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Transient response analysis of laminated stiffened plates
Composite Structures 58 (2002) 97–107 www.elsevier.com/locate/compstruct
Transient response analysis of laminated stiffened plates Y.V. Satish Kumar a, Madhujit Mukhopadhyay b
b,*
a Indian Register of Shipping, Powai, Mumbai 400 072, India Department of Ocean Engineering and Naval Architecture, Indian Institute of Technology, Kharagpur 721 302, India
Abstract The paper presents the linear transient response analysis of laminated stiffened plates. The work is based on a new laminated stiffened plate element recently developed which can accommodate any number of arbitrarily oriented stiffeners within the plate element and obviates the need to use the mesh lines along the stiffeners. Thus the mesh generation is not constrained by the disposition of the stiffener and is based more on the choice of stress resolution sought. The formulation is based on the first order shear deformation theory. The dynamic response analysis of a few laminated plates stiffened with blade/I-section/hat stiffeners under step load are carried out. Though some investigations have been reported on transient response of unstiffened laminated plates, the literature on laminated stiffened plates has not been reported. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Finite element; Stiffened plate; Plate; Stiffener; Hat stiffener; Transverse shear; Transient response
1. Introduction Laminated stiffened plates have found extensive applications in aerospace and ship-building industries due to their high stiffness-weight ratio and high strengthweight ratio. These structures operate under harsh environments and are subjected to severe dynamic loading. However, to the best of the knowledge of the authors only one investigation has been reported on the response analysis of stiffened structures and that too on stiffened shells and not on stiffened plates, though some research has been conducted on laminated bare plates and shells [1–14]. Gong and Kam [15] have carried out the transient response analysis of a stiffened composite submersible hull. The structure is modelled using layered shell elements for both plate and stiffener in MSC/Patran and LS-DYNA3D. However, the stiffener is modelled with shell elements and a large number of equations are required to analyse the complete structure. In order to reduce the number of equations, a new stiffened plate element based on first order shear deformation theory is developed for transient response analysis of laminated
*
Corresponding author. E-mail address: [email protected] (M. Mukhopadhyay).
stiffened plates. The reduction in the number of equations stems from the elegance of the plate element to accommodate any number of arbitrarily oriented stiffeners. This is unlike the existing finite element techniques which require the stiffener to pass along the plate nodal lines and the mesh division is based more on the users’ convenience for refinement. The basic plate element is a combination of Allman’s plane stress triangular element [16] and the Discrete Kirchhoff-Mindlin triangular plate bending element [17]. The displacement compatibility between the stiffener and the plate is ensured by using the same shape functions for both elements and the stiffener characteristics are interpreted only at the plate element nodes. Therefore, the stiffener element does not introduce any additional nodes. This facility does not exist in any of the commercial finite element packages. Moreover, the element adapts numerically both to thin and thick plate analyses and does not lock under shear. In the present paper, the validity of the new formulation is demonstrated for the transient response analysis of isotropic and laminated stiffened plates under step and air-blast loads. The results are found to be in agreement with those of published literature and the finite element software ANSYS. The observations of response behaviour of laminated plates stiffened with various stiffener sections such as hat/I-section and blade stiffeners are also discussed.
0263-8223/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 0 2 ) 0 0 0 3 6 - 3
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Nomenclature a, b, h length, width and thickness of the plate respectively ds , bs depth and width of the stiffener respectively lk length of the edge of the plate element having the midside node k nnod number of nodes in the regenerated plate element (nnod ¼ 3) nns number of nodes of stiffener element (nns ¼ 2) np number of plies u, m, w, hx , hy and hz nodal displacements u, v, hx , hy stiffener displacements zk height of the k lamina from mid surface of the laminate Ae area of the plate element Ast , Ist area and second moment of area of the stiffener respectively ½Bbh st , ½BbDh st strain displacement matrices of the stiffener in bending Bst strain displacement matrix of the stiffener Ck , Sk cosine and sine of the angle of the edge of the element, containing the mid side k, with x-axis D, Dst rigidity matrices of plate and stiffener respectively Db , Dm , Ds bending, membrane and shear rigidity matrices of plate element respectively E1 , E2 longitudinal and transverse moduli of the lamina respectively G12 , G23 , G31 shear moduli of the lamina Kb , Km stiffness matrices of DKMT and Allman’s elements respectively
Kp , Kst stiffness matrices of plate and stiffener elements respectively Lk length of the kth lamina of the stiffener M, Mst mass matrices of plate and stiffener elements respectively Mk , Ik mass and mass moment of inertia of the kth lamina of the plate element respectively st , Mst mass matrices of the stiffener at stiffener nodes M in local and plate coordinate systems respectively Ni , Pk quadratic and linear shape functions respectively Nx , Ny , Nxy membrane forces T condensation matrix Ts strain transformation matrix of the stiffener d displacement vector of plate element db , dm displacement vector of DKMT and membrane elements respectively dst displacement vector of the stiffener a , vst , cst axial, bending and shear strains of the stiffener respectively st , st strains in the stiffener in stiffener local coordinates and plate coordinates respectively m12 Poissons ratio of the lamina n, g, f area coordinates of the plate element qk density of the kth lamina /k shear factor Dhsk nodal rotation of midside node k
2. Theory
presented in this paper except for the expression of rigidity matrix ½D. The rigidity matrix ½D is given by
2.1. Stiffness matrix of the stiffened plate element
½D ¼ d ½Dm
The stiffness matrix of the stiffened plate Fig. 1 consists of the stiffness matrix of the plate and that of the stiffener. The stiffness matrix of the plate element is given by
where ½Dm is the rigidity matrix corresponding to membrane forces (Nx , Ny and Nxy ), ½Db is the bending rigidity matrix of the DKMT element and ½Ds is the shear rigidity matrix of the DKMT element. The total rigidity matrix [D] for a general laminate is given by 3 2 0 0 7 6 A11 A12 A16 B11 B12 B16 6 A12 A22 A26 B12 B22 B26 0 0 7 7 6 6 A16 A26 A66 B16 B26 B66 0 0 7 7 6 6 B11 B12 B16 D11 D12 D16 0 0 7 7 ½D ¼ 6 6 B12 B22 B26 D12 D22 D26 0 0 7 7 6 6 B16 B26 B66 D16 D26 D66 0 0 7 7 6 6 0 0 0 0 0 0 k12 A44 0 7 5 4 0 0 0 0 0 0 0 k22 A55
½Km ½Kp ¼ ½0
½0 ½Kb
ð1Þ
where ½Km is the membrane stiffness matrix of the plate corresponding to (u, v, hz ) at each of the three corner nodes of the triangle as developed by Allman [16] and ½Kb is the stiffness matrix of the Discrete Kirchhoff– Mindlin triangular plate bending element (DKMT) corresponding to (w, hx and hy ) in each of the three corner nodes of the triangle [17]. The plate elements are discussed in detail for isotropic materials in [16,17]. Hence, their formulations are not
½Db ½Ds c
ð2Þ
ð3Þ
Y.V. Satish Kumar, M. Mukhopadhyay / Composite Structures 58 (2002) 97–107
99
Fig. 2. Allman’s triangle.
Fig. 1. Stiffened plate.
Fig. 3. DKMT element.
The coefficients Aij , Bij and Dij are found in any standard text book of fibre reinforced plastics and composites [18]. k12 and k22 are the shear correction factors [19]. Moreover, the element uses a shear factor /k [20] which ensures that the element takes account of transverse shear effects in thick plates. The shear factor reduces to a negligibly low value in thin plates and eliminates the effects of transverse shear.
The membrane and rotational displacements (Figs. 2 and 3) at any point within the plate element are interpolated using the relations X 6 u fdm g ¼ ¼ ½Ni fdmi g ð4Þ v i¼1 hx ¼
2.1.1. Stiffness matrix of the stiffener The middle plane of the plate is taken as the reference plane for the stiffener in its stiffness matrix formulation. The adoption of common shape functions for both the plate and the stiffener elements allows the stiffness matrix of the stiffener to be expressed in terms of the nodal parameters of the plate element.
hy ¼
3 X
Pi hxi þ
6 X
i¼1
k¼4
3 X
6 X
i¼1
Pi hyi þ
Nk Ck Dhsk ð5Þ Nk Sk Dhsk
k¼4
where Pi are linear shape functions and Nk are quadratic shape functions [21], fdmi g are the nodal inplane displacements, hxi and hyi are the nodal rotations of the
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where ½Bbh st and ½BbDh are given in Appendix A. fdb g is the displacement vector of DKMT plate element given as T
fdb g ¼ f hx1
hy1
hy3
Dhs4
Dhs5
Dhs6 g ð13Þ
Hence, the generalised strains of the stiffener in the plate element coordinate system are expressed as fst g ¼ ½Bst fdg
ð14Þ
where ½Bst is the strain–displacement matrix of the stiffener and T
fdg ¼ f u1
v1
hx1
hy1
Fig. 4. Stiffener inside the element.
hy g
hx
ð6Þ
The generalised strains of the stiffener in the local coordinate system are given by n oT n o ohx oh y u ow st ¼ o þ hx ð7Þ ox ox ox ox Applying the chain rule of partial differentiation, the generalised strains in the local coordinates are transformed into the plate element coordinates as n o h i st ¼ Ts fst g ð8Þ where strains strains
½Ts is the transformation matrix of stiffener (Appendix A) and fst g is the vector of stiffener in the plate element coordinate system.
T fst g ¼
u;x
v;y þhx
1 ðu;y 2
þ v;x Þ w;y þ hy
hx;x
hy;y
hy;x
hx;y
w;x
fst g consists of the strains due to axial, bending and shear deformations and the strain vector is given as where fa g ¼
u;x
fvst g v;y
fcst g g
1 ðu;y 2
þ v;x Þ
ð10Þ
fvst gT ¼ f hx;x hy;y hx;y hy;x g fcst g ¼ f w;x þ hx w;y þ hy g
v4
Dhs4
Dhs6 g
The stiffness matrix of the stiffener is thus given as Z T T ½Kst ¼ ½T ½Bst ½Dst ½Bst ½T dx ð16Þ ½T is the condensation matrix which condenses the six node triangular plate element to a three node element [16,17]. ½Dst is the rigidity matrix of the stiffener based on average laminate properties. The rigidity of the stiffener is a function of the orientation of the stiffener laminations which may be either parallel or perpendicular to the laminations of the plate (Fig. 5). In the present paper, the formulation of Ref. [22] is used to compute the rigidity matrix of the stiffener based on the stacking sequence. The rigidity matrix of a stiffener with parallel laminations: The rigidity matrix for the kth lamina of the stiffener [22] is expressed as 2 3 B16 bs 0 7 6 A11 bs B11 bs 6 B11 bs D 11 bs 16 bs D 0 7 6 7 ½Dst ¼ 6 1 3 0 7 4 B16 bs D16 bs 6 ðQ66 þ Q44 Þds bs 5 0 0 0 k12 A44 ð17Þ
ð9Þ
fst g ¼ f fa g
u4
ð15Þ
corner nodes and Dhsk are the nodal rotations of the mid-side node k of edge s of the element. Let the stiffener be arbitrarily oriented at an angle u with respect to plate element coordinate system as shown in Fig. 4. The nodal displacements of the stiffener in its local coordinates x and y are T fdst g ¼ f u w
where ij ¼ Dij þ 2eBij þ e2 Aij D Bij ¼ Bij þ eAij
ð18Þ
ds and bs are the depth and width of the kth lamina of the stiffener. e is the eccentricity of the stiffener with respect to the midplane of the plate (Fig. 1) and k12 is the
ð11Þ
The axial a and shear cst strain expressions of the stiffener are similar to those of plate elements and the bending strains vst of the stiffener are expressed as fvst g ¼ ð½Bbh st þ ½BbDh st Þfdb g
ð12Þ
Fig. 5. Lamina orientations of the stiffener (a) parallel laminations (b) perpendicular laminations.
Y.V. Satish Kumar, M. Mukhopadhyay / Composite Structures 58 (2002) 97–107
shear correction factor of the stiffener [19]. Q44 and Q66 are the stiffness coefficients as found in any standard book on composites [18]. The rigidity matrix of a stiffener with perpendicular laminations: The rigidity matrix for the kth lamina of the stiffener [22] is expressed as ½Dst 2 3 d eA ds B16 ds A16 ds A11 3 s 11 6 7 ds þ ds e2 A11 ds eB16 ds eA16 7 6 ds eA11 12 7 ¼6 6 7 1 4 ds eB16 ds eB16 ðQ44 þ Q66 Þds b3s ds B66 5 6 ds A16 ds eA16 ds B66 ds A66 ð19Þ where ds and bs are the depth and width of the kth lamina of the stiffener and e is the eccentricity of the kth lamina of the stiffener with respect to the midplane of the plate (Fig. 1). 2.2. Mass matrices 2.2.1. Mass matrix of the plate element The nodal mass of the plate element at any node is given as np Ae X ½M ¼ d Mk Mk Mk Ik Ik Ik c ð20Þ 3 k¼1
3. Results and discussion The laminated plates are assumed to be simply supported in all the numerical examples. The restraints are as given below Along x-axis: u ¼ w ¼ hx ¼ hz ¼ 0 Along y-axis: v ¼ w ¼ hy ¼ hz ¼ 0 3.1. Transient response of an isotropic DRES stiffened panel under airblast load To observe the performance of the element with stiffeners inside it, the response analysis of a rectangular clamped stiffened plate, with four identical stiffeners in one direction only (Fig. 6) is carried out. The stiffened plate is subjected to blast loads in a blast chamber by the Defense Research Establishment Suffield (DRES), Canada [23]. The variation of average pressure, due to blast load, with time is as shown in Fig. 7. The Young’s modulus, Poisson’s ratio and density of material are
Mk ¼ qk ðzk zk 1 Þ qk ðz3k z3k 1 Þ 3 qk is the density of the kth lamina and zk and zk 1 are the top and the bottom z-coordinates of the kth lamina from the midplane of the plate. np is the number of plies/ laminae. Mass matrix of the stiffener: The mass matrix of the stiffener in local coordinates is given as Ik ¼
st T ¼ ½M
np X qk Akst Lkst d1 1 2 k¼1
0
0
1 1
0
0c
ð21Þ
Fig. 6. Five bay clamped DRES stiffened panel.
where Akst and Lkst are the area and the length of the kth lamina of the stiffener element. The mass matrix of the stiffener is transferred into the plate element coordinates from local coordinates. Based on the position of the stiffener in the plate element, the corresponding natural coordinates (n, g and f) at each node of the stiffener are evaluated and the stiffener mass matrix is distributed to the plate element nodes using the linear shape functions of the element [22]. Pj2 ðni ; gi ; fi Þ ½Mst j ¼ ½M st i P3 2 k¼1 Pk ðni ; gi ; fi Þ i ¼ 1; nns where nns ¼ 2
j ¼ 1; 2; 3 ð22Þ
where ½M st is the nodal mass of the stiffener in plate element coordinates and ½Mst is the equivalent mass of the stiffener distributed at the plate element nodes.
101
Fig. 7. Airblast pressure vs time.
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t ¼ 0 to a uniform pressure (step load) of magnitude 100 kPa. The material properties are E1 ¼ 25:0 GPa E2 ¼ 1:0 GPa G12 ¼ 0:05 GPa G23 ¼ 0:2 GPa G31 ¼ 0:5 Gpa m12 ¼ 0:25 q ¼ 1:0 kg=mm3
Fig. 8. Displacement response at the centre of a five bay DRES stiffened panel.
E ¼ 2:0684 1011 N/m2 , m ¼ 0:3 and q ¼ 7839:81 kg/m3 respectively. Sinha and Mukhopadhyay [24] have used a high precision triangular stiffened shallow shell element to study the response of the DRES stiffened panel. The linear response analysis is carried out using an 8 8 mesh over one half of the plate using a time step of 0.05 ms. The time variation of deflection at the centre of the plate is compared with that of Sinha and Mukhopadhyay [24] in Fig. 8. The comparison shows good agreement between the two results and also shows that the present formulation accommodating the stiffeners within the plate element is fairly accurate.
The response analysis of the unstiffened laminate has been analysed previously by Meimaris and Day [10] using 20-noded parabolic isoparametric solid element. The deflection is studied at the centre of the plate using a 16 16 mesh (over the whole plate) and with a time step of 1 ls. Fig. 10 shows a fair agreement in the displacement response between the present formulation and those of Meimaris and Day [10]. The present plate is a thick plate and the effects of transverse shear are considerable. The results indicate that the element considers the effects of transverse shear and is capable in analysing thick plates. Moreover, its efficiency in analysing thick plates is comparable to that of a 20-noded parabolic isoparametric solid element.
3.2. Transient response analysis of a laminated plate under step load The transient response analysis of a rectangular cross-ply laminate ½0°=90°T is carried out with the present element to observe its performance in FRP structures. The laminate is simply supported along all the edges (Fig. 9, ignore stiffener) and is subjected at time
Fig. 10. Displacement response of the laminated unstiffened plate.
Fig. 9. Simply supported stiffened plate with a central eccentric stiffener.
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103
3.3. Transient response analysis of a laminated stiffened plate under step load The transient response analysis of a cross-ply laminate ½0°=90°T stiffened centrally with a cross-ply blade stiffener ½0°=90°T as shown in Fig. 9 is carried out with the present element plate using a 16 16 mesh over the whole plate. The analysis is also carried out with the general purpose finite element software ANSYS using eight noded layered shell elements (SHELL99) for both plate and stiffener. The material properties are similar to those given in example 3.2. The ply orientations of the stiffener are perpendicular with respect to plate laminae. The simply supported stiffened plate is subjected to an uniformly distributed step load of 100 kPa and the time step used in the response study is 1 lsec. Fig. 11 shows the variation of the centre deflection response with time compared with that obtained using ANSYS. The results are in fair agreement. The analysis is carried out with 5665 d.o.f. and 352 elements in ANSYS whereas only 1734 d.o.f and 256 elements are used in the present analysis. To test the validity of the arbitrarily oriented stiffener formulation, the plate shown in Fig. 9 has been analysed using the irregular mesh (Fig. 12). The ply orientations of the stiffener are aligned parallel to the plate laminae. The displacement response of the plate at point P (x ¼ 109:375 mm and y ¼ 125 mm) using the arbitrary mesh (15 16) is compared with that of the regular mesh (16 16) as shown in Fig. 13. The comparison shows a good agreement between the two and indicates that the present formulation with arbitrarily oriented stiffener is as efficient as that with stiffener along the plate element boundaries. 3.4. Transient response analysis of laminated plates stiffened with hat/I-section/rectangular stiffeners The response study of a simply supported laminated plate stiffened with various stiffener sections (hat/I-sec-
Fig. 12. Finite element mesh of a laminated plate with arbitrarily oriented stiffener inside the plate element.
Fig. 13. Displacement response of a laminated plate with arbitrarily oriented stiffener inside the plate element.
tion/blade) having the same bending rigidity (EIst ) is carried out. The flexural rigidity of the stiffener takes into account of the plate rigidity also and is calculated using Eqs. (17) and (19). The orientation in the stiffener laminae with respect to plate laminae are as shown in Fig. 14. The plate is subjected to a uniformly distributed step load of 100 kPa. The material properties are: E1 ¼ 130:0 Gpa E2 ¼ 10:0 Gpa G12 ¼ 4:85 GPa G23 ¼ 3:62 Gpa G31 ¼ 4:85 Gpa m12 ¼ 0:31 q ¼ 1389:23 kg=mm3
Fig. 11. Displacement response of the laminated stiffened plate.
The stacking sequence in the plate is ½0°=90°=45°=0°= 45°s whereas that in the stiffener flanges and webs is ½0°=90°=45°= 45°s . Fig. 15 shows the centre deflection response of hat/I-section/blade stiffened plates with
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Fig. 14. A laminated plate stiffened with various stiffener sections––hat/I-section/blade.
respect to time. The maximum displacement of blade stiffened plates is higher than the hat and I-section stiffened plates. The central displacement of the blade stiffened plate is higher than that of plates stiffened with hat and I-section stiffeners. This is due to the high axial-bending coupling component of the rigidity matrix of the blade stiffener whereas it is minimum for the
I-section stiffener for the given dimensions. However, the period of response of I-section stiffened plates is high whereas that of the blade stiffened plate is moderate and hat stiffened plate is minimum. This is due to the low torsional rigidity of the I-section stiffener for the given dimensions whereas that of the hat stiffener is highest. Therefore, the period of response of the plates with
Y.V. Satish Kumar, M. Mukhopadhyay / Composite Structures 58 (2002) 97–107
Fig. 15. Displacement response of a laminated plate stiffened with various stiffener sections––hat/I-section/blade.
I-section stiffeners is higher and that of hat stiffened plates is minimum. Fig. 16 shows the response of stress rx at the centre of the stiffened plates. The stress in the hat stiffened plate is minimum than that of plates stiffened with I-section and blade stiffeners.
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Fig. 16. Stress response of a laminated plate stiffened with various stiffener sections––hat/I-section/blade.
the amplitude of displacement whereas the torsional rigidity influences the period of response. Appendix A
2
cos2 u 6 0 ½Tx 6 4 0 0
sin2 u 0 0 0
1 2
sin 2u 0 0 0
0 cos2 u 12 sin 2u 0
0 sin2 u 1 sin 2u 2 0
0 1 sin 2u 2 cos2 u 0
0 1 sin 2u 2 sin2 u 0
0 0 0 cos u
3 0 0 7 7 0 5 sin u
ð23Þ
4. Conclusions ½Bbh st
The transient response analysis of the laminated composite stiffened plates is presented in the paper. The analysis is carried out using a new stiffened plate element which has the elegance of accommodating arbitrarily oriented stiffeners within the plate element. This enables the structural analysis of stiffened structures to be carried out with fewer d.o.f. than the conventional approach (discrete plate and beam) and with no loss of accuracy. The transient response of an isotropic stiffened plate and a laminated plate subjected to step load/airblast load is carried out. The displacement response is found to be in agreement with other published results. The displacement response of a laminated stiffened plate is compared with that obtained using the ANSYS software and the agreement between the two results is reasonable. The displacement and stress response of laminated plates stiffened with hat/I-section/blade stiffeners having same bending rigidity is studied in the present investigation. The axial-bending coupling component of the rigidity matrix of the stiffener influences
¼
2
y32
1 6 6 0 6 2Ae 4 0 x32
3
0
y13
0
y21
0
0
0
0
x32
0
x13
0
x21
0
0
y32 0
0 x13
y13 0
0 x21
y21 0
0 0
0 0
07 7 7 05 0 ð24Þ
where x and y are the nodal coordinates of the plate element and xji ¼ xj xi
yji ¼ yj yi
i; j ¼ 1; 2; 3
2 60 1 6 60 ½BbDh st ¼ 2Ae 6 40 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
P14 P24 P34 P44
P15 P25 P35 P45
3 P16 7 P26 7 7 P36 7 5 P46 ð25Þ
Ae is the area of the plate element.
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Y.V. Satish Kumar, M. Mukhopadhyay / Composite Structures 58 (2002) 97–107
P14 ¼ ðN4;n y13 þ N4;g y21 ÞC4 ;
P15 ¼ ðN5;n y13 þ N5;g y21 ÞC5
P16 ¼ ðN6;n y13 þ N6;g y21 ÞC6 ;
P24 ¼ ðN4;n y21 þ N4;g x21 ÞS4
P25 ¼ ðN5;n y21 þ N5;g x21 ÞS5 ;
P26 ¼ ðN6;n y21 þ N6;g x21 ÞS6
P34 ¼ ðN4;n y13 þ N4;g y21 ÞS4 ;
P35 ¼ ðN5;n y13 þ N5;g y21 ÞS5
P36 ¼ ðN6;n y13 þ N6;g y21 ÞS6 ;
P44 ¼ ðN4;n y21 þ N4;g x21 ÞC4
P45 ¼ ðN5;n y21 þ N5;g x21 ÞC5 ;
Nk;n ¼
oNk ; on
Nk;g ¼
oNk og
P46 ¼ ðN6;n y21 þ N6;g x21 ÞC6
k ¼ 4; 5; 6
Ck , Sk are cosine and sine of the angle between the edge of the plate element containing the midside node k and x-axis, k ¼ 4,5,6.
Appendix B. Transient response analysis of stiffened plate The response of an eccentrically stiffened simply supported square plate with a central stiffener (Fig. 17) has been analysed to study the effects of axial-bending rigidity and torsional rigidity of the stiffener. The stiffened plate is subjected to a suddenly applied uniform pressure of 0.3 MPa and the response of the plate is predicted with a time step of 0.05 msec. The Young’s modulus, Poissson’s ratio and density of material are E ¼ 6:89 1010 N/m2 , m ¼ 0:3 and q ¼ 2670 kg/m3 respectively. The time history curves of the displacement at the stiffener (A), as shown in Fig. 18 is computed using different axial-bending coupling rigidity of the stiffener. Three different values of axial-bending cou-
Fig. 17. Two bay centrally stiffened plate.
pling rigidity have been considered i.e. 0:5 Ds ð1; 2Þ, 1:0 Ds ð1; 2Þ and 1:25 Ds ð1; 2Þ. Ds ð1; 2Þ has been computed using the dimensions of the stiffener as shown in Fig. 17. The central deflection has been found to increase with increasing the axial-bending rigidity of the stiffener. The period of response has also been found to increase with increase in the coupling rigidity of the stiffener. Fig. 19 shows the variation of transient deflection at point A, for varying toriosnal rigidity of the stiffener i.e 0:5 Ds ð4; 4Þ, 1:0 Ds ð4; 4Þ and 1:5 Ds ð4; 4Þ. Ds ð4; 4Þ has been calculated using the dimensions of the stiffener as shown in Fig. 17. The torsional rigidity of the stiffener has no significant influence on the response behaviour of the stiffened plate.
Fig. 18. Transient response of stiffened plate for varying axial-bending coupling rigidity of the stiffener.
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Fig. 19. Transient response of stiffened plate for varying torsional rigidity of the stiffener.
References [1] Dobyns AL. Analysis of simply-supported orthotropic plates subject to static and dynamic loads. AIAA 1981;19:642–50. [2] Craig TJ, Dawe DJ. Flexural vibration of symmetrically laminated composite rectangular plates including transverse shear effects. Int J Solids and Struct 1986;22:155–69. [3] Khdeir AA, Reddy JN. Dynamic response of antisymmetric angle-ply laminated plates subjected to arbitrary loading. J Sound and Vib 1988;126:437–45. [4] Reddy JN. On the solutions to forced motions of rectangular composite plates. J Appl Mech 1982;49:403–8. [5] Reddy JN. Dynamic (transient) analysis of layered anisotropic composite material plates. Int J Numer Meth Eng 1983;19:237– 55. [6] Librescu L, Nosier A. Response of laminated composite flat panels to sonic boom and explosive blast loadings. AIAA 1990;28: 345–52. [7] Kant T, Varaiya JH, Arora CP. A finite element transient analysis of composite and sandwich plates based on a refined theory and implicit time integration techniques. Comput Struct 1990;36(3): 401–20. [8] Taylor TW, Nayfeh AH, Wade JE. Dynamic response of thick anisotropic composite plates. Compos Eng 1994;4(5):475–86. [9] Khdeir AA. Transient response of refined cross-ply laminated plates for various boundary conditions. J Acoust Soc Am 1995; 97(3):1664. [10] Meimaris C, Day JD. Dynamic response of laminated anisotropic plates. Comput Struct 1995;55(2):269–78. [11] Chen J, Dawe DJ. Linear transient analysis of rectangular laminated plates by a finite strip superposition method. Compos Struct 1996;35(2):213–28. [12] Wang S, Chen J, Dawe DJ. Linear transient analysis of rectangular laminates using spline finite strips. Compos Struct 1998; 41(1):57–66.
[13] Lam KY, Zhang ZJ, Gong SW, Chan ES. The transient response of submerged orthotropic cylindrical shells exposed to underwater shock. Compos Struct 1999;43(3):179–93. [14] Wang YY, Lam KY, Liu GR. The effect of rotary inertia on the dynamic response of laminated composite plate. Compos Struct 2000;48(4):265–73. [15] Gong SW, Lam KY. Transient response of stiffened composite sub-mersible hull subjected to underwater explosive shock. Compos Struct 1998;41(1):27–37. [16] Allman DJ. A compatible triangular element including vertex rotations for plane elasticity analysis. Comput Struct 1984;19 (1/2):1–8. [17] Katili I. A new Discrete Kirchhoff–Mindlin element based on Mindlin–Reissner plate theory and assumed shear strain fields––Part I: An extended DKT element for thick-plate bending analysis. Int J Numer Meth Eng 1993;36(11):1859–83. [18] Daniel Isaac M, Ishai Ori. Engineering mechanics of composite materials. New York: Oxford University Press; 1994. [19] Whitney JM. Shear correction factors for orthotropic laminates under static load. J Appl Mech, Trans Am Soc Mech Eng 1973;40(1):302–4. [20] Satish Kumar YV, Mukhopadhyay M. A new finite element for buckling analysis of laminated stiffened plates. Compos Struct 1999;46(4):321–31. [21] Zienkiewicz OC. The finite element method. 3rd ed. New Delhi: Tata McGraw-Hill; 1979. [22] Chattopadhyay B. Linear and non-linear static and dynamic analysis of composite stiffened plates by the finite element method. PhD thesis, Indian Institute of Technology, Kharagpur, India, 1993. [23] Jiang J, Olson MD. Nonlinear dynamic analysis of blast loaded cylindrical shell structures. Comput Struct 1991;41(1):41–52. [24] Sinha G, Mukhopadhyay M. Static, freee and forced vibration analysis of arbitrary non-uniform shells with tapered stiffeners. Comput Struct 1997;62(5):919–33.
Transient response analysis of laminated stiffened plates Y.V. Satish Kumar a, Madhujit Mukhopadhyay b
b,*
a Indian Register of Shipping, Powai, Mumbai 400 072, India Department of Ocean Engineering and Naval Architecture, Indian Institute of Technology, Kharagpur 721 302, India
Abstract The paper presents the linear transient response analysis of laminated stiffened plates. The work is based on a new laminated stiffened plate element recently developed which can accommodate any number of arbitrarily oriented stiffeners within the plate element and obviates the need to use the mesh lines along the stiffeners. Thus the mesh generation is not constrained by the disposition of the stiffener and is based more on the choice of stress resolution sought. The formulation is based on the first order shear deformation theory. The dynamic response analysis of a few laminated plates stiffened with blade/I-section/hat stiffeners under step load are carried out. Though some investigations have been reported on transient response of unstiffened laminated plates, the literature on laminated stiffened plates has not been reported. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Finite element; Stiffened plate; Plate; Stiffener; Hat stiffener; Transverse shear; Transient response
1. Introduction Laminated stiffened plates have found extensive applications in aerospace and ship-building industries due to their high stiffness-weight ratio and high strengthweight ratio. These structures operate under harsh environments and are subjected to severe dynamic loading. However, to the best of the knowledge of the authors only one investigation has been reported on the response analysis of stiffened structures and that too on stiffened shells and not on stiffened plates, though some research has been conducted on laminated bare plates and shells [1–14]. Gong and Kam [15] have carried out the transient response analysis of a stiffened composite submersible hull. The structure is modelled using layered shell elements for both plate and stiffener in MSC/Patran and LS-DYNA3D. However, the stiffener is modelled with shell elements and a large number of equations are required to analyse the complete structure. In order to reduce the number of equations, a new stiffened plate element based on first order shear deformation theory is developed for transient response analysis of laminated
*
Corresponding author. E-mail address: [email protected] (M. Mukhopadhyay).
stiffened plates. The reduction in the number of equations stems from the elegance of the plate element to accommodate any number of arbitrarily oriented stiffeners. This is unlike the existing finite element techniques which require the stiffener to pass along the plate nodal lines and the mesh division is based more on the users’ convenience for refinement. The basic plate element is a combination of Allman’s plane stress triangular element [16] and the Discrete Kirchhoff-Mindlin triangular plate bending element [17]. The displacement compatibility between the stiffener and the plate is ensured by using the same shape functions for both elements and the stiffener characteristics are interpreted only at the plate element nodes. Therefore, the stiffener element does not introduce any additional nodes. This facility does not exist in any of the commercial finite element packages. Moreover, the element adapts numerically both to thin and thick plate analyses and does not lock under shear. In the present paper, the validity of the new formulation is demonstrated for the transient response analysis of isotropic and laminated stiffened plates under step and air-blast loads. The results are found to be in agreement with those of published literature and the finite element software ANSYS. The observations of response behaviour of laminated plates stiffened with various stiffener sections such as hat/I-section and blade stiffeners are also discussed.
0263-8223/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 0 2 ) 0 0 0 3 6 - 3
98
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Nomenclature a, b, h length, width and thickness of the plate respectively ds , bs depth and width of the stiffener respectively lk length of the edge of the plate element having the midside node k nnod number of nodes in the regenerated plate element (nnod ¼ 3) nns number of nodes of stiffener element (nns ¼ 2) np number of plies u, m, w, hx , hy and hz nodal displacements u, v, hx , hy stiffener displacements zk height of the k lamina from mid surface of the laminate Ae area of the plate element Ast , Ist area and second moment of area of the stiffener respectively ½Bbh st , ½BbDh st strain displacement matrices of the stiffener in bending Bst strain displacement matrix of the stiffener Ck , Sk cosine and sine of the angle of the edge of the element, containing the mid side k, with x-axis D, Dst rigidity matrices of plate and stiffener respectively Db , Dm , Ds bending, membrane and shear rigidity matrices of plate element respectively E1 , E2 longitudinal and transverse moduli of the lamina respectively G12 , G23 , G31 shear moduli of the lamina Kb , Km stiffness matrices of DKMT and Allman’s elements respectively
Kp , Kst stiffness matrices of plate and stiffener elements respectively Lk length of the kth lamina of the stiffener M, Mst mass matrices of plate and stiffener elements respectively Mk , Ik mass and mass moment of inertia of the kth lamina of the plate element respectively st , Mst mass matrices of the stiffener at stiffener nodes M in local and plate coordinate systems respectively Ni , Pk quadratic and linear shape functions respectively Nx , Ny , Nxy membrane forces T condensation matrix Ts strain transformation matrix of the stiffener d displacement vector of plate element db , dm displacement vector of DKMT and membrane elements respectively dst displacement vector of the stiffener a , vst , cst axial, bending and shear strains of the stiffener respectively st , st strains in the stiffener in stiffener local coordinates and plate coordinates respectively m12 Poissons ratio of the lamina n, g, f area coordinates of the plate element qk density of the kth lamina /k shear factor Dhsk nodal rotation of midside node k
2. Theory
presented in this paper except for the expression of rigidity matrix ½D. The rigidity matrix ½D is given by
2.1. Stiffness matrix of the stiffened plate element
½D ¼ d ½Dm
The stiffness matrix of the stiffened plate Fig. 1 consists of the stiffness matrix of the plate and that of the stiffener. The stiffness matrix of the plate element is given by
where ½Dm is the rigidity matrix corresponding to membrane forces (Nx , Ny and Nxy ), ½Db is the bending rigidity matrix of the DKMT element and ½Ds is the shear rigidity matrix of the DKMT element. The total rigidity matrix [D] for a general laminate is given by 3 2 0 0 7 6 A11 A12 A16 B11 B12 B16 6 A12 A22 A26 B12 B22 B26 0 0 7 7 6 6 A16 A26 A66 B16 B26 B66 0 0 7 7 6 6 B11 B12 B16 D11 D12 D16 0 0 7 7 ½D ¼ 6 6 B12 B22 B26 D12 D22 D26 0 0 7 7 6 6 B16 B26 B66 D16 D26 D66 0 0 7 7 6 6 0 0 0 0 0 0 k12 A44 0 7 5 4 0 0 0 0 0 0 0 k22 A55
½Km ½Kp ¼ ½0
½0 ½Kb
ð1Þ
where ½Km is the membrane stiffness matrix of the plate corresponding to (u, v, hz ) at each of the three corner nodes of the triangle as developed by Allman [16] and ½Kb is the stiffness matrix of the Discrete Kirchhoff– Mindlin triangular plate bending element (DKMT) corresponding to (w, hx and hy ) in each of the three corner nodes of the triangle [17]. The plate elements are discussed in detail for isotropic materials in [16,17]. Hence, their formulations are not
½Db ½Ds c
ð2Þ
ð3Þ
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99
Fig. 2. Allman’s triangle.
Fig. 1. Stiffened plate.
Fig. 3. DKMT element.
The coefficients Aij , Bij and Dij are found in any standard text book of fibre reinforced plastics and composites [18]. k12 and k22 are the shear correction factors [19]. Moreover, the element uses a shear factor /k [20] which ensures that the element takes account of transverse shear effects in thick plates. The shear factor reduces to a negligibly low value in thin plates and eliminates the effects of transverse shear.
The membrane and rotational displacements (Figs. 2 and 3) at any point within the plate element are interpolated using the relations X 6 u fdm g ¼ ¼ ½Ni fdmi g ð4Þ v i¼1 hx ¼
2.1.1. Stiffness matrix of the stiffener The middle plane of the plate is taken as the reference plane for the stiffener in its stiffness matrix formulation. The adoption of common shape functions for both the plate and the stiffener elements allows the stiffness matrix of the stiffener to be expressed in terms of the nodal parameters of the plate element.
hy ¼
3 X
Pi hxi þ
6 X
i¼1
k¼4
3 X
6 X
i¼1
Pi hyi þ
Nk Ck Dhsk ð5Þ Nk Sk Dhsk
k¼4
where Pi are linear shape functions and Nk are quadratic shape functions [21], fdmi g are the nodal inplane displacements, hxi and hyi are the nodal rotations of the
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where ½Bbh st and ½BbDh are given in Appendix A. fdb g is the displacement vector of DKMT plate element given as T
fdb g ¼ f hx1
hy1
hy3
Dhs4
Dhs5
Dhs6 g ð13Þ
Hence, the generalised strains of the stiffener in the plate element coordinate system are expressed as fst g ¼ ½Bst fdg
ð14Þ
where ½Bst is the strain–displacement matrix of the stiffener and T
fdg ¼ f u1
v1
hx1
hy1
Fig. 4. Stiffener inside the element.
hy g
hx
ð6Þ
The generalised strains of the stiffener in the local coordinate system are given by n oT n o ohx oh y u ow st ¼ o þ hx ð7Þ ox ox ox ox Applying the chain rule of partial differentiation, the generalised strains in the local coordinates are transformed into the plate element coordinates as n o h i st ¼ Ts fst g ð8Þ where strains strains
½Ts is the transformation matrix of stiffener (Appendix A) and fst g is the vector of stiffener in the plate element coordinate system.
T fst g ¼
u;x
v;y þhx
1 ðu;y 2
þ v;x Þ w;y þ hy
hx;x
hy;y
hy;x
hx;y
w;x
fst g consists of the strains due to axial, bending and shear deformations and the strain vector is given as where fa g ¼
u;x
fvst g v;y
fcst g g
1 ðu;y 2
þ v;x Þ
ð10Þ
fvst gT ¼ f hx;x hy;y hx;y hy;x g fcst g ¼ f w;x þ hx w;y þ hy g
v4
Dhs4
Dhs6 g
The stiffness matrix of the stiffener is thus given as Z T T ½Kst ¼ ½T ½Bst ½Dst ½Bst ½T dx ð16Þ ½T is the condensation matrix which condenses the six node triangular plate element to a three node element [16,17]. ½Dst is the rigidity matrix of the stiffener based on average laminate properties. The rigidity of the stiffener is a function of the orientation of the stiffener laminations which may be either parallel or perpendicular to the laminations of the plate (Fig. 5). In the present paper, the formulation of Ref. [22] is used to compute the rigidity matrix of the stiffener based on the stacking sequence. The rigidity matrix of a stiffener with parallel laminations: The rigidity matrix for the kth lamina of the stiffener [22] is expressed as 2 3 B16 bs 0 7 6 A11 bs B11 bs 6 B11 bs D 11 bs 16 bs D 0 7 6 7 ½Dst ¼ 6 1 3 0 7 4 B16 bs D16 bs 6 ðQ66 þ Q44 Þds bs 5 0 0 0 k12 A44 ð17Þ
ð9Þ
fst g ¼ f fa g
u4
ð15Þ
corner nodes and Dhsk are the nodal rotations of the mid-side node k of edge s of the element. Let the stiffener be arbitrarily oriented at an angle u with respect to plate element coordinate system as shown in Fig. 4. The nodal displacements of the stiffener in its local coordinates x and y are T fdst g ¼ f u w
where ij ¼ Dij þ 2eBij þ e2 Aij D Bij ¼ Bij þ eAij
ð18Þ
ds and bs are the depth and width of the kth lamina of the stiffener. e is the eccentricity of the stiffener with respect to the midplane of the plate (Fig. 1) and k12 is the
ð11Þ
The axial a and shear cst strain expressions of the stiffener are similar to those of plate elements and the bending strains vst of the stiffener are expressed as fvst g ¼ ð½Bbh st þ ½BbDh st Þfdb g
ð12Þ
Fig. 5. Lamina orientations of the stiffener (a) parallel laminations (b) perpendicular laminations.
Y.V. Satish Kumar, M. Mukhopadhyay / Composite Structures 58 (2002) 97–107
shear correction factor of the stiffener [19]. Q44 and Q66 are the stiffness coefficients as found in any standard book on composites [18]. The rigidity matrix of a stiffener with perpendicular laminations: The rigidity matrix for the kth lamina of the stiffener [22] is expressed as ½Dst 2 3 d eA ds B16 ds A16 ds A11 3 s 11 6 7 ds þ ds e2 A11 ds eB16 ds eA16 7 6 ds eA11 12 7 ¼6 6 7 1 4 ds eB16 ds eB16 ðQ44 þ Q66 Þds b3s ds B66 5 6 ds A16 ds eA16 ds B66 ds A66 ð19Þ where ds and bs are the depth and width of the kth lamina of the stiffener and e is the eccentricity of the kth lamina of the stiffener with respect to the midplane of the plate (Fig. 1). 2.2. Mass matrices 2.2.1. Mass matrix of the plate element The nodal mass of the plate element at any node is given as np Ae X ½M ¼ d Mk Mk Mk Ik Ik Ik c ð20Þ 3 k¼1
3. Results and discussion The laminated plates are assumed to be simply supported in all the numerical examples. The restraints are as given below Along x-axis: u ¼ w ¼ hx ¼ hz ¼ 0 Along y-axis: v ¼ w ¼ hy ¼ hz ¼ 0 3.1. Transient response of an isotropic DRES stiffened panel under airblast load To observe the performance of the element with stiffeners inside it, the response analysis of a rectangular clamped stiffened plate, with four identical stiffeners in one direction only (Fig. 6) is carried out. The stiffened plate is subjected to blast loads in a blast chamber by the Defense Research Establishment Suffield (DRES), Canada [23]. The variation of average pressure, due to blast load, with time is as shown in Fig. 7. The Young’s modulus, Poisson’s ratio and density of material are
Mk ¼ qk ðzk zk 1 Þ qk ðz3k z3k 1 Þ 3 qk is the density of the kth lamina and zk and zk 1 are the top and the bottom z-coordinates of the kth lamina from the midplane of the plate. np is the number of plies/ laminae. Mass matrix of the stiffener: The mass matrix of the stiffener in local coordinates is given as Ik ¼
st T ¼ ½M
np X qk Akst Lkst d1 1 2 k¼1
0
0
1 1
0
0c
ð21Þ
Fig. 6. Five bay clamped DRES stiffened panel.
where Akst and Lkst are the area and the length of the kth lamina of the stiffener element. The mass matrix of the stiffener is transferred into the plate element coordinates from local coordinates. Based on the position of the stiffener in the plate element, the corresponding natural coordinates (n, g and f) at each node of the stiffener are evaluated and the stiffener mass matrix is distributed to the plate element nodes using the linear shape functions of the element [22]. Pj2 ðni ; gi ; fi Þ ½Mst j ¼ ½M st i P3 2 k¼1 Pk ðni ; gi ; fi Þ i ¼ 1; nns where nns ¼ 2
j ¼ 1; 2; 3 ð22Þ
where ½M st is the nodal mass of the stiffener in plate element coordinates and ½Mst is the equivalent mass of the stiffener distributed at the plate element nodes.
101
Fig. 7. Airblast pressure vs time.
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t ¼ 0 to a uniform pressure (step load) of magnitude 100 kPa. The material properties are E1 ¼ 25:0 GPa E2 ¼ 1:0 GPa G12 ¼ 0:05 GPa G23 ¼ 0:2 GPa G31 ¼ 0:5 Gpa m12 ¼ 0:25 q ¼ 1:0 kg=mm3
Fig. 8. Displacement response at the centre of a five bay DRES stiffened panel.
E ¼ 2:0684 1011 N/m2 , m ¼ 0:3 and q ¼ 7839:81 kg/m3 respectively. Sinha and Mukhopadhyay [24] have used a high precision triangular stiffened shallow shell element to study the response of the DRES stiffened panel. The linear response analysis is carried out using an 8 8 mesh over one half of the plate using a time step of 0.05 ms. The time variation of deflection at the centre of the plate is compared with that of Sinha and Mukhopadhyay [24] in Fig. 8. The comparison shows good agreement between the two results and also shows that the present formulation accommodating the stiffeners within the plate element is fairly accurate.
The response analysis of the unstiffened laminate has been analysed previously by Meimaris and Day [10] using 20-noded parabolic isoparametric solid element. The deflection is studied at the centre of the plate using a 16 16 mesh (over the whole plate) and with a time step of 1 ls. Fig. 10 shows a fair agreement in the displacement response between the present formulation and those of Meimaris and Day [10]. The present plate is a thick plate and the effects of transverse shear are considerable. The results indicate that the element considers the effects of transverse shear and is capable in analysing thick plates. Moreover, its efficiency in analysing thick plates is comparable to that of a 20-noded parabolic isoparametric solid element.
3.2. Transient response analysis of a laminated plate under step load The transient response analysis of a rectangular cross-ply laminate ½0°=90°T is carried out with the present element to observe its performance in FRP structures. The laminate is simply supported along all the edges (Fig. 9, ignore stiffener) and is subjected at time
Fig. 10. Displacement response of the laminated unstiffened plate.
Fig. 9. Simply supported stiffened plate with a central eccentric stiffener.
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103
3.3. Transient response analysis of a laminated stiffened plate under step load The transient response analysis of a cross-ply laminate ½0°=90°T stiffened centrally with a cross-ply blade stiffener ½0°=90°T as shown in Fig. 9 is carried out with the present element plate using a 16 16 mesh over the whole plate. The analysis is also carried out with the general purpose finite element software ANSYS using eight noded layered shell elements (SHELL99) for both plate and stiffener. The material properties are similar to those given in example 3.2. The ply orientations of the stiffener are perpendicular with respect to plate laminae. The simply supported stiffened plate is subjected to an uniformly distributed step load of 100 kPa and the time step used in the response study is 1 lsec. Fig. 11 shows the variation of the centre deflection response with time compared with that obtained using ANSYS. The results are in fair agreement. The analysis is carried out with 5665 d.o.f. and 352 elements in ANSYS whereas only 1734 d.o.f and 256 elements are used in the present analysis. To test the validity of the arbitrarily oriented stiffener formulation, the plate shown in Fig. 9 has been analysed using the irregular mesh (Fig. 12). The ply orientations of the stiffener are aligned parallel to the plate laminae. The displacement response of the plate at point P (x ¼ 109:375 mm and y ¼ 125 mm) using the arbitrary mesh (15 16) is compared with that of the regular mesh (16 16) as shown in Fig. 13. The comparison shows a good agreement between the two and indicates that the present formulation with arbitrarily oriented stiffener is as efficient as that with stiffener along the plate element boundaries. 3.4. Transient response analysis of laminated plates stiffened with hat/I-section/rectangular stiffeners The response study of a simply supported laminated plate stiffened with various stiffener sections (hat/I-sec-
Fig. 12. Finite element mesh of a laminated plate with arbitrarily oriented stiffener inside the plate element.
Fig. 13. Displacement response of a laminated plate with arbitrarily oriented stiffener inside the plate element.
tion/blade) having the same bending rigidity (EIst ) is carried out. The flexural rigidity of the stiffener takes into account of the plate rigidity also and is calculated using Eqs. (17) and (19). The orientation in the stiffener laminae with respect to plate laminae are as shown in Fig. 14. The plate is subjected to a uniformly distributed step load of 100 kPa. The material properties are: E1 ¼ 130:0 Gpa E2 ¼ 10:0 Gpa G12 ¼ 4:85 GPa G23 ¼ 3:62 Gpa G31 ¼ 4:85 Gpa m12 ¼ 0:31 q ¼ 1389:23 kg=mm3
Fig. 11. Displacement response of the laminated stiffened plate.
The stacking sequence in the plate is ½0°=90°=45°=0°= 45°s whereas that in the stiffener flanges and webs is ½0°=90°=45°= 45°s . Fig. 15 shows the centre deflection response of hat/I-section/blade stiffened plates with
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Y.V. Satish Kumar, M. Mukhopadhyay / Composite Structures 58 (2002) 97–107
Fig. 14. A laminated plate stiffened with various stiffener sections––hat/I-section/blade.
respect to time. The maximum displacement of blade stiffened plates is higher than the hat and I-section stiffened plates. The central displacement of the blade stiffened plate is higher than that of plates stiffened with hat and I-section stiffeners. This is due to the high axial-bending coupling component of the rigidity matrix of the blade stiffener whereas it is minimum for the
I-section stiffener for the given dimensions. However, the period of response of I-section stiffened plates is high whereas that of the blade stiffened plate is moderate and hat stiffened plate is minimum. This is due to the low torsional rigidity of the I-section stiffener for the given dimensions whereas that of the hat stiffener is highest. Therefore, the period of response of the plates with
Y.V. Satish Kumar, M. Mukhopadhyay / Composite Structures 58 (2002) 97–107
Fig. 15. Displacement response of a laminated plate stiffened with various stiffener sections––hat/I-section/blade.
I-section stiffeners is higher and that of hat stiffened plates is minimum. Fig. 16 shows the response of stress rx at the centre of the stiffened plates. The stress in the hat stiffened plate is minimum than that of plates stiffened with I-section and blade stiffeners.
105
Fig. 16. Stress response of a laminated plate stiffened with various stiffener sections––hat/I-section/blade.
the amplitude of displacement whereas the torsional rigidity influences the period of response. Appendix A
2
cos2 u 6 0 ½Tx 6 4 0 0
sin2 u 0 0 0
1 2
sin 2u 0 0 0
0 cos2 u 12 sin 2u 0
0 sin2 u 1 sin 2u 2 0
0 1 sin 2u 2 cos2 u 0
0 1 sin 2u 2 sin2 u 0
0 0 0 cos u
3 0 0 7 7 0 5 sin u
ð23Þ
4. Conclusions ½Bbh st
The transient response analysis of the laminated composite stiffened plates is presented in the paper. The analysis is carried out using a new stiffened plate element which has the elegance of accommodating arbitrarily oriented stiffeners within the plate element. This enables the structural analysis of stiffened structures to be carried out with fewer d.o.f. than the conventional approach (discrete plate and beam) and with no loss of accuracy. The transient response of an isotropic stiffened plate and a laminated plate subjected to step load/airblast load is carried out. The displacement response is found to be in agreement with other published results. The displacement response of a laminated stiffened plate is compared with that obtained using the ANSYS software and the agreement between the two results is reasonable. The displacement and stress response of laminated plates stiffened with hat/I-section/blade stiffeners having same bending rigidity is studied in the present investigation. The axial-bending coupling component of the rigidity matrix of the stiffener influences
¼
2
y32
1 6 6 0 6 2Ae 4 0 x32
3
0
y13
0
y21
0
0
0
0
x32
0
x13
0
x21
0
0
y32 0
0 x13
y13 0
0 x21
y21 0
0 0
0 0
07 7 7 05 0 ð24Þ
where x and y are the nodal coordinates of the plate element and xji ¼ xj xi
yji ¼ yj yi
i; j ¼ 1; 2; 3
2 60 1 6 60 ½BbDh st ¼ 2Ae 6 40 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
P14 P24 P34 P44
P15 P25 P35 P45
3 P16 7 P26 7 7 P36 7 5 P46 ð25Þ
Ae is the area of the plate element.
106
Y.V. Satish Kumar, M. Mukhopadhyay / Composite Structures 58 (2002) 97–107
P14 ¼ ðN4;n y13 þ N4;g y21 ÞC4 ;
P15 ¼ ðN5;n y13 þ N5;g y21 ÞC5
P16 ¼ ðN6;n y13 þ N6;g y21 ÞC6 ;
P24 ¼ ðN4;n y21 þ N4;g x21 ÞS4
P25 ¼ ðN5;n y21 þ N5;g x21 ÞS5 ;
P26 ¼ ðN6;n y21 þ N6;g x21 ÞS6
P34 ¼ ðN4;n y13 þ N4;g y21 ÞS4 ;
P35 ¼ ðN5;n y13 þ N5;g y21 ÞS5
P36 ¼ ðN6;n y13 þ N6;g y21 ÞS6 ;
P44 ¼ ðN4;n y21 þ N4;g x21 ÞC4
P45 ¼ ðN5;n y21 þ N5;g x21 ÞC5 ;
Nk;n ¼
oNk ; on
Nk;g ¼
oNk og
P46 ¼ ðN6;n y21 þ N6;g x21 ÞC6
k ¼ 4; 5; 6
Ck , Sk are cosine and sine of the angle between the edge of the plate element containing the midside node k and x-axis, k ¼ 4,5,6.
Appendix B. Transient response analysis of stiffened plate The response of an eccentrically stiffened simply supported square plate with a central stiffener (Fig. 17) has been analysed to study the effects of axial-bending rigidity and torsional rigidity of the stiffener. The stiffened plate is subjected to a suddenly applied uniform pressure of 0.3 MPa and the response of the plate is predicted with a time step of 0.05 msec. The Young’s modulus, Poissson’s ratio and density of material are E ¼ 6:89 1010 N/m2 , m ¼ 0:3 and q ¼ 2670 kg/m3 respectively. The time history curves of the displacement at the stiffener (A), as shown in Fig. 18 is computed using different axial-bending coupling rigidity of the stiffener. Three different values of axial-bending cou-
Fig. 17. Two bay centrally stiffened plate.
pling rigidity have been considered i.e. 0:5 Ds ð1; 2Þ, 1:0 Ds ð1; 2Þ and 1:25 Ds ð1; 2Þ. Ds ð1; 2Þ has been computed using the dimensions of the stiffener as shown in Fig. 17. The central deflection has been found to increase with increasing the axial-bending rigidity of the stiffener. The period of response has also been found to increase with increase in the coupling rigidity of the stiffener. Fig. 19 shows the variation of transient deflection at point A, for varying toriosnal rigidity of the stiffener i.e 0:5 Ds ð4; 4Þ, 1:0 Ds ð4; 4Þ and 1:5 Ds ð4; 4Þ. Ds ð4; 4Þ has been calculated using the dimensions of the stiffener as shown in Fig. 17. The torsional rigidity of the stiffener has no significant influence on the response behaviour of the stiffened plate.
Fig. 18. Transient response of stiffened plate for varying axial-bending coupling rigidity of the stiffener.
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Fig. 19. Transient response of stiffened plate for varying torsional rigidity of the stiffener.
References [1] Dobyns AL. Analysis of simply-supported orthotropic plates subject to static and dynamic loads. AIAA 1981;19:642–50. [2] Craig TJ, Dawe DJ. Flexural vibration of symmetrically laminated composite rectangular plates including transverse shear effects. Int J Solids and Struct 1986;22:155–69. [3] Khdeir AA, Reddy JN. Dynamic response of antisymmetric angle-ply laminated plates subjected to arbitrary loading. J Sound and Vib 1988;126:437–45. [4] Reddy JN. On the solutions to forced motions of rectangular composite plates. J Appl Mech 1982;49:403–8. [5] Reddy JN. Dynamic (transient) analysis of layered anisotropic composite material plates. Int J Numer Meth Eng 1983;19:237– 55. [6] Librescu L, Nosier A. Response of laminated composite flat panels to sonic boom and explosive blast loadings. AIAA 1990;28: 345–52. [7] Kant T, Varaiya JH, Arora CP. A finite element transient analysis of composite and sandwich plates based on a refined theory and implicit time integration techniques. Comput Struct 1990;36(3): 401–20. [8] Taylor TW, Nayfeh AH, Wade JE. Dynamic response of thick anisotropic composite plates. Compos Eng 1994;4(5):475–86. [9] Khdeir AA. Transient response of refined cross-ply laminated plates for various boundary conditions. J Acoust Soc Am 1995; 97(3):1664. [10] Meimaris C, Day JD. Dynamic response of laminated anisotropic plates. Comput Struct 1995;55(2):269–78. [11] Chen J, Dawe DJ. Linear transient analysis of rectangular laminated plates by a finite strip superposition method. Compos Struct 1996;35(2):213–28. [12] Wang S, Chen J, Dawe DJ. Linear transient analysis of rectangular laminates using spline finite strips. Compos Struct 1998; 41(1):57–66.
[13] Lam KY, Zhang ZJ, Gong SW, Chan ES. The transient response of submerged orthotropic cylindrical shells exposed to underwater shock. Compos Struct 1999;43(3):179–93. [14] Wang YY, Lam KY, Liu GR. The effect of rotary inertia on the dynamic response of laminated composite plate. Compos Struct 2000;48(4):265–73. [15] Gong SW, Lam KY. Transient response of stiffened composite sub-mersible hull subjected to underwater explosive shock. Compos Struct 1998;41(1):27–37. [16] Allman DJ. A compatible triangular element including vertex rotations for plane elasticity analysis. Comput Struct 1984;19 (1/2):1–8. [17] Katili I. A new Discrete Kirchhoff–Mindlin element based on Mindlin–Reissner plate theory and assumed shear strain fields––Part I: An extended DKT element for thick-plate bending analysis. Int J Numer Meth Eng 1993;36(11):1859–83. [18] Daniel Isaac M, Ishai Ori. Engineering mechanics of composite materials. New York: Oxford University Press; 1994. [19] Whitney JM. Shear correction factors for orthotropic laminates under static load. J Appl Mech, Trans Am Soc Mech Eng 1973;40(1):302–4. [20] Satish Kumar YV, Mukhopadhyay M. A new finite element for buckling analysis of laminated stiffened plates. Compos Struct 1999;46(4):321–31. [21] Zienkiewicz OC. The finite element method. 3rd ed. New Delhi: Tata McGraw-Hill; 1979. [22] Chattopadhyay B. Linear and non-linear static and dynamic analysis of composite stiffened plates by the finite element method. PhD thesis, Indian Institute of Technology, Kharagpur, India, 1993. [23] Jiang J, Olson MD. Nonlinear dynamic analysis of blast loaded cylindrical shell structures. Comput Struct 1991;41(1):41–52. [24] Sinha G, Mukhopadhyay M. Static, freee and forced vibration analysis of arbitrary non-uniform shells with tapered stiffeners. Comput Struct 1997;62(5):919–33.