1 INTRODUCTION
Composite laminates belong to a category of thin-walled structures that are commonly being used in many applications like aerospace, civil, marine, and mechanical engineering structures. These laminates have been extensively used in weight sensitive aircrafts and aerospace industries since their inception, and recently in civil engineering structures such as bridge decks, bridge girders, strengthening and retrofitting of existing structures.
Higher strength/weight ratio, high specific stiffness, enhanced fatigue life are some of the well-known characteristics of composite materials. These characteristics are further enhanced by adding stiffener to a plate without considerably affecting its overall weight. For a negligible weight penalty, there is an enormous increase in the strength and stability characteristics throughout the structures.
Plates with an array of stiffeners are commonly found in fuselage and wings of an aircraft, a ship's hull and its deck, offshore drilling rigs, pressure vessels, roofing units, launching pedestal of rocket, etc. The diverse applications of such stiffened plates are found to be exposed to in-plane edge loading in many situations. The presence of these loadings may significantly alter the free vibration response of such stiffened plates. In fact, a situation may arise wherein the natural frequency of a component becomes zero, thereby causing instability of the component at a fairly low-stress field. Hence, the problem of elastic stability may be considered as a special case of vibration problems, and such problems are of considerable importance and interest to researchers for many years.
Many past studies have reported on stability and/or vibration characteristics of stiffened/unstiffened plates with uniform compressive edge loads ((^{Kamruzzaman et al., 2006}; ^{Khedmati and Edalat, 2010}; ^{Mirzaei et al., 2015}; ^{Sayyad and Ghugal, 2014}; ^{Sayyad et al., 2016}; ^{Singh and Chakrabarti, 2012}). However, such uniform edge compressions are unusual in practice. Many practical situations demand the stiffened plate subjected to concentrated and partial edge loads.
The problem of elastic stability of a simply supported plate with in-plane concentrated loads acting opposite to each other was first attempted by (^{Sommerfield, 1906}). The results were based on simplified approximate plane stress analysis and hence lead to a considerable error. (^{Yamaki, 1953}) obtained the buckling loads for a simply supported plate with partially distributed edge loads by employing assumptions similar to (^{Sommerfield, 1906}). (^{Leissa and Ayoub, 1988}) investigated the vibration and stability of circular and rectangular plates under concentrated edge loads acting opposite to each other using Ritz method. They utilized finite element and plane elasticity solutions for obtaining the initial pre-buckling stresses, and they were the first to analyze vibration characteristics under concentrated forces. (^{Kaldas and Dickinson, 1981}) also employed Rayleigh-Ritz technique to study the vibration and stability behaviour of thin plates with complicated pre-buckled stress distribution. (^{Jana and Bhaskar, 2006}) examined the effect of different edge loads on stability behaviour of plates using a semi-analytical method. The initial in-plane stresses were found by using Airy's stress function and extended Kantorovich technique. The critical loads were hence determined by Galerkin's method. (^{Hashemi et al., 2008}) have contributed analytical solutions for the stability behaviour of isotropic plate under uni-/bidirectional in-plane compressive edge loadings. (^{Deolasi et al., 1995}) have employed finite element (FE) technique for analyzing the buckling and vibration problems of plates with different edge loading conditions. (^{Singh et al., 2012}) imparted some buckling results for rectangular plates under different non-uniform edge load distributions using FE technique. Stability behaviour of clamped and simply supported square laminated plates under uni-/bidirectional partial edge loading conditions using FE method was investigated by (^{Sundaresan et al., 1998}). The same problem was further explored by (^{Sahu et al., 2001}) with an extension to the vibration behaviour of laminated plates under different edge loads. (^{Srivastava et al., 2003}) examined the critical buckling behaviour of isotropic stiffened plates with partial and concentrated loads using FE approach. The free vibration characteristics of isotropic stiffened plates are studied by (^{Hamedani et al., 2012}) using super finite element technique. The same problem and techniques were further extended to get buckling results by (^{Hamedani and Ranji, 2013}).
It has been observed that, in finite element formulation, most of the authors have used eight-node serendipity elements by considering the effect of shear deformation. Some authors (^{Palani et al., 1989}; ^{Palani et al., 1992}; ^{Pugh et al., 1978}) have noticed that for certain mesh configuration and boundary conditions, the eight-node plate element locks in shear, even when selective or reduced integration techniques are used for a thin plate configuration. It is also stated that the eight-node serendipity element is comparatively more sensitive to element aspect ratios and element shape distortions (^{Palani et al., 1989}) whereas, the nine-node Lagrange element exactly interpolates the quadratic displacement fields and has nearly optimal performance (^{Palani et al., 1989}; ^{Pugh et al., 1978}). However, some of the authors (^{Butalia et al., 1990}; ^{Hughes and Cohen, 1978}) have also found that the nine-node Lagrange element locks in shear, even when reduced or selective integration techniques are employed for thin plates. In fact, the stiffness matrix exhibits rank deficiency resulting in the appearance of spurious mechanisms, i.e. zero energy modes (^{Butalia et al., 1990}; ^{Hinton and Owen, 1984}; ^{Hughes and Cohen, 1978}). These communicable mechanisms may result in erratic solution. These shortcomings, namely locking and spurious mechanisms, have led to the development of a heterosis element by employing serendipity shape functions for transverse degree of freedom w, and Lagrange shape functions for rotations θ_{x} and θ_{y} (^{Butalia et al., 1990}; ^{Hughes and Cohen, 1978}). This element exhibits improved characteristics as compared to each of the previous quadratic elements and offers high accuracy for extremely thin plate configuration (^{Hughes and Cohen, 1978}). The same element has been used in this study by considering the effect of in-plane displacements u, v in addition to transverse and rotational displacements.
From this brief overview of the past literature, it is observed that a large body of research work has been carried out on the buckling characteristics of plates under uniform edge compression. On the contrary, relatively less amount of research works deal with the partial in-plane edge compression. Further, the buckling behaviour of isotropic stiffened plates under partial in-plane edge compression is sparsely treated in the literature. To the best knowledge of the authors, no comprehensive work is identified in the literature regarding in-plane partial edge load for stiffened laminated plates. The present work deals with the problem of vibration and buckling of stiffened composite laminates under the action of in-plane partial edge loads.
Analytical solution to this class of problems under such loadings is extremely difficult. Therefore, a nine-node heterosis element and a three node beam element are employed within the framework of finite element method. In this work, a detailed parametric study is carried out to investigate the effect of different ply-angles, boundary conditions, stiffener sizes, partial edge loads, and their positions on the vibration and stability behaviour of stiffened panels.
2 FORMULATION OF EQUATIONS WITH THEORY
The geometrical configuration of the bidirectional stiffened laminated plate of plan-form (a x b) with stiffeners parallel to x and y directions is shown in Figure 1. The middle-plane displacement fields corresponding to x, y, and z directions are u, v, and w, respectively.
2.1 Strain-Displacement Relations
Lagrange's strain displacement equations are used throughout the formulation. The expressions pertaining to the linear strain relation of the plate by considering the effect of shear deformation are given by
where ε_{xl}, ε_{yl} and (_{xyl} are the linear in-plane normal and shear strains; (_{xz} and (_{yz} are transverse shear strains; z is the distance of any layer from the middle plane of the plate and χ are the curvatures.
The expression for the non-linear strain displacement relation of the plate is as follows (^{Bathe, 1996})
where in, θ_{x} and θ_{y} are the rotations of normal to the un-deformed mid-plane of the plate about y and x axes, respectively.
The stress-strain relation for a lamina with reference to the plate axes is given by
By using Equations ^{(1)} and ^{(3)}, the constitutive relation for the laminated plate can be given by
in short form,
where { ^{ Np } _{i} } = [ ^{ Np } _{x} ^{ Np } _{y} ^{ Np } _{xy} ] ^{T} are stress resultants, { ^{ MP } _{i} } = [ ^{ MP } _{x} ^{ MP } _{y} ^{ MP } _{xy} ] ^{T} are moment resultants, { ^{ QP } _{i} } = [ ^{ QP } _{xz} ^{ QP } _{yz} ] ^{T} are transverse shear stress resultants; likewise, {ε ^{P} _{j} } = [ε ^{P} _{x} ε ^{P} _{y} ε ^{P} _{xy} ] ^{T} are the middle plane strains, {χ ^{P} _{j} } = [χ ^{P} _{x} χ ^{P} _{y} χ ^{P} _{xy} ] ^{T} are the middle plane curvatures and {γ ^{P} _{j} } = [γ ^{P} _{xz} γ ^{P} _{yz} ] ^{T} are the transverse shear strains. From Eq. ^{(4)}, the extension-extension [ ^{ AP } _{ij} ], extension-bending [ ^{ BP } _{ij} ] and bending-bending [ ^{ DP } _{ij} ] of the stiffness components can be expressed as
and
in which n is the number of layers, α is the shear correction factor, which is given by 5/6 (^{Kolli and Chandrashekhara, 1996}; ^{Reddy, 1996}) and is the stiffness matrix of k^{th} lamina with reference to plate axes (^{Reddy, 1996}).
The constitutive relationship for the laminated stiffeners can be written compactly as follows:
where { ^{ Ns } _{x} } = [ ^{ Nsx } _{x} ^{ Msx } _{x} ^{ Msx } _{xy} ^{ Qxx } _{xz} ] ^{T} and {ε ^{s} _{x} } = [ε ^{sx} _{x} χ ^{sx} _{x} χ ^{sx} _{xy} γ ^{sx} _{xz} ] ^{T} are the stress resultants and middle plane strains for the stiffener, respectively; [ ^{ Dsx } ] is the modified reduced constitutive matrix, which is obtained by ignoring the resultant stresses ( ^{ Nsx } _{y} = ^{ Nsx } _{xy} = ^{ Msx } _{y} = ^{ Qsx } _{yz} = 0) but not strains (ε ^{sx} _{y} ≠ γ ^{sx} _{xy} ≠ χ ^{sx} _{y} ≠ γ ^{sx} _{yz} ≠ 0) in the Eq. ^{(4)} and upon simplification, the reduced constitutive matrix of x-stiffener is obtained (^{Kolli and Chandrashekhara, 1996}). A similar procedure is adopted to obtain the reduced constitutive matrix, [ ^{ Dsy } ] of y-stiffener.
2.2 Finite Element Formulation
In the present work, the plate has been discretized using nine-node (9-N) heterosis plate element with five degrees of freedom (DOF) given as u, v, w, θ_{x} and θ_{y} at all edge nodes and four DOF such as u, v, θ_{x} and θ_{y} at the inside node. The serendipity shape functions have been used for transverse DOF w, and Lagrange shape functions for remaining DOF that includes u, v, θ_{x}, θ_{y} as shown in Figure 2. In a discretization of stiffener, three-node beam element with four DOF at each node has been used viz., u, w, θ_{x}, θ_{y} for x-stiffener and v, w, θ_{x}, θ_{y} for y-stiffener. However, the other two kinds of quadratic elements such as eight-node (8-N) serendipity and nine-node (9-N) Lagrange elements have also been considered in this study as illustrated in Figure 2. The results of these quadratic elements are given only in comparison studies along with the results of heterosis element.
2.2.1 Elastic Stiffness Matrices
The total strain energy due to linear strains in the entire stiffened plate is given by
where L is the number of elements of the plate or the x-stiffener or the y-stiffener involved in the formulation. After substituting Eqs ^{(4)}, ^{(7)} and ^{(8)} in Eq. ^{(9)} and expressing linear strains, {ε _{l} } =[B]{d_{e} } in terms of derivatives of nodal shape functions and its corresponding displacement vectors, the linear part of the total strain energy can be obtained as,
where {d_{e} }and {d} are the elemental and global displacement vectors, respectively; [ ^{ kp } ] [ ^{ ksx } ]and [ ^{ ksy } ]are the element elastic stiffness matrices for plate, x-stiffener and y-stiffener, respectively; [K] is the global stiffness matrix of the entire stiffened plate and are expressed as,
where[ ^{ Esx } ]and[ ^{ Esy } ]are the transformation matrices; [ ^{ BP } ],[ ^{ Bsx } ] and [ ^{ Bsy } ]are the elemental strain displacement matrices for plate, x-stiffener and y-stiffener, respectively, wherein
and
and also
in which e_{sx =} ( d_{sx} + h )/2 and e_{sy} = (d_{sy} + h)/2 are stiffener eccentricities.
2.2.2 Geometric Stiffness Matrices
The total strain energy due to non-linear strains in the entire stiffened plate is given by
Substituting the non-linear strain displacement of plate [Eq. ^{(2)}] and the corresponding stiffeners in the above equation and upon simplification, we get the final form of Eq. ^{(17)} as given below:
Where [ ^{ kP } _{G} ], [ ^{ ksx } _{G} ] and [ ^{ ksy } _{G} ] are the elemental geometric stiffness matrices for plate, x-stiffener and y-stiffener, respectively; [K_{G} ] is the overall geometric stiffness matrix of the entire stiffened plate, which are expressed as,
where [ ^{ BP } _{G} ], [ ^{ Bsx } _{G} ] and [ ^{ Bsy } _{G} ] are the elemental non-linear strain displacement matrices for plate, x-stiffener and y-stiffener, respectively; [ ^{ SP } ], [ ^{ Ssx } ] and [ ^{ Ssy } ]are the corresponding initial pre-buckling stress matrices, in which
and
and
in which
2.2.3 Consistent Mass Matrices
The abbreviated form of total kinetic energy in the entire stiffened plate by including the effect of in-plane inertia, transverse inertia and rotational inertia is as follows:
where as [ ^{ mP } ], [ ^{ msx } ] and [ ^{ msy } ]are the elemental mass matrices for plate, x-stiffener and y-stiffener, respectively; [M] is the overall mass matrix of the entire stiffened plate, which are expressed as,
in which
and
with
2.3 Governing Equations
The governing equation of motion for vibration and stability problems derived using extended Hamilton's principle is given below:
where T is the kinetic energy for entire stiffened plate, U is the strain energy due to linear and non-linear strains; t_{1}, t_{2} are two arbitrary values of time t and Δ denotes the variational operator.
Substituting Eqns ^{(10)}, ^{(18)} and ^{(27)} into the Eq. ^{(34)} and upon simplification, the following most general governing differential equation of motion is obtained as follows:
The governing equations for the vibration and buckling can be obtained separately by reducing Eq. ^{(35)} as follows:
Vibration problem: When the stiffened/unstiffened plate vibrates harmonically with a given in-plane load, Eq. ^{(35)} reduces to
Buckling problem: when Eq. ^{(35)} reduces to
where P_{cr} is the critical load and P is the applied edge compression.
Both Eqs ^{(36)} and ^{(37)} represent the eigenvalue problems. The square of frequency, ^{ ω2 } , for an applied in-plane load P is the eigenvalue in Eq. ^{(36)} and the critical load P_{cr} is the eigenvalue in Eq. ^{(37)}.
Eigenvector {q} represents the corresponding mode shapes of vibration or buckling problems.
3 NUMERICAL RESULTS AND DISCUSSIONS
3.1 Computer Program
The solution of stability problems using either dynamic or static approach involves extraction of eigenvalues and eigenvectors, as evidenced in Eqs ^{(36)} and ^{(37)} respectively. The large sized matrices appearing in these two equations have to be stored in skyline form. (^{Bathe, 1996}) has given the time-tested code in FORTRAN for extraction of the eigenvalues using the subspace iteration technique. To be compatible with this, the FORTRAN code has been developed to generate various element level and global matrices and finally stored these matrices in skyline assembled form.
In the code, a selective integration scheme is incorporated for the generation of element elastic stiffness matrix. A 3 x 3 Gauss rule is adopted for membrane as well as bending effects while 2 x 2 Gauss rule is adopted for shear effects. As the geometric stiffness matrix is a function of the pre-buckling stress field in an element due to the application of any kind of in-plane loads, it is a prerequisite to perform the plane stress analysis before the generation of geometric stiffness matrix. Hence, code is done for the plane stress analysis to evaluate the stress field at 3 x 3 Gauss points. Accordingly, the integration for the generation of geometric stiffness matrix has been done using 3 x 3 Gauss rule (full integration). Similarly, A 3 x 3 Gauss rule is adopted to evaluate element level mass matrix also.
3.2 Boundary Conditions, Stiffener Parameters and Material Constants
In the present work, for the configuration of the stiffened laminated plate shown in Figure 1, two types of boundary conditions have been employed are as follows:
for initial pre-buckled stress analysis: along the edges (a) x = 0, a; w = θ_{y} = 0, (b) y = 0, b; w = θ_{x} = 0 and at x = a/2, u = 0; at y = b/2, v = 0
for buckling analysis: along the edges (a) x = 0, a; u = w = θ_{y} = 0, (b) y = 0, b; v = w= θ_{x}=0.
for initial pre-buckled stress analysis: along the edges (a) x = 0, a; w = θ_{x} = θ_{y} = 0, (b) y = 0, b; w = θ_{x} = θ_{y} = 0 and at x = a/2, u = 0; at y = b/2, v = 0
for buckling analysis: along the edges (a) x = 0, a; u = v = w = θ_{x} = θ_{y} = 0, (b) y = 0, b; u = v = w = θ_{x} = θ_{y}= 0.
The stiffener dimensions are decided according to the non-dimensional parameters, in which δ and β are defined as: δ = nA_{s} /bh which is the ratio of total area of the stiffeners to that of the plate, where n is the number of stiffeners and A_{s} is the cross-sectional area of each stiffener; similarly, β = nE_{2}I_{s} /bD is the ratio of total bending stiffness of the stiffeners to that of the plate, where I_{s} is the moment of inertia of the stiffener about the plate centroidal axis and D is the flexural rigidity of the plate, which is D = Eh ^{3}/12(1 - v ^{2)} for isotropic case and D = E _{2} h ^{3}/12(1 - υ^{2} _{12}) for composite case. The ratio of thickness of plate to the width of the plate (h/b) is considered as 0.01 unless otherwise stated.
The vibration frequency and the critical loads are represented in non-dimensional form as given in Table 1 and the elastic material properties with reference to the principal axes have been given in Table 2 unless otherwise mentioned.
Material | Stiffened/unstiffened plates | |||||
---|---|---|---|---|---|---|
E _{11} | E _{22} | G _{12} | G _{13} | G _{23} | v _{12} | |
Isotropic | 1.0 | 1.0 | 0.3854 | 0.3854 | 0.3854 | 0.30 |
Composite | 25.0 | 1.0 | 0.50 | 0.50 | 0.20 | 0.25 |
In Table 1, ω is the absolute value of frequency, ρ is the material density, D is the flexural rigidity of the plate and P_{cr} is the absolute critical buckling load.
3.3 Comparative Studies
The validation of the finite element formulation described above is necessary to ascertain the correctness of the development of various matrices involved in the analysis of vibration and buckling problems. From the comparative studies of deflections and rotations for unstiffened/stiffened plates subjected to lateral loads, one can ascertain the validity of the elastic stiffness matrix. After ascertaining this, if the vibration frequencies are compared, the formulation of the mass matrix can be validated. Finally, if the buckling loads are compared with the classical/numerical solutions, the generation of the geometric stiffness matrix can be validated. Validation of all these matrices has been carried out as described in the following sub-sections.
3.3.1 Static Analysis of Laminated Stiffened and Unstiffened Plates
The values of deflection of unstiffened/stiffened laminated plates under lateral loads are determined and the results are represented in Table 3 and Table 4, respectively. The effect of uniformly distributed load on the central deflection of a laminated plate is tabulated in Table 3 along with the results of (^{Reddy, 1996}) and 3D-FEM solutions ^{Xiao et al., 2008}). The results show very good agreement with the analytical solution. Table 4 shows the central deflection of a laminated square plate with centrally placed bidirectional stiffeners subjected to various boundary conditions with the material properties pertaining to AS4/3501 graphite/epoxy composite: E_{11} = 144.8 GPa; E_{22} = 9.65 GPa; G_{12} = G_{13} = 4.14 GPa; G_{23} = 3.45 GPa; v_{12} = 0.3; ρ = 1389.23 kg/m^{3}. Similarly, the geometrical dimensions of the stiffened plate [see Figure 1] are as follows: a = 254 mm, b = 254 mm, h = 12.7 mm, b_{sx} = b_{sy} = 6.35 mm and d_{sx} = d_{sy} = 25.4 mm. The results using nine-node heterosis element and eight-node serendipity element are depicted in Table 4. It can be observed that both the elements give satisfactory results. However, the results of heterosis element seem to be more accurate when compared to those obtained from eight-node element and nine-node element, which was previously used by (^{Kolli and Chandrashekhara, 1996}). The satisfactory agreement of the results with the literature clearly shows the correctness in the formulation of element elastic stiffness matrix for both plate [ ^{ kP } ] as well as stiffeners[ ^{ ksx } ] and[ ^{ ksy } ].
3.3.2 Vibration Study of Laminated Unstiffened and Stiffened Plates
To further validate the formulation of the mass matrix, the free vibration characteristics of an unstiffened square composite plate are studied using nine-node heterosis element (9-NHE); the results are presented in Table 5 along with the closed-form solutions (CFS) given by (^{Reddy and Phan, 1985}). However, to ascertain the performance of other kinds of elements, the authors have also studied the free vibration behaviour using nine-node Lagrange element (9-NLE) and eight-node serendipity element (8-NSE). It is concluded that the adoption of nine node heterosis element in the formulation is a better choice than the use of other elements.
b/h | 2 layers (45/-45) | 8 layers (45/-45/45...) | ||||||
---|---|---|---|---|---|---|---|---|
Present results | CFS (Reddy and Phan, 1985) | Present results | CFS (Reddy and Phan, 1985) | |||||
9-NHE | 9-NLE | 8-NSE | 9-NHE | 9-NLE | 8-NSE | |||
5 | 10.335 | 10.244 | 10.243 | 10.335 | 12.892 | 12.863 | 12.862 | 12.892 |
10 | 13.044 | 12.975 | 12.975 | 13.044 | 19.289 | 19.235 | 19.235 | 19..289 |
20 | 14.179 | 14.154 | 14.153 | 14.179 | 23.259 | 23.225 | 23.225 | 23.259 |
25 | 14.338 | 14.322 | 14.321 | 14.338 | 23.924 | 23.899 | 23.899 | 23.924 |
50 | 14.561 | 14.557 | 14.556 | 14.561 | 24.909 | 24.902 | 24.901 | 24.909 |
100 | 14.618 | 14.917 | 14.617 | 14.918 | 25.176 | 25.174 | 25.174 | 25.176 |
B.C | Mode no. | 0/90 | 45/-45 | |||||
---|---|---|---|---|---|---|---|---|
(Satish Kumar and Mukhopadhyay, 2000) | (Thinh and Khoa, 2008) | (Thinh and Quoc, 2008) | Present (heterosis element) | (Satish Kumar and Mukhopadhyay, 2000) | (Thinh and Khoa, 2008) | Present (heterosis element) | ||
1 | 1076.0 | 1014.0 | 1053.6 | 1030.1 | 1005.7 | 1007.1 | 980.9 | |
S-S-S-S | 2 | 2059.6 | 2139.5 | 2083.7 | 2221.3 | 2254.4 | 2284.7 | 2343.3 |
3 | 2302.7 | 2397.9 | 2327.6 | 2446.3 | 2358.7 | 2434.5 | 2352.9 | |
4 | 2635.8 | 2683.0 | 2556.9 | 2689.1 | 3247.4 | 3208.3 | 3293.6 | |
1 | 1666.5 | 1542.1 | 1609.5 | 1783.4 | 1714.2 | 1573.8 | 1662.5 | |
C-C-C-C | 2 | 2929.2 | 2848.3 | 2926.3 | 3070.8 | 3049.3 | 2909.4 | 3046.6 |
3 | 3140.1 | 3041.4 | 3141.2 | 3253.8 | 3077.4 | 2967.7 | 3048.6 | |
4 | 3666.3 | 3653.5 | 3639.2 | 3665.6 | 3943.9 | 3896.3 | 3948.5 | |
1 | 1445.8 | 1333.7 | 1427.5 | 1453.9 | 1380.9 | 1304.0 | 1340.9 | |
C-C-S-S | 2 | 2107.7 | 2259.6 | 2083.9 | 2441.8 | 3471.6 | 2483.6 | 2485.7 |
3 | 3054.0 | 2929.5 | 2896.7 | 3042.2 | 2912.6 | 2835.2 | 2914.3 | |
4 | 3196.8 | 3221.7 | 3209.9 | 3232.6 | 3609.6 | 3557.0 | 3635.1 |
Similar to the comparative analysis carried out for an unstiffened plate, the studies are further extended to a laminated plate with centrally placed bidirectional stiffeners under various edge conditions and the results are tabulated in Table 6 along with those available in the literature (^{Satish Kumar and Mukhopadhyay, 2000}; ^{Thinh and Khoa, 2008}; ^{Thinh and Quoc, 2008}). The geometric and material properties of the stiffened plate are similar to the case 3.3.1. The comparison of results establishes the correctness of the mass matrix formulation for both the plate as well as the stiffeners.
3.3.3 Buckling Characteristics of Unstiffened/Stiffened Plates
For the buckling analysis, as in Eqs 36 and 37, one more matrix called geometric stiffness matrix appears in the formulation of the problem. It is essential to validate the formulation of such matrix. In this regard, a non-dimensional quantity η (load bandwidth ratio) is introduced, which is either the ratio of the position of the concentrated load to the width of the plate or width of edge compression to the width of the plate [refer Figures 3, ^{5} and ^{6}]. The critical loads for an unstiffened plate under concentrated load [see Figure 3 (a)] are found for η = 0.25 and 0.5 and compared in Table 7 with those available in the literature (^{Deolasi et al., 1995}; ^{Leissa and Ayoub, 1988}). Similarly, a comparative study has been carried out for an S-S-S-S edged laminated plate with partial in-plane load [see Figure 3 (b)] for various η and ply-orientations as shown in Figure 4. The non-dimensionalised critical loads are validated with the results given by (^{Sundaresan et al., 1998}). It is to be noted here that ( is the critical load for a fully loaded plate and (_{cr} is the critical load for a partially loaded plate.
Further, the buckling behaviour of S-S-S-S edged plate with a central single stiffener under uniformly distributed edge load over width b is studied for different bending rigidity of the stiffener. The value of δ is varied from 0.05 to 0.20, and that of β from 5 to 15. The results are presented in Table 8 in non-dimensionalised form as Y_{cr} = P_{cr}b ^{2} / π^{2} D), and compared with the classical solution of (^{Timoshenko and Gere, 1961}), numerical solutions of (^{Mukhopadhyay and Mukherjee, 1990}), and (^{Hamedani and Ranji, 2013}). In all the cases, the present results agree well with those available in the literature confirming the correctness of the geometric stiffness matrix for the plate as well as the stiffeners.
a/b | η | (Leissa and Ayoub, 1988) | (Deolasi et al., 1995) | Present |
---|---|---|---|---|
0.5 | 0.25 | 32.282 | 37.608 | 37.249 |
0.50 | 30.061 | 29.852 | 29.630 | |
1.0 | 0.25 | 37.000 | 36.405 | 36.316 |
0.50 | 25.814 | 25.720 | 25.658 | |
2.0 | 0.25 | 36.854 | 36.832 | 36.659 |
0.50 | 26.523 | 28.851 | 28.690 |
β | δ | Non-dimensional buckling load | |||||
---|---|---|---|---|---|---|---|
Neglecting T and e | Considering T and e | ||||||
(Timoshenko and Gere, 1961) | (Mukhopadhyay and Mukherjee, 1990) | (Jafarpour and Ranji, 2013) | Present (heterosis) | (Jafarpour and Ranji, 2013) | Present (heterosis) | ||
0.05 | 12.00 | 11.72 | 11.89 | 11.80 | 11.91 | 11.86 | |
5 | 0.10 | 11.10 | 10.93 | 11.08 | 10.98 | 11.09 | 11.27 |
0.20 | 9.72 | 9.70 | 9.85 | 9.69 | 9.86 | 10.01 | |
0.05 | 16.00 | 16.00 | 16.01 | 15.97 | 18.16 | 17.91 | |
10 | 0.10 | 16.00 | 16.00 | 16.01 | 15.97 | 16.97 | 17.20 |
0.20 | 15.80 | 15.44 | 15.01 | 15.97 | 15.04 | 15.77 | |
0.05 | 16.00 | 16.00 | 16.01 | 15.97 | 20.41 | 20.30 | |
15 | 0.10 | 16.00 | 16.00 | 16.01 | 15.97 | 20.41 | 20.33 |
0.20 | 16.00 | 16.00 | 16.01 | 15.97 | 19.36 | 20.33 |
T and e = Torsional rigidity and eccentricity of the stiffener
3.4 Convergence Study
In the finite element method, it is essential to appropriately discretize the structure for proper convergence of results. In this regard, the plate is discretized into m, number of rows and n, number of columns, i.e. m x n plate elements as shown in Figure 5. Table 9 shows the convergence of the non-dimensional critical loads for an unstiffened/stiffened plate under the action of uniaxial compressive edge load from one edge. The Table 9 also shows the non-dimensional vibration frequencies under the action of in-plane edge loads P/P_{cr} = +0.5 (compressive) and P/P_{cr} = 0 (no in-plane load). It is observed that the convergence of results is satisfactory for the mesh size of 10 x 10 and hence this mesh size is maintained throughout the work.
3.5 Vibration and Buckling Behaviour of Laminated Stiffened Plate under Variety of Partial Edge Loading
In the present study, the buckling and vibration characteristics of a plate with centrally placed stiffener/s subjected to uni-/bidirectional partial in-plane compressive loads have been investigated. Figure 6 shows the configuration of the stiffened plates with different kinds of in-plane edge loads. For buckling analysis, an initial pre-buckling stress field has to be determined where the total load, P acting on the plate is constant irrespective of the type and width of loading. Also, the total volume of stiffened plate is same irrespective of the number of stiffeners in any directions.
3.5.1 Vibration Characteristics of Laminated Stiffened Plate under Uniaxial Edge Loading
The effect of non-dimensional load λ on the vibration behaviour of S-S-S-S edged laminated square plate with a centrally placed single stiffener has been studied for δ = 0.1 and β = 10. This load case is shown in Figure 6 (a). The results presented in Figures 7 (a) and ^{(b)} indicate that the frequency decreases with an increase in the value of edge load. As the load approaches the critical value, the vibration frequency reduces to zero and this load is called the buckling load. This dynamic method of analysis overcomes the shortcomings of the evaluation of critical loads by the static approach, which generally happens where the in-plane stress distribution within an element is significantly non-uniform. This method of analysis has been adopted to determine the critical loads for various problems discussed henceforth.
3.5.2 Laminated Stiffened Plate under Uniaxial and Biaxial Partial Edge Load from One Edge
The effect of a four layered anti-symmetric ply-orientation (±θ^{o)} _{2} and the load bandwidth ratio η on the stability behaviour of a unidirectional loaded square plate with a central single stiffener [refer Figure 6 (a)] for S-S-S-S and C-C-C-C edged stiffened plates are depicted in Figures 8 (a) and ^{(b)}, respectively. It is noticed from Figure 8 (a) that the buckling resistance is generally found to be maximum for a plate with (±30^{o)} _{2} layup irrespective of the values of η. This may be attributed to the higher flexural stiffness of the anti-symmetric stiffened plate with (±30^{o)} _{2} layup. Similarly, it is noticed from Figure 8 (b) that the critical load is found to be maximum for ply-orientations between (±30^{o)} _{2} and (±40^{o)} _{2} depending on values of η for the C-C-C-C edged stiffened plate. From these two figures, it is also inferred that the buckling load generally increases with increase in the value of η. This may be due to the fact that the contribution of the stiffener in resisting critical load increases as the line of action of the resultant load approaches the stiffener center line. It may be stated that the stiffened plate with around (±30^{o)} _{2} layup can be a better choice for this type of loading condition.
Similar studies are carried out for the same plate with biaxial edge loading and centrally placed biaxial stiffeners [refer Figure 6 (b)] and the results are depicted in Figures 9 (a) and ^{(b)}. In Figure 9 (a), it is noticed that the buckling load steadily increases to a maximum as the ply-orientation varies from (±0^{o)} _{2} to (±30^{o)} _{2} and starts decreasing from (±30^{o)} _{2} to (±45^{o)} _{2}. As the value of ply-orientation increases from (±45^{o)} _{2} to (±60^{o)} _{2}, the critical loads again continue to increase to their maximum values. From (±60^{o)} _{2} to (±90^{o)} _{2}, the critical loads are found to decrease to a minimum. This wavy variation is symmetrical in nature and the maximum buckling values are found at both (±30^{o)} _{2} and (±60^{o)} _{2} ply-angles.
Similarly, it can be noticed in Figure 9 (b) that the critical loads monotonically increases as the value of ply-orientation is increased from (±0^{o)} _{2} to (±45^{o)} _{2} and decreases from (±45^{o)} _{2} to (±90^{o)} _{2}. The decreasing trend of critical loads from (±30^{o)} _{2} to (±45^{o)} _{2} to an intermediate value as observed in Figure 9 (a) is not reflected in this case, i.e. wavy variation does not exist in the C-C-C-C edged stiffened plate. It may be attributed to the fact that the edge rotational restraint increases the stiffness of the stiffened plate predominantly and hence the critical loads continue to increase for all values of η as seen in Figure 9 (b). For this kind of loading, the stiffened plate with (±30^{o)} _{2} or (±60^{o)} _{2} layup may be more useful for S-S-S-S edge conditions and the stiffened plate with (±45^{o)} _{2} layup may be a better choice for C-C-C-C edge conditions.
3.5.3 Laminated Stiffened Plate under Uniaxial and Biaxial Partial Edge Loading from Both Edges
The effect of load case shown in Figure 6 (c) on the critical loads of unidirectional stiffened plate is examined and the results are depicted in Figures 10 (a) and ^{(b)} for different ply-orientations and boundary conditions (S-S-S-S, C-C-C-C). The buckling load is found to be maximum at (±30^{o)} _{2} layup irrespective of the values of η as can be seen in Figure 10 (a). The same phenomenon has been observed in Figure 8 (a), the possible reason being that the effect of loading from one edge [Figure 6 (a)] holds true for loading from both edges [Figure 6 (c)]. Further, it is interesting to mention that for a particular ply-angle of about (( 48^{o)} _{2}, the critical loads are practically same irrespective of the values of η, and similar phenomenon is again observed at (( 90^{o)} _{2} ply-angle in Figure 10 (a). The buckling behaviour of C-C-C-C edged stiffened plate, as shown in Figure 10 (b), is almost similar to that of S-S-S-S edged stiffened plate. However, in this case, the critical loads are independent of the value of η for only one particular ply-orientation, i.e. at (( 90^{o)} _{2}.
The variation of critical loads for the bidirectional loaded square plate with bidirectional stiffeners [refer Figure 6 (d)] is now investigated and the results are presented in Figures 11 (a) and ^{(b)} for S-S-S-S and C-C-C-C edge conditions, respectively. The wavy variation of buckling loads observed in Figure 11 (a) is similar as that observed in Figure 9 (a). In both the cases, it is observed that the curves have two peaks, i.e. at (±30^{o)} _{2} as well as (±60^{o)} _{2} layups. Therefore, it is appropriate to choose the stiffened plate with either of these layups for these types of loading cases. The buckling variation of a stiffened plate with C-C-C-C edge condition is shown in Figure 11 (b). The variation is slightly different when compared to the one shown in Figure 9 (b). In this case, the curves for η = 0.2 to 0.6 have double peaks, i.e. at (±30^{o)} _{2} and (±60^{o)} _{2} ply-angles and beyond η = 0.6, the curves are having a single peak at (±45^{o)} _{2} ply-angle. However, in this case, the choice of ply-orientation is purely dependent on the width of edge compression. For this type of loading, the stiffened plate with (±45^{o)} _{2} layup may be suitable for η = 0.8 - 1.0, whereas, for the remaining values of η, the stiffened plate with (±30^{o)} _{2} or (±60^{o)} _{2} layup may be an appropriate choice.
3.5.4 Laminated Stiffened Plate with Uniaxial and Biaxial Partial Edge Loading at the Center
The variation of γ_{cr} with the ply-orientation angles (±θ^{o)} _{2} for S-S-S-S and C-C-C-C square plate with a centrally placed single stiffener subjected to uniaxial loading [refer Figure 6 (e)] is shown in Figures 12 (a) and ^{(b)}. It is noticed from Figure 12 (a) that, in general, the critical load γ_{cr} is maximum for a ply-angle of (±30^{o)} _{2} except for lower values of η. Further, the variation of buckling load after (±55^{o)} _{2} ply-orientations is not significant with respect to η. Similar behaviour can also be observed in a C-C-C-C edged stiffened plate as shown in Figure 12 (b). However, in this case, the buckling variation is not significant after (±75^{o)} _{2} ply-orientation for all values of η.
Similar studies are carried out for the same plate with a centrally placed biaxial stiffener and biaxial loading conditions and the results are depicted in Figures 13 (a) and ^{(b)}. The study indicates that the critical load generally increases with an increased value of η. It may be due to increasing of higher edge restraints as the width of load increases towards the corner of the plate. However, in case of S-S-S-S edged stiffened plate as shown in Figure 13 (a), the critical load γ_{cr} is found to be maximum for both (±30^{o)} _{2} and (±60^{o)} _{2} ply-orientations only for higher values of η. However, the maximum shifts towards (±20^{o)} _{2} and (±70^{o)} _{2} ply-orientations as the value of η decrease towards η = 0.2. To sum up, the buckling behaviour with the wavy variation of this case is almost similar to the case shown in Figure 9 (a). The buckling variation of C-C-C-C edged stiffened plate as shown in Figure 13 (b) displays similar behaviour when compared to the one shown in Figure 11 (b). For this type of loading and edge conditions, (±45^{o)} _{2} ply-angle shows a higher buckling resistance for all values of η except η = 0.1.
4 CONCLUSION
The results from the present investigation of the vibration and stability characteristics of S-S-S-S and C-C-C-C edged angle-ply stiffened plate under a variety of partial edge compressions can be summarized as follows:
It is found that for any particular ply-orientation and loading condition, the vibration frequency decreases as the intensity of edge load increases and becomes zero at the corresponding critical loads.
For a plate with a single stiffener subjected to unidirectional partial edge loading (extending from one edge and both edges), the critical load is found to be maximum at around (±30^{o}) _{2} layup irrespective of boundary conditions. The same phenomenon is also applicable for the centrally loaded stiffened plate only for higher values of η.
It is also noticed that the boundary conditions play a significant role in the stability behaviour of a stiffened plate under bidirectional partial edge loads. The stiffened plate with (± 30^{o}) _{2} and (± 60^{o}) _{2} layups show maximum buckling resistance for S-S-S-S edge condition irrespective of loading conditions. However, for C-C-C-C edge condition, the critical load is found to be maximum at (± 45^{o}) _{2} ply-orientation in the case of bidirectional load acting from one edge of the plate (un-symmetrical loading case). For the same C-C-C-C edged stiffened plate subjected to symmetrical biaxial loading, the critical loads are found to be maximum at (± 45^{o}) _{2} ply-angle only for higher η.
It is interesting to note that when an S-S-S-S edged stiffened plate is subjected to a unidirectional edge load acting from both edges, the critical buckling loads are found to be practically same at an ply-angle of (± 48^{o}) _{2} irrespective of load bandwidth ratio η. This phenomenon is again observed at (± 90^{o}) _{2} ply-orientation. However, a similar trend is observed in C-C-C-C edged stiffened plate only at a ply-orientation of (90^{o}) _{2}.