- Email: [email protected]
Vibration analysis of anisotropic plates with eccentric stiffeners
ks@a
Cm/mvr
Pergamon
0045-7949(94)00593-l
& Smt
VIBRATION ANALYSIS OF ANISOTROPIC WITH ECCENTRIC STIFFENERS
t
PLATES
Dong-Min Lee and In Lee Department
of Aerospace Engineering, Korea Advanced Institute of Science and Technology, 373-l Kusong-dong, Yusong-gu, Taejon, Korea (Receirred IO April 1994)
Abstract-The analysis of vibration characteristics of anisotropic plates with eccentric stiffeners has been performed using the finite element method based on the shear deformable plate theory. The stiffeners are modeled as a beam element based on Timoshenko beam theory. The present analysis presents the effects of fiber orientation, dimension and location of the stiffener on the vibration characteristics of anisotropic stiffened plates. The stiffened plates are composed of graphite
the result of the present beam model for the stiffener made of composite material is compared with that of the shell model. This result shows that the present model for the stiffened plate gives quite accurate results. The size, fibre orientation angle and location mode shapes of the stiffened anisotropic plate.
of the stiffener
affect the natural
frequencies
and
However, the papers on the vibration analysis of the stiffened anisotropic plate have rarely been published. Attaf and Hollaway [6] studied vibrational analyses of eccentrically stiffened and unstiffened GRP composite plates subjected to in-plane forces. Thus the present paper concentrates on the vibration analyses of eccentrically stiffened rectangular composite plates with clamped boundary conditions. In this study, Timoshenko beam for stiffeners and shear deformable plate for skin plates are used for stiffened anisotropic plate modeling.
INTRODUCTION
Various composite plates are being used in structural applications in the aerospace, automotive and shipbuilding industries. Most of composite materials have good properties-the high stiffness-weight ratio, high strength-weight ratio and high damping etc. If, in addition, exposed to a dynamic situation, these materials offer better dynamic characteristics than any other conventional materials. The weight saving is an important consideration for high performance applications. Also, for the dynamic problem of plates, it is often necessary to minimize the maximum deflections of plates without introducing any considerable weight penalty. This can be achieved by adding stiffeners to the plates. A number of papers on isotropic stiffened plates have been proposed. Olson and Hazel1 [I] have presented results from a theoretical and experimental comparison study on the vibration characteristics of all clamped and eccentrically stiffened isotropic plates. They used a triangular finite element in the calculations. Deb and Booton [2] used an eight-noded finite element model for static analysis of stiffened plates. Mukherjee and Mukhopadhyay [3] extended this model for vibration analysis. Recently, Palani et al. [4] have published performance studies of the two models for static and vibration analysis of stiffened plates with various boundary conditions using four mass lumping schemes. Liu and Chen [.5] investigated the free vibrations of a skew cantilevered plate with a stiffener by a finite element method.
FINITE ELEMENT FORMULATION I. Skin plate modeling
The skin plate is modeled using plate element with five degrees of freedom (DOF) at a node. The displacement field in a shear deformable plate is assumed to be as follows:
U(x, I’, z, t)=vo(x,y,t)+zll/(x,y,t) U’(S, J, 2, 1) = %(X, ,v, t)
(1)
where U. c, NJare displacements in the x-, y-, z-directions, respectively; uO, L’~,and W, are the associated midplane displacements; 4 and $ are rotations in the s and JZ planes; t is the time. The coordinates of the stiffened plate are given in Fig. 1. The linear strain and displacement relations can be written as 99
Dong-Min Lee and In Lee where N, are the shape functions. Using the first variational form (6L = 0) with respect to the Lagrangian (L = T - U). the equations of the finite element can be formulated into algebraic forms. Thus the element mass and stiffness matrices for the anisotropic plates can be easily obtained from the variational principle.
-X
2. St#ener
modeling
The present anistropic stiffener is modeled using the Timoshenko beam element. Here the stiffener, along the r-axis only, is formulated for convenience. A beam finite element for a laminated anisotropic stiffener is formulated using the shear deformable theory [7, 81. To maintain the compatibility between the plate and the stiffener (Fig. l), the displacement field of the stiffener is assumed to be as follows:
stiffener
Fig. 1. Geometry of anisotropic plate with stiffener.
L’h =
(2) Also, the constitutive be written as
equations
v. + ;I//
for the lamina can
(3)
where u,, , [lb, wb are displacements of the stiffener in the x-, J-. :-directions, respectively; 4 and I/I are rotations defined in the previous section; uO, uO, and H‘”are displacements at a neutral axis of the stiffener. Considering eccentricity of the stiffener. the displacements at the neutral axis of the stiffener can be written as
u,=u-ecp where the Q,, denotes the transformed reduced stiffness. The strain and kinetic energies can be written as follows:
I’,, =
11’” =
(4)
:
I, are defined
(9)
Ii’
&#I
as’
c,
=c.=o
as
7 “& p(l,?)dz.
(1,,13)=
e*
where u, L’and ~4’are the displacements of the plate given in eqn (1). The linear strain and displacement relations in Timoshenko beam can be written as
c,2!!+;_ as where the inertias
L’ -
(6)
(‘0)
Substituting eqns (l)-(3) into eqn (4), we can obtain three components of strain energy; extensional, bending and transverse shear energies. The generalized displacements (u, u, ~1,4, +!I) are interpolated by the shape functions of a nine-node quadrilateral element.
When the width and the height of the stiffener are much smaller than the length of the stiffener, we can assume fl, I z 0 and 0;; x 0 in the stiffener. Taking the coordinates of the composite stiffener into account (Fig. I), the constitutive equations for the stiffener can be obtained from eqn (3).
I
,Z
Anisotropic plates with eccentric stiffeners
KM= 4
$01
s
w’&$ dx - eE,
w’b4’dx
‘I
modeling
iegti
q$Sd, dx -eEi
K44= E3
s+
._-___*_m.,s______-_
T
._r_“.lf_s,l______-_
+(F, +e2E,)
101.5
sJ
6’dqS’d.x
1
Fig. 2. Stiffened isotropic plate with aft clamped boundary conditions.
where the stiffness matrix, [Cl, is called modified re$ce; ;;tffness PI; Cn = Pn - Q:,ii?,. C,, = Qta I2 26 121and C,, = &, - $I&/&. The mistanee to the faterat motion of the stiffener is usually negligibly smatl compared to that of the plate, and the strain energy due to the lateral motion of the stiffener is ignored. However, the kinetic energy due to the lateral motion of the stiffener cannot be ignored. Thus, the variational form of strain and kinetic energies of the stiffener can be written as fohows:
where the upper prime denotes derivative with respect to x, and E, =
C,,dA,
E2=
sA
S,=
C,,dA sA
C,b(-y)dA,
S,=
C,,z*dA
Fz =
sA
fA
C,,Y’ dA 04)
Substituting eqns (8X-(tt) into eqn (12), we can obtain the fotfowing equations for the stiffness matrix from the variational principle as in the previous section. For the rectangular cross-section of the stiffener, the effective torsional moment of inertia .I, is used like the isotropic case.
where the value of the coefficient /? is determined according to the cross-section shape and the value of an isotropic case are employed for @. Here, h is the height of the stiffener. Similarly, we can obtain the equations for the mass matrix as follows:
where p is the material density; IV is the second moment of area; J is the polar moment of area. The displacement field fu, t’, w, 4%JI) of the stiffener is interpolated by the shape function of the threenode quadratic beam element,
Dong-Min
102 Table
1. Comparison
of natural frequencies square plate
Lee and In Lee
for all clamped
Frequencies
Olson and Hazel1 [l] __~~
Number of modes
Experiment
Analysis
I
718.1 751.4 997.4 1007.1 1419.8 1424.3 1631.5 1853.9 2022.8 2025.0
2 3 4 5 6 7 8 9 10
689 725 961 986 I376 1413 1512 1770 1995 2069
Present 711.1 143.4 975.2 993.4 1414.5 1423.0 1552.9 1886.6 2024.6 2064. I
Substituting eqn (16) into eqns (13) and (15), the element stiffness matrix [Kelb and mass matrix [IV.& for the stiffener can be obtained. In order to evaluate the natural frequencies of stiffened composite plate, we have to solve the eigenvalue problem. (K-/IM)u=O
(17)
where K = [K], + [Q, and M = [Ml, + [M&. Here, subscript p is for the plate and subscript b is for the stiffener. The global mass and stiffness matrices are assembled using the skyline matrix scheme [9]. The subspace iteration method [lo], which is widely used for large-scale finite element calculations, is used to solve eqn (17). This method is an iterative strategy to obtain the lower modes of the generalized eigenvalue problem accurately.
RESULTS
Ver$cation
AND
DISCUSSION
of beam model for composite stifSener
To verify the accuracy of the present formulations for the eccentrically stiffened plate, an isotropic case is selected as an example for comparison. The stiffened plate with all clamped boundary conditions (Fig. 2) is investigated as the example of the isotropic case, which was studied by Olson and Hazel1 [ 11. They have carried out an experiment on the stiffened plate of annealed 65ST6 aluminum alloy using real time holographic technique and presented numerical calculations by the finite element method. The present results and Olson and Hazell’s results are listed in Table I. The present results were obtained using 7 x 12 elements, considering the symmetry condition along the central line perpendicular to the stiffener.
Table 2. Material Material Graphite+poxy (AS1/3501-6)
~
The influence of the stiffener size on the natural frequencies is examined. The natural frequencies for ten models with different size of composite stiffeners are given in Table 4. The dimensions of stiffened plates are shown in Fig. 3 and are identical with the previous model. The boundary conditions are all
properties G,
128 [GPa]
4.48 [GPal
1.53 [GPa]
of the present
stiffener.
Eflkct of the dimension qf stiflener
G,z(G,,)
sequence
plate with cross-ply
The present results are in agreement with the results of Olson and Hazel]. To verify the present beam model for a composite stiffener using the modified reduced stiffness given in eqn (1 I), we compared the results of the present beam model with those of a shell model for the identical stiffened anisotropic plate. The alternative and reasonable model for the stiffener is a shell model (or the plate model with six degrees of freedom per node). In that model, the skin plate and stiffener are modeled using the same shell elements. However the disadvantage of the shell model is that the total problem size is much larger than that of the beam model for the stiffener. Generally, the symmetric vibration mode in the isotropic plates does not always become a symmetric mode in the anisotropic plates. Thus the full finite element modeling should be used for stiffened anisotropic plates. The numerical results are obtained for the stiffened plate made of a graphiteeepoxy composite materials (AS1/3501-6). The properties of that material are listed in Table 2. The stacking sequence of the skin plate is [O/ +45/90], A stiffened anisotropic plate as shown in Fig. 3 is chosen as an example for the composite plate with a cross-ply composite stiffener. The dimension of the stiffener is t, x h = 3.64 x 10.5 (mm’). The mesh of the skin plate is 10 x 6 for both models. The results are listed in Table 3. The beam model for the stiffener has 1365 DOF and the shell model has 1596 DOF. However the beam model gives quite accurate results for the natural frequencies of the stiffened composite plate. Therefore the beam model for the stiffener is used in this analysis.
El
Stacking
500 (mm)
Fig. 3. Anisotropic
L‘l? 0.25
skin plate model: [O/ *45/90],
P 1500 [kg m’l
Ply thickness 0.13
[mm1
Anisotropic Table 3. Comparison Mode no.
of natural
frequencies
Beam model
1
Total DOF
stiffeners
103
for two models Shell model
213.8 229.4 270.2 313.8 354.0
2 3 4 5
plates with eccentric
QJS
e PlY
213.6 222.2 270.2 312.4 354.6
1365
2 34 1
I596
-500
(mm)-
Fig. 5. Anisotropic
(CCCC). The stacking sequence of the stiffener is [0,,/90,,],. The stiffeners are the cross-ply laminated beam which has the same lamination ratio of 90-O” ply. The lowest five frequencies for each stiffened plate are given in Table 4, and the first three mode shapes of specific models are shown in Fig. 4. Referring to Table 4 and Fig. 4, it is concluded that the increase of the cross-section dimension of stiffeners induces the change of mode shapes and the corresponding natural frequencies. As the cross-section of the stiffener becomes larger, every frequency increases and especially, the mode shape change of the first two modes is significant. The symmetric
plate with angle ply stiffener.
clamped
Table 4. Natural Model no.
1 2 3 4 5 6 7 8 9 10
fourth
frequencies
Size of stiffener i,, x h
0x0 1.04 I .56 2.08 2.60 3.12 3.64 4.16 4.68 5.20
x x x x x x x x x
3.0 4.5 6.0 7.5 9.0 10.5 12.0 13.5 15.0
model Fig. 4. Typical
mode about the stiffener (the first mode in model 5) becomes the antisymmetric mode (the first mode in model 7) as the size of the stiffener increases. EfSect of the fiber
with different
Frequencies
stiffeners
of five modes (Hz)
I
2
3
4
85. I 91.2 108.3 136.9 170.6 203.0 213.8 217.5 222.2 227.8
134.0 162.3 207.3 208.2 209.2 211.0 229.4 248.6 261.6 270.2
207.4 207.6 214.9 254.2 257.7 263.1 270.2 278.4 286.8 294.5
216.1 250.7 252.3 264.3 292.9 306.7 313.8 317.8 320.2 321.8
fifth mode shapes
of st@ener
The [O, / + Q,], composite stiffener is used to investigate the effect of the fiber orientation of the stiffener on the natural frequency. The boundary conditions are two-side clamped and two-side free conditions (CFCF) as shown in Fig. 5. Figure 6 shows the effect of the fiber orientation, Q of the stiffener upon the
of 10 models
~_
orientation
model of anisotropic
stiffened
plates
__~ 5 252.5 274.8 329.2 332.6 338.4 345.8 354.0 361.7 368.4 373.7
Dong-Min Lee and In Lee
104
Table 5. Natural
frequencies
Frequencies
Stiffener location (d/b) 0.2 0.4 0.6 0.8
2 3 f? 6
I .o
of doubly
stiffened
plates
(Hz)
of five modes
1
2
3
4
5
138.2 170.3 189.8 144.2 110.7
141.2 200.7 301.7 266.2 227.9
252.3 294.8 326.4 375.2 269.1
259.2 318.1 382.3 438.5 372.7
424.1 471.4 434.6 499.0 409.2
? 70
40
0
30 Angle
60
90
(i-e degree)
Mode
Shape
Fig. 6. Effects of fiber orientation (0) of stiffener on natural frequencies of anisotropic stiffened plates.
frequencies. The ratio of thickness to height for the stiffener is 2.34-30. The first and third modes are antisymmetric modes about the stiffener and the second and fourth modes are symmetric modes. The frequencies of antisymmetric modes are dependent on the fiber orientation. However. the frequencies of symmetric modes nearly do not change. As shown in Fig. 6, the stiffener with 30 ply angle has the maximum values of the frequencies.
The dimensions of the doubly stiffened plate are presented in Fig. 7. The stacking sequences of the skin plate and stiffener are [OJk452/902], and [0,/90,],, respectively. Figure 8 shows the effect of the location of stiffeners on the frequencies. As the location of stiffeners becomes farther from the center line, the frequency of the lowest four modes goes up and down. The frequencies are severely dependent on the location of the stiffeners. Thus. in order to restrain the frequency of the specific mode, the appropriate location of the stiffeners can be determined. As shown in Fig. 8, the proper location to increase the fundamental frequency is a position about 60% from the center line. In the case of the square plate with double stiffeners, the optimal location to increase the fundamental frequency was calculated to be a position nearly 60-65% from the center line. The optimal location is dependent on the dimension of skin plate, boundary conditions and the stacking sequence.
100
/ 0.2
I
04
d/b
0.6
(location
Fig. 8. Effects of location
0.8
of
1 .o
Stiffener)
of stiffener on natural
frequencies.
CONCLUSION
In the present study, a simple model for the vibration analysis of the stiffened anisotropic plate has been proposed. The shear deformable plate and beam element give very accurate results. The effect of the size of the stiffener on the natural frequencies has been investigated. The change of stiffener size influences the natural frequencies and mode shapes of stiffened plates. The fiber orientation angle of [O/ + 01, stiffener affects the natural frequencies of antisymmetric mode about the stiffener. The location of the stiffener influences the natural frequency. In the case of double stiffeners, the appropriate location of stiffener to increase the fundamental frequency is a position about 60% from the center line in the stiffened plate with CFCF boundary conditions. Therefore, the size, fiber orientation angle and location of the stiffener in stiffened plates should be selected properly to control the specific frequency and mode shape.
REFERENCES
M. D. Olson some integral
and C. R. Hazell. Vibration studies on rib-stiffened plates. J. Sound Vibr. 50(I),
43 61 (1977).
L
500
Fig. 7. Anisotropic
(mm) 2
plate with double
stiffeners.
A. Deb and M. Booton, Finite element model for stiffened plates under transverse loading. Compur. St,_uct. 28(3), 361-372 (1988). A. Mukherjee and M. Mukhopadhyay, Finite element free vibration of eccentrically stiffened plates. Comput. Struct. 30(6), 1303- 1317 (1988). G. S. Palani et cd.. An efficient finite element model for static and vibration analysis of eccentrically stiffened plates-shells. Comput. Struct. 43(4). 651-661 (1992).
Anisotropic
plates with eccentric
5. W. H. Liu and W. C. Chen, Vibration analysis of skew cantilever plates with stiffeners. J. Sound Vihr. 159(l), 1-I I (1992). 6. B. Attaf and L. Hollaway, Vibrational analyses of stiffened and unstiffened composite plates subjected to in-mane loads. Com~~osires 21(2). 117-126 (1990). 7. A.’ T. Chen and T. Y. Yang. Static and dynamic formulaton of a symmetrically laminated beam finite element for a microcomputer. J. Compos. Mater. 19, 4599475 (1985).
stiffeners
105
8. E. C. Smith and I. Chopra, Formulation and evaluation of an analytical model for composite box-beams. AIAA/ASME/ASCE/AHS/ACE 31.71 SDM Cot~f.. pp. 759-782 (1990). 9. G. Dhatt and G. Touzot, The Finite Element Method Displayed. Wiley, New York (1984). IO. K. J. Bathe, Finite Element Procedures in Engineering Analysis. Prentice-Hall, Englewood Cliffs, NJ (1982). I I. J. N. Reddy, Energy and Variational Methods in Applied Mechanics. Wiley, New York (1984).
Cm/mvr
Pergamon
0045-7949(94)00593-l
& Smt
VIBRATION ANALYSIS OF ANISOTROPIC WITH ECCENTRIC STIFFENERS
t
PLATES
Dong-Min Lee and In Lee Department
of Aerospace Engineering, Korea Advanced Institute of Science and Technology, 373-l Kusong-dong, Yusong-gu, Taejon, Korea (Receirred IO April 1994)
Abstract-The analysis of vibration characteristics of anisotropic plates with eccentric stiffeners has been performed using the finite element method based on the shear deformable plate theory. The stiffeners are modeled as a beam element based on Timoshenko beam theory. The present analysis presents the effects of fiber orientation, dimension and location of the stiffener on the vibration characteristics of anisotropic stiffened plates. The stiffened plates are composed of graphite
the result of the present beam model for the stiffener made of composite material is compared with that of the shell model. This result shows that the present model for the stiffened plate gives quite accurate results. The size, fibre orientation angle and location mode shapes of the stiffened anisotropic plate.
of the stiffener
affect the natural
frequencies
and
However, the papers on the vibration analysis of the stiffened anisotropic plate have rarely been published. Attaf and Hollaway [6] studied vibrational analyses of eccentrically stiffened and unstiffened GRP composite plates subjected to in-plane forces. Thus the present paper concentrates on the vibration analyses of eccentrically stiffened rectangular composite plates with clamped boundary conditions. In this study, Timoshenko beam for stiffeners and shear deformable plate for skin plates are used for stiffened anisotropic plate modeling.
INTRODUCTION
Various composite plates are being used in structural applications in the aerospace, automotive and shipbuilding industries. Most of composite materials have good properties-the high stiffness-weight ratio, high strength-weight ratio and high damping etc. If, in addition, exposed to a dynamic situation, these materials offer better dynamic characteristics than any other conventional materials. The weight saving is an important consideration for high performance applications. Also, for the dynamic problem of plates, it is often necessary to minimize the maximum deflections of plates without introducing any considerable weight penalty. This can be achieved by adding stiffeners to the plates. A number of papers on isotropic stiffened plates have been proposed. Olson and Hazel1 [I] have presented results from a theoretical and experimental comparison study on the vibration characteristics of all clamped and eccentrically stiffened isotropic plates. They used a triangular finite element in the calculations. Deb and Booton [2] used an eight-noded finite element model for static analysis of stiffened plates. Mukherjee and Mukhopadhyay [3] extended this model for vibration analysis. Recently, Palani et al. [4] have published performance studies of the two models for static and vibration analysis of stiffened plates with various boundary conditions using four mass lumping schemes. Liu and Chen [.5] investigated the free vibrations of a skew cantilevered plate with a stiffener by a finite element method.
FINITE ELEMENT FORMULATION I. Skin plate modeling
The skin plate is modeled using plate element with five degrees of freedom (DOF) at a node. The displacement field in a shear deformable plate is assumed to be as follows:
U(x, I’, z, t)=vo(x,y,t)+zll/(x,y,t) U’(S, J, 2, 1) = %(X, ,v, t)
(1)
where U. c, NJare displacements in the x-, y-, z-directions, respectively; uO, L’~,and W, are the associated midplane displacements; 4 and $ are rotations in the s and JZ planes; t is the time. The coordinates of the stiffened plate are given in Fig. 1. The linear strain and displacement relations can be written as 99
Dong-Min Lee and In Lee where N, are the shape functions. Using the first variational form (6L = 0) with respect to the Lagrangian (L = T - U). the equations of the finite element can be formulated into algebraic forms. Thus the element mass and stiffness matrices for the anisotropic plates can be easily obtained from the variational principle.
-X
2. St#ener
modeling
The present anistropic stiffener is modeled using the Timoshenko beam element. Here the stiffener, along the r-axis only, is formulated for convenience. A beam finite element for a laminated anisotropic stiffener is formulated using the shear deformable theory [7, 81. To maintain the compatibility between the plate and the stiffener (Fig. l), the displacement field of the stiffener is assumed to be as follows:
stiffener
Fig. 1. Geometry of anisotropic plate with stiffener.
L’h =
(2) Also, the constitutive be written as
equations
v. + ;I//
for the lamina can
(3)
where u,, , [lb, wb are displacements of the stiffener in the x-, J-. :-directions, respectively; 4 and I/I are rotations defined in the previous section; uO, uO, and H‘”are displacements at a neutral axis of the stiffener. Considering eccentricity of the stiffener. the displacements at the neutral axis of the stiffener can be written as
u,=u-ecp where the Q,, denotes the transformed reduced stiffness. The strain and kinetic energies can be written as follows:
I’,, =
11’” =
(4)
:
I, are defined
(9)
Ii’
&#I
as’
c,
=c.=o
as
7 “& p(l,?)dz.
(1,,13)=
e*
where u, L’and ~4’are the displacements of the plate given in eqn (1). The linear strain and displacement relations in Timoshenko beam can be written as
c,2!!+;_ as where the inertias
L’ -
(6)
(‘0)
Substituting eqns (l)-(3) into eqn (4), we can obtain three components of strain energy; extensional, bending and transverse shear energies. The generalized displacements (u, u, ~1,4, +!I) are interpolated by the shape functions of a nine-node quadrilateral element.
When the width and the height of the stiffener are much smaller than the length of the stiffener, we can assume fl, I z 0 and 0;; x 0 in the stiffener. Taking the coordinates of the composite stiffener into account (Fig. I), the constitutive equations for the stiffener can be obtained from eqn (3).
I
,Z
Anisotropic plates with eccentric stiffeners
KM= 4
$01
s
w’&$ dx - eE,
w’b4’dx
‘I
modeling
iegti
q$Sd, dx -eEi
K44= E3
s+
._-___*_m.,s______-_
T
._r_“.lf_s,l______-_
+(F, +e2E,)
101.5
sJ
6’dqS’d.x
1
Fig. 2. Stiffened isotropic plate with aft clamped boundary conditions.
where the stiffness matrix, [Cl, is called modified re$ce; ;;tffness PI; Cn = Pn - Q:,ii?,. C,, = Qta I2 26 121and C,, = &, - $I&/&. The mistanee to the faterat motion of the stiffener is usually negligibly smatl compared to that of the plate, and the strain energy due to the lateral motion of the stiffener is ignored. However, the kinetic energy due to the lateral motion of the stiffener cannot be ignored. Thus, the variational form of strain and kinetic energies of the stiffener can be written as fohows:
where the upper prime denotes derivative with respect to x, and E, =
C,,dA,
E2=
sA
S,=
C,,dA sA
C,b(-y)dA,
S,=
C,,z*dA
Fz =
sA
fA
C,,Y’ dA 04)
Substituting eqns (8X-(tt) into eqn (12), we can obtain the fotfowing equations for the stiffness matrix from the variational principle as in the previous section. For the rectangular cross-section of the stiffener, the effective torsional moment of inertia .I, is used like the isotropic case.
where the value of the coefficient /? is determined according to the cross-section shape and the value of an isotropic case are employed for @. Here, h is the height of the stiffener. Similarly, we can obtain the equations for the mass matrix as follows:
where p is the material density; IV is the second moment of area; J is the polar moment of area. The displacement field fu, t’, w, 4%JI) of the stiffener is interpolated by the shape function of the threenode quadratic beam element,
Dong-Min
102 Table
1. Comparison
of natural frequencies square plate
Lee and In Lee
for all clamped
Frequencies
Olson and Hazel1 [l] __~~
Number of modes
Experiment
Analysis
I
718.1 751.4 997.4 1007.1 1419.8 1424.3 1631.5 1853.9 2022.8 2025.0
2 3 4 5 6 7 8 9 10
689 725 961 986 I376 1413 1512 1770 1995 2069
Present 711.1 143.4 975.2 993.4 1414.5 1423.0 1552.9 1886.6 2024.6 2064. I
Substituting eqn (16) into eqns (13) and (15), the element stiffness matrix [Kelb and mass matrix [IV.& for the stiffener can be obtained. In order to evaluate the natural frequencies of stiffened composite plate, we have to solve the eigenvalue problem. (K-/IM)u=O
(17)
where K = [K], + [Q, and M = [Ml, + [M&. Here, subscript p is for the plate and subscript b is for the stiffener. The global mass and stiffness matrices are assembled using the skyline matrix scheme [9]. The subspace iteration method [lo], which is widely used for large-scale finite element calculations, is used to solve eqn (17). This method is an iterative strategy to obtain the lower modes of the generalized eigenvalue problem accurately.
RESULTS
Ver$cation
AND
DISCUSSION
of beam model for composite stifSener
To verify the accuracy of the present formulations for the eccentrically stiffened plate, an isotropic case is selected as an example for comparison. The stiffened plate with all clamped boundary conditions (Fig. 2) is investigated as the example of the isotropic case, which was studied by Olson and Hazel1 [ 11. They have carried out an experiment on the stiffened plate of annealed 65ST6 aluminum alloy using real time holographic technique and presented numerical calculations by the finite element method. The present results and Olson and Hazell’s results are listed in Table I. The present results were obtained using 7 x 12 elements, considering the symmetry condition along the central line perpendicular to the stiffener.
Table 2. Material Material Graphite+poxy (AS1/3501-6)
~
The influence of the stiffener size on the natural frequencies is examined. The natural frequencies for ten models with different size of composite stiffeners are given in Table 4. The dimensions of stiffened plates are shown in Fig. 3 and are identical with the previous model. The boundary conditions are all
properties G,
128 [GPa]
4.48 [GPal
1.53 [GPa]
of the present
stiffener.
Eflkct of the dimension qf stiflener
G,z(G,,)
sequence
plate with cross-ply
The present results are in agreement with the results of Olson and Hazel]. To verify the present beam model for a composite stiffener using the modified reduced stiffness given in eqn (1 I), we compared the results of the present beam model with those of a shell model for the identical stiffened anisotropic plate. The alternative and reasonable model for the stiffener is a shell model (or the plate model with six degrees of freedom per node). In that model, the skin plate and stiffener are modeled using the same shell elements. However the disadvantage of the shell model is that the total problem size is much larger than that of the beam model for the stiffener. Generally, the symmetric vibration mode in the isotropic plates does not always become a symmetric mode in the anisotropic plates. Thus the full finite element modeling should be used for stiffened anisotropic plates. The numerical results are obtained for the stiffened plate made of a graphiteeepoxy composite materials (AS1/3501-6). The properties of that material are listed in Table 2. The stacking sequence of the skin plate is [O/ +45/90], A stiffened anisotropic plate as shown in Fig. 3 is chosen as an example for the composite plate with a cross-ply composite stiffener. The dimension of the stiffener is t, x h = 3.64 x 10.5 (mm’). The mesh of the skin plate is 10 x 6 for both models. The results are listed in Table 3. The beam model for the stiffener has 1365 DOF and the shell model has 1596 DOF. However the beam model gives quite accurate results for the natural frequencies of the stiffened composite plate. Therefore the beam model for the stiffener is used in this analysis.
El
Stacking
500 (mm)
Fig. 3. Anisotropic
L‘l? 0.25
skin plate model: [O/ *45/90],
P 1500 [kg m’l
Ply thickness 0.13
[mm1
Anisotropic Table 3. Comparison Mode no.
of natural
frequencies
Beam model
1
Total DOF
stiffeners
103
for two models Shell model
213.8 229.4 270.2 313.8 354.0
2 3 4 5
plates with eccentric
QJS
e PlY
213.6 222.2 270.2 312.4 354.6
1365
2 34 1
I596
-500
(mm)-
Fig. 5. Anisotropic
(CCCC). The stacking sequence of the stiffener is [0,,/90,,],. The stiffeners are the cross-ply laminated beam which has the same lamination ratio of 90-O” ply. The lowest five frequencies for each stiffened plate are given in Table 4, and the first three mode shapes of specific models are shown in Fig. 4. Referring to Table 4 and Fig. 4, it is concluded that the increase of the cross-section dimension of stiffeners induces the change of mode shapes and the corresponding natural frequencies. As the cross-section of the stiffener becomes larger, every frequency increases and especially, the mode shape change of the first two modes is significant. The symmetric
plate with angle ply stiffener.
clamped
Table 4. Natural Model no.
1 2 3 4 5 6 7 8 9 10
fourth
frequencies
Size of stiffener i,, x h
0x0 1.04 I .56 2.08 2.60 3.12 3.64 4.16 4.68 5.20
x x x x x x x x x
3.0 4.5 6.0 7.5 9.0 10.5 12.0 13.5 15.0
model Fig. 4. Typical
mode about the stiffener (the first mode in model 5) becomes the antisymmetric mode (the first mode in model 7) as the size of the stiffener increases. EfSect of the fiber
with different
Frequencies
stiffeners
of five modes (Hz)
I
2
3
4
85. I 91.2 108.3 136.9 170.6 203.0 213.8 217.5 222.2 227.8
134.0 162.3 207.3 208.2 209.2 211.0 229.4 248.6 261.6 270.2
207.4 207.6 214.9 254.2 257.7 263.1 270.2 278.4 286.8 294.5
216.1 250.7 252.3 264.3 292.9 306.7 313.8 317.8 320.2 321.8
fifth mode shapes
of st@ener
The [O, / + Q,], composite stiffener is used to investigate the effect of the fiber orientation of the stiffener on the natural frequency. The boundary conditions are two-side clamped and two-side free conditions (CFCF) as shown in Fig. 5. Figure 6 shows the effect of the fiber orientation, Q of the stiffener upon the
of 10 models
~_
orientation
model of anisotropic
stiffened
plates
__~ 5 252.5 274.8 329.2 332.6 338.4 345.8 354.0 361.7 368.4 373.7
Dong-Min Lee and In Lee
104
Table 5. Natural
frequencies
Frequencies
Stiffener location (d/b) 0.2 0.4 0.6 0.8
2 3 f? 6
I .o
of doubly
stiffened
plates
(Hz)
of five modes
1
2
3
4
5
138.2 170.3 189.8 144.2 110.7
141.2 200.7 301.7 266.2 227.9
252.3 294.8 326.4 375.2 269.1
259.2 318.1 382.3 438.5 372.7
424.1 471.4 434.6 499.0 409.2
? 70
40
0
30 Angle
60
90
(i-e degree)
Mode
Shape
Fig. 6. Effects of fiber orientation (0) of stiffener on natural frequencies of anisotropic stiffened plates.
frequencies. The ratio of thickness to height for the stiffener is 2.34-30. The first and third modes are antisymmetric modes about the stiffener and the second and fourth modes are symmetric modes. The frequencies of antisymmetric modes are dependent on the fiber orientation. However. the frequencies of symmetric modes nearly do not change. As shown in Fig. 6, the stiffener with 30 ply angle has the maximum values of the frequencies.
The dimensions of the doubly stiffened plate are presented in Fig. 7. The stacking sequences of the skin plate and stiffener are [OJk452/902], and [0,/90,],, respectively. Figure 8 shows the effect of the location of stiffeners on the frequencies. As the location of stiffeners becomes farther from the center line, the frequency of the lowest four modes goes up and down. The frequencies are severely dependent on the location of the stiffeners. Thus. in order to restrain the frequency of the specific mode, the appropriate location of the stiffeners can be determined. As shown in Fig. 8, the proper location to increase the fundamental frequency is a position about 60% from the center line. In the case of the square plate with double stiffeners, the optimal location to increase the fundamental frequency was calculated to be a position nearly 60-65% from the center line. The optimal location is dependent on the dimension of skin plate, boundary conditions and the stacking sequence.
100
/ 0.2
I
04
d/b
0.6
(location
Fig. 8. Effects of location
0.8
of
1 .o
Stiffener)
of stiffener on natural
frequencies.
CONCLUSION
In the present study, a simple model for the vibration analysis of the stiffened anisotropic plate has been proposed. The shear deformable plate and beam element give very accurate results. The effect of the size of the stiffener on the natural frequencies has been investigated. The change of stiffener size influences the natural frequencies and mode shapes of stiffened plates. The fiber orientation angle of [O/ + 01, stiffener affects the natural frequencies of antisymmetric mode about the stiffener. The location of the stiffener influences the natural frequency. In the case of double stiffeners, the appropriate location of stiffener to increase the fundamental frequency is a position about 60% from the center line in the stiffened plate with CFCF boundary conditions. Therefore, the size, fiber orientation angle and location of the stiffener in stiffened plates should be selected properly to control the specific frequency and mode shape.
REFERENCES
M. D. Olson some integral
and C. R. Hazell. Vibration studies on rib-stiffened plates. J. Sound Vibr. 50(I),
43 61 (1977).
L
500
Fig. 7. Anisotropic
(mm) 2
plate with double
stiffeners.
A. Deb and M. Booton, Finite element model for stiffened plates under transverse loading. Compur. St,_uct. 28(3), 361-372 (1988). A. Mukherjee and M. Mukhopadhyay, Finite element free vibration of eccentrically stiffened plates. Comput. Struct. 30(6), 1303- 1317 (1988). G. S. Palani et cd.. An efficient finite element model for static and vibration analysis of eccentrically stiffened plates-shells. Comput. Struct. 43(4). 651-661 (1992).
Anisotropic
plates with eccentric
5. W. H. Liu and W. C. Chen, Vibration analysis of skew cantilever plates with stiffeners. J. Sound Vihr. 159(l), 1-I I (1992). 6. B. Attaf and L. Hollaway, Vibrational analyses of stiffened and unstiffened composite plates subjected to in-mane loads. Com~~osires 21(2). 117-126 (1990). 7. A.’ T. Chen and T. Y. Yang. Static and dynamic formulaton of a symmetrically laminated beam finite element for a microcomputer. J. Compos. Mater. 19, 4599475 (1985).
stiffeners
105
8. E. C. Smith and I. Chopra, Formulation and evaluation of an analytical model for composite box-beams. AIAA/ASME/ASCE/AHS/ACE 31.71 SDM Cot~f.. pp. 759-782 (1990). 9. G. Dhatt and G. Touzot, The Finite Element Method Displayed. Wiley, New York (1984). IO. K. J. Bathe, Finite Element Procedures in Engineering Analysis. Prentice-Hall, Englewood Cliffs, NJ (1982). I I. J. N. Reddy, Energy and Variational Methods in Applied Mechanics. Wiley, New York (1984).