection is maximum at the centre of the plate, which decreases as the foundation parameter Kf increases, keeping other plate parameters fixed. Thus the elastic stability of the plate under loading is more, if some elastic foundation to the structural component is provided. Figure also shows that the transverse deflection for taper parameter α=0.3 (centrally thicker plate) is greater than that for α=-0.3 (centrally thinner plate), the reason being greater mass attribution at the centre in case of centrally thicker plate. The centrally thicker plate having higher amplitude of deflection can be protected from being damaged using foundation. This type of structural components is highly used in civil, mechanical, aeronautical, and modern technology. The idea of structural stability in these fields is very essential for design engineer to know before. Figure 2(b) presents that transverse deflection for clamped plate under disk loading for the same plate parameters as in Figure 2(a). It has been observed that the transverse deflection gradually decreases as the flexibility parameter Kϕ increases for elastically restrained plate. Thus the transverse deflection for clamped plate is always less than that for simply supported plate, keeping other plate parameters fixed. The effect of foundation parameter Kf on transverse deflection for clamped plate shows that deflection at the centre decreases more rapidly for centrally thicker plate than that for centrally thinner plate, the reason being more mass attribution for centrally thicker plate. Though values are nearly close, the foundation parameter Kf increases nature of variation of transverse deflection of CL-plate which is found to be different. Figures 2(c) and 2(d) present the transverse deflection when load is annular extending from R0=0.3 to R1=0.7 for SS- and CL-plates, respectively. Transverse deflection for annular loaded plate is found to be greater than that for disk loaded plate; of course it depends on the position of annular region of loading. The effect on deflection for simply supported plate is more than that of clamped plate. Transverse deflection for uniformly loaded plate is observed to be maximum as compared to disk and annular loaded plates as shown in Figures 3(a) and 3(b).
Radius vector versus deflection parameter W/a4q0Dr0 for taper parameter α=-0.3: — and α=0.3: …. Keys foundation parameter Kf=0.01: •••, Kf=0.02: ▪▪▪, and Kf=0.03: ▴▴▴.
Disk loading (SS-plate)
Disk loading (CL-plate)
Annular loading (SS-plate)
Annular loading (CL-plate)
Radius vector versus radial bending moment Mr/a2q0 for taper parameter α=-0.3: — and α = 0.3: …. Keys foundation parameter Kf=0.01: •••, Kf=0.02: ▪▪▪, and Kf=0.03: ▴▴▴.
Uniform loading (SS-plate)
Uniform loading (CL-plate)
Disk loading (SS-plate)
Disk loading (CL-plate)
Radius vector versus radial bending moment Mr/a2q0 for taper parameter α=-0.3: — and α = 0.3: …. Keys foundation parameter Kf=0.01: •••, Kf=0.02: ▪▪▪, and Kf=0.03: ▴▴▴.
Annular loading (SS-plate)
Annular loading (CL-plate)
Uniform loading (SS-plate)
Uniform loading (CL-plate)
Radius vector versus tangential bending moment Mθ/a2q0 for taper parameter α=-0.3: — and α=0.3: …. Keys foundation parameter Kf=0.01: •••, Kf=0.02: ▪▪▪, and Kf=0.03: ▴▴▴.
Disk loading (SS-plate)
Disk loading (CL-plate)
Annular loading (SS-plate)
Annular loading (CL-plate)
(•) values in Figures 6(c) and 6(d) are taken from [26] for SS-plate: — and CL-plate: …, respectively.
Uniform loading (SS-plate)
Uniform loading (CL-plate)
Radius vector versus transverse deflection
Radius vector versus bending moment
The radial bending moments for disk, annular, and uniformly loaded SS- and CL-plates are presented in Figures 3(c), 3(d), 4(a), 4(b), 4(c), and 4(d). Radial bending moment for disk loaded plate is presented in Figures 3(c) and 3(d). Figures show that radial bending moment for α=-0.3 is greater than that for α=0.3, other plate parameters being fixed. It is observed that peak of the bending moment decreases as the foundation parameter Kf increases. Maximum radial bending moment occurs at R=0.35 for disk loaded plate for these plate parameter values, whereas in case of annular loaded plate peak of the bending moment occurs at R=0.5. Peak of the bending moment shifts towards the edge R=1.0 for uniformly loaded plate (shown in Figures 4(a) and 4(b)). Radial bending moment at the edge increases as the flexibility parameter Kϕ increases and is maximum for C1 plate. Radial bending moment for α=-0.3 is greater than that for α=0.3, which decreases as the foundation parameter Kf increases. Maximum bending moment at the edge occurs when the plate is uniformly loaded (Figures 4(c) and 4(d)) compared to those of disk and annular loaded plates. Tangential bending moments are also presented in Figures 5(a), 5(b), 5(c), 5(d), 6(a), and 6(b) under three types of loadings. Radial bending moment at the edge for SS orthotropic plate is zero, but the tangential bending moment is nonzero. Tangential bending moment at the edge for SS-plate is greater than that of C1 plate for corresponding value of plate parameters. The present choice of function considered in the study has faster rate of convergence as compared to the polynomial coordinate function used by earlier researchers [13, 21, 26, 27] (see Figure 7).
Normalised frequency parameter Ω/Ω∗ for first three natural frequencies in fundamental mode: □□□, second Mode: ⋄⋄⋄, and third mode: △△△ for CL-plate with Eθ/Er=5.0, n=1, α = 0.3. Keys present method: — and polynomial coordinate method: …, where Ω∗ is the frequency obtained using 15 terms.
Table 1 compares the deflection and radial bending moment at the centre of the plate for Ω<Ω00 and Ω00<Ω<Ω01 for uniform isotropic plate obtained by Laura et al. [24] using Ritz method. Table 2 compares the transverse deflection of simply supported plate of variable thickness. Figure 6(c) compares the transverse deflection with the result of Laura et al. [26] for uniformly loaded isotropic circular plate of constant thickness with simply supported and clamped edge, respectively, obtained by Glarkin’s method. The bending moments for same plate parameter values as taken by Laura et al. [26] have been also compared and presented in Figure 6(d). Comparison of these obtained results shows an excellent agreement with the result available in the literature.
Comparison of displacement and radial bending moment at the centre as a function of η (=Ω < Ω00) and (=Ω00 < Ω < Ω01) for uniform isotropic circular plate.
η
Ω < Ω00
Ω00 < Ω < Ω01
W/a4q0Dr0
Mr/a2q0
W/a4q0Dr0
Mr/a2q0
Reference [24]
Present
Reference [24]
Present
Reference [24]
Present
Reference [24]
Present
CL-plateΩ00 = 10.2158Ω01 = 39.771
0.2
0.0160
0.01630
0.085
0.08524
0.0110
0.01181
0.084
0.08468
0.4
0.01872
0.01873
0.099
0.09952
0.0055
0.00553
0.051
0.05121
0.6
0.02479
0.02471
0.135
0.13533
0.0040
0.00404
0.039
0.03840
0.8
0.0446
0.04461
0.253
0.25281
0.0044
0.00446
0.080
0.08001
SS-plateΩ00 = 4.9351Ω01 = 29.721
0.2
0.0664
0.06640
0.215
0.21554
0.0226
0.02263
0.094
0.09407
0.4
0.0760
0.07603
0.248
0.24874
0.0094
0.00946
0.052
0.05224
0.6
0.1001
0.10010
0.331
0.33179
0.0062
0.006238
0.049
0.04948
0.8
0.1787
0.17877
0.603
0.60332
0.0063
0.006329
0.076
0.07653
Comparison of displacements for simply supported circular plate of linearly varying thickness for νθ = 0.25.
Taper parameter α
Reference [26]
Present
−0.5708
0.1817
0.1815
−0.0
0.06568
0.0656
−0.33333
0.1143
0.1127
6. Conclusion
The forced vibrational characteristics of polar orthotropic circular plates resting on Winkler type of elastic foundation have been studied using Ritz method. The function based on static deflection of polar orthotropic plates has been used to approximate the transverse deflection of the plate. The present choice of function has faster rate of convergence as compared to the polynomial coordinate function used by earlier researchers. Study shows that the deflection attains maximum value at the centre of the plate under uniform loading. The effect is influenced by thickness variation. Centrally thicker plate has more deflection than that of centrally thinner plate, the region being the more mass attribution at the centre in case of centrally thicker plate. While designing a structural component, stability is one of the important factors to be considered. The result analysis shows that the consideration of elastic foundation can protect the system from fatigue failure under heavy loads.
Competing Interests
The author declares that they have no competing interests.
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