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Dynamic and static analysis of skew sandwich plates
Journal of Sound and Vibration (1985) 99(3), 393-401
DYNAMIC
AND
STATIC
SANDWICH S. S.
Department
F. NC
of Civil Engineering,
(Received
AND
ANALYSIS
OF SKEW
PLATES D. K. Y. LAM
University of Ottawa, Ottawa, Ontario, Canada
6 July 1983, and in revised form 3 July 1984)
A finite element displacement model is presented for the dynamic and static analysis of clamped and simply supported skew sandwich plates. The geometric admissibility conditions of the principle of minimum total potential energy are satisfied by representing the assumed displacement pattern by a polynomial function. For static analysis, the sandwich plate is assumed to be uniformly loaded although point loads can be handled with very little modification in the computer program. For the dynamic analysis, results are presented for free flexural vibrations of skew sandwich plates with different plate aspect ratios, angles of skew and core rigidities.
1.
INTRODUCTION
Static and dynamic analysis of rectangular sandwich plates have been performed by a number of research workers [ 1,2] in the past using the variational approach and obtaining numerical results by a series types solution. A series solution was also presented by Kennedy [3] for static deformations of parallelogrammic clamped sandwich panels. More recently, the finite element method has been applied to single core or multilayered sandwich plates by several research workers [4-lo]. However scant data is available on the dynamic analysis of skew sandwich plates. Although in general the series solution method can yield accurate results, successful application is limited to sandwich structures with very simple boundary conditions. For solutions involving more general types of boundary conditions, it has been necessary to resort to numerical methods such as finite difference or finite elements. In the present paper, the finite element method is used for the static and dynamic analyses of skew sandwich plates incorporating a parallelogrammic element having five degrees of freedom per node. The parallelogrammic sandwich plate under consideration consists of a relatively thick homogeneous orthotropic core sandwiched between two thin homogeneous isotropic faces of equal thickness. The other assumptions used in this investigation, which are standard for sandwich plates, can be obtained elsewhere [ 111. The stiffness matrix is represented in terms of the element geometry, the stiffness of the faces (membrane, bending) and the transverse shear stiffness of the orthotropic core. The consistent mass matrix includes both rotary inertia and translatory inertia.
2. DERIVATION
OF THE STIFFNESS
MATRIX
Because of the unique behaviour of sandwich plates, it is convenient to build up each individual sandwich plate element by using a set of sub-elements, two skin layers and one core layer (see Figure 1). The stiffness matrices of the skin layers and the core layer are developed separately and then combined to form the total elemental stiffness matrix 393 0022-460X/85/070393 + 09 %03.OQ/O @ 1985 Academic Press Inc (London) Limited
394
S. S. F. NC
AND
D. K. Y. LAM
Figure 1. A typical skew plate.
for the sandwich plate element. In dealing with skew plates, a parallelogrammic-shaped element has been found to be very useful, as it has one additional node over a triangular element which permits more accurate representation of the displacement function and the varying stress field. Use of these elements is obviously more restricted than that of triangular elements in other types of problems.
Figure 2. A typical parallelogrammic element.
A typical parallelogrammic sandwich element is shown in Figure 2 and the displacement vector for this element consists of the in-plane displacements u, u and the lateral displacement w. Although a parallelogrammic element with three degrees of freedom per node has been found to yield sufficiently accurate results for static analysis, experience with this element on dynamic analysis has demonstrated the need for a more refined element. In the present study, a parallelogrammic element with five degrees of freedom per node (u, v, w, WC,wq) was used. The assumed displacement function after introducing the dimensionless
skew co-ordinates
(3 = v/b,
c= [/a),
is -
v =
u=a,+a2~+a,ij+a&,
a5 +
ah<+
a,ij
+
as&j,
~=a,+a,,~+a,,7i+a,,F’+a,,577+a,,iS~+a,,~~+a,6~4 +
aIT&?+
a18fj3+
a19i?ii
+
a205;;3
(1)
ANALYSIS
OF SKEW
SANDWICH
395
PLATES
It can be seen that continuity of element displacements is maintained but continuity of the normal slopes is violated for this element. However this displacement function can still represent rigid body modes for frequency analysis. Using the principle of virtual work gives the stiffness matrix as
[KS1= w-q
[ [~lT[‘hllT[~IIW[~l dfd +XL-‘I,
where [L] is the matrix relating the generalized co-ordinates to the nodal displacements, [B] is the matrix relating the strain in the skew co-ordinate system to the generalized co-ordinates, [H] is the matrix relating the strain in the skew co-ordinate system to the strain in the rectangular co-ordinate system, [O] is the stress-strain matrix, and J is the Jacobian for transforming the integration to the dimensionless skew co-ordinates. The skew co-ordinates for the elements are shown in Figure 2. For the sake of brevity, the matrices [L], [B], [H] and [O] are not shown here but can be obtained elsewhere [ 111. 3. THE CONSISTENT LOAD VECTOR The load vector, which is obtained by calculating the virtual work done by the lateral uniformly applied load of intensity q, can be written as
[SJ = ahin 444 0, f, Zi,A, O,O,$, -&ii, O,O,$, &, -&GO, O,& -4, -&}T. TABLE
(3)
1
Comparison of centre deflection of a clamped rhombic sandwich plate with various shear rigidities; ES = 10’psi, v, = vc = 0.32, t, = 1.0 in, t, = 0.25 in, a = b = 40 in, G,, = Gyz = G,_., q = 1 psi
Centre deflection (in) Kwok 191 Skew angle
,3D.0.F. (psi)
\
Monforton 1101 (16X16)
D.O.F. (16X16)
0.248 3 0.023 48
0.241 7 0.033 7 0.023 02
0.245 8 0.032 1 0.020 98
0.233 6 0.031 2 0.020 9
0.227 3 0.030 57 0.020 4
0.2007 0.0223 -
0.203 1 0.023 3 0.014 54
0.192 5 0.023 1 0.014 46
0.187 2 0.022 5 1 0.014 24
0.1324 0.0126 -
0.1402 0.0130 -
0.142 3 0.013 8 0.007 33
0.134 3 0.013 3 0.007 28
0.130 6 0.01296 0.007 2
0.0711 0.0052 -
0.0753 ox@55 -
0.076 7 0.005 8 0.002 3 1
0.072 2 oxlO 5 0.002 24
0.070 2 0.005 4 0.002 2
5D.0.F.
(8x8)
t
$
(12x12)
4
G
90
500 10000 100000
-
-
-
75
500 10 000 100000
0.2452 0.0302 -
0.2301 0.0295 -
0.2433 0.0302 -
60
500 1000 10000
0.2020 0.0223 -
0.1897 0.0217 -
45
500 10000 100000
0.1406 0.0129 -
30
500 10000 100000
0.0749 0.0054 -
t Fully clamped edge. $ Reduced clamped edge.
Kennedy 131 -
5
396
S. S. F. NG AND D. K. Y. LAM TABLE 2
Natural frequencies of a clamped skew orthotropic sandwich plate; Es = lo7 psi, vS = O-34, t, = 0.25 in, t, = 0.016 in, a = 67.75 in, b = 43.5 in, G,, = 19500 psi, G,,, = 7500 psi, pC= 4.02 X lop6 lb s2/in4, pS = 248 x 10P6 lb s2/in4 Skew angle 4
Element with five degrees of freedom m
n
4x4
6X6
75
1 2 1 3 2
1 1 2 1 2
76 143 209 284 539
66 104 166 177 205
63 96 155 156 188
62 92 145 149 178
60
1 2 1 3 2
1 1 2 1 2
91 164 247 320 574
79 121 199 200 249
76 112 173 189 229
75 107 160 181 216
45
1 2 1 3 2
1 1 2 1 2
127 215 338 401 642
113 163 253 283 348
109 150 219 266 304
106 143 200 253 273
30
1 2 1 3 2
1 1 2 1 2
226 345 563 583 815
204 272 392 489 541
197 250 339 439 478
192 236 306 384 485
TABLE
8x8
12 x 12
3
Natural frequencies of a simply supported skew sandwich plate; Es = lo7 psi, v = O-3, t, = 0.25 in, t, = 0.016 in, a = 72 in, b =48 in, G,, = 19500 psi, G,,== 7500 psi, PC= 11.4 X low6 lb s2/in4, pS = 259 x 10e6 lb s2/in4 Element with five degrees of freedom Skew angle
4x4
6X6
8X8
12x12
1 1 2 1 2
28.42 62.06 87.51 124.22 141-84
26.45 52.08 79.59 94.19 109.49
25.94 49.87 77.65 87.66 104.05
25.23 48.29 76.32 83-98 101.09
1 2 1 3 2
1 1 2 1 2
37.79 82.15 110.63 149.75 192.75
33.94 65.01 99.77 109.68 141.52
32.82 60.80 96.83 99.16 133.49
31.69 57.81 93.29 94.71 127.33
1 2 1 3 2
1 1 2 1 2
54.49 111.56 161.89 193.72 271.01
48.92 87.87 140.46 147.49 207.24
47.00 81.01 124.36 142.18 178.42
45.50 76.34 114.79 138.05 161.06
4
m
n
75
1 2 1 3 2
60
45
ANALYSIS
4. THE
OF SKEW
SANDWICH
CONSISTENT
The mass matrix can be computed structure and is given by
MASS
by considering
[M] = Jp[ L-‘]* ”
397
PLATES
MATRIX
the kinetic energy of the sandwich
[CITITIT[~IICl d i!d ii d z[L-‘I,
(4)
Gc =
100000 psi
G,= 10 000 psi
G,= 5000 psi
Cc= 1000 psi
G,= 500 psi
0 9o”
I
75O
I
60”
I
45O
1
3o”
’
Skew angle $ Figure 3. Lowest natural frequencies of a clamped skew sandwich plate ratio = 1.5. E,_ = IO’ psi, V, = 0.34, !, =0.25 in, 1, =O.O16in, (I =60in. 4.02 x 10m6 lb ?/in”, pS = 248 x 10e6 lb s2/in4.
with various skew angle. Aspect h =4Oin, G,, = G,., = G, pe =
398
S. S. F. NC
AND
k
b
Q
aa
II
‘j II
D. K. Y. LAM
B II
m
0
C’
399
ANALYSIS OF SKEW SANDWICH PLATES
where [IL] is the matrix relating the generalized co-ordinates to the nodal displacements, [ Tj is the matrix relating the displacement of the individual layers to the global displacements, [C J is the matrix relating the assumed displacement field to the matrix of undetermined constants in the interpolation function, and p is the mass density of the layer. Again, the mass matrix of each individual layer is formulated separately and then combined to form the elemental mass matrix [ 111. 5. BOUNDARY
CONDITIONS
The boundary conditions for the obique boundary edge of the skew sandwich plates are applied after transforming the global co-ordinate system into an orthogonal system so that the generalized displacements become components normal and tangential to the
Figure 4. Lowest natural frequencies of a clamped skew sandwich plate with shear rigidity. various Aspect ratio = 1.5. E, = 1, = V, = 0.34, 10’psi, 0.25 6. ,nini = f;o=,,“.Oz i:G a 1 G, ’ ’ I&4$x 10e6 lb s2/in4, Ps = 248 x 10e6 lb sz/in4.
1
1.0
I
1.5
I
2.0
Aspect ratio
Figure 5. Lowest natural frequencies of a clamped skew sandwich plate with various as ct ratios. (A,). p = 4.02 x 10-G s2/ in4 Gxy = G,,= = G, ?, = 0.016 in, r, = 0.25 in, u, = 0.34, 24gx10-61bs2/in4. a=4Oin, b=4Oin, A,=l.O; a=60in, b=4Oin, A,=1.51 a=80in, b=4Oin, ;\,=?z G,=1ooOOOOpsi,-, G,=XIOpsi. --1 Es = 10’psi,
400
S. S. F. NG
AND
D. K. Y.
LAM
oblique boundary. The symmetries in the structural matrices are maintained. When particular natural or kinematic boundary conditions are known, the boundary conditions are enforced by zeroing the corresponding row and column and placing unity in the leading diagonal of the structural matrices. In this way shifting of the resulting matrix is avoided.
6. NUMERICAL
RESULTS
AND DISCUSSIONS
A comparison of the centre deflection of clamped rhombic sandwich plates obtained from the five degrees of freedom model for various skey angles and shear rigidities is shown in Table 1, and it can be seen from the table that excellent agreement with the results of previous investigators is obtained. Also it can be observed that as the skew angle is increased the centre deflection decreases; similar decrease in deflection is also observed as the rigidities of the core are increased. As there are no results available in the technical literature for the vibration of skew sandwich plates, no comparisons can be made. Some numerical results are presented in Tables 2 and 3 for vibration of skew sandwich plates with various skew angles. It can be observed that the convergence of the solutions for high skew angles is as good as for low skew angles although it is again noted that the rate of convergence is slightly slower for high angles of skew. This may be in part due to the presence of unbounded bending stresses at the obtuse corners whose edges are not completely clamped or free (12). As the skew angle increases this singularity is strengthened and the accuracy of the results usually decreases proportionately. However, in this analysis meaningful results are also obtained for plates with relatively high skew angles. In Figures 3-5, lowest natural frequencies of clamped skew sandwich plates with various skew angle, shear rigidities and aspect ratios are presented, respectively. As can be observed the lowest natural frequency of skew sandwich plates increases with increase in the skew angles or the shear rigidity of the core layer. It should also be noted that as the aspect ratio of the skew sandwich plate increases the lowest natural frequency decreases proportionately.
7. CONCLUSIONS
A finite element method for computing the displacements and natural frequencies of clamped and simply supported skew sandwich plates has been presented. Stiffness and mass matrices for the skew sandwich plate element have been derived. The numerical examples indicate that the method presented gives an excellent approximation for the displacements and natural frequencies of skew sandwich plates. By examination of the analytical results obtained, the following conclusions can be drawn. (a) E#ect of the angle of skew. Under constant aspect ratio, and identical physical properties and geometrical dimensions, and identical boundary conditions, the higher the skew angle the higher is the lowest natural frequency and the less is the deflection. (b) E@ct of shear rigidity. Under identical properties and dimensions, and identical boundary conditions and skew angle, the greater the shear rigidity the higher is the lowest natural frequency and the less is the deflection. (c) Eflect of aspect ratio. For a given skew angle, for sandwich plates with identical properties .and thickness, the greater the aspect ratio the smaller is the lowest natural frequency and the larger is the deflection.
ANALYSIS
OF SKEW SANDWICH
PLATES
401
REFERENCES
1. K. T. YEN, S. GENTURKUM and V. P. POHLE 1951 NACA TN 2581. Deflections of a simply supported rectangular sandwich plate subjected to transverse loads. 2. B. D. LIAW and R. W. LITTLE 1967 American Institute ofAeronautics and Astronautics Journal, 5, 301-304. Bending of multilayer sandwich plates. 3. J. B. KENNEDY 1970 7&e Aeronautical Journal of the Royal Aeronautical Society 74, 496-501. On the deformation of parallelogrammic sandwich panels. 4. G. R. MONFORTON and L. A. SCHMIT, JR 1968 Proceedings of the Conference on Matrix Method in Structural Mechanics, Wright-Patterson Air Force Base, Ohio, AFFDL-TR-68-150, 573-616. Finite element analysis of sandwich plates and cylindrical shells with laminated faces. 5. J. F. AEEL and E. P. POPOV, 1968 Proceedings of the Conference on Matrix Method in Structural Mechanics Wright-Patterson Air Force Base AFFDL-TR-68-150, 213-243. Static and dynamic
finite element analysis of sandwich structures. 6. K. M. AHMED 1970 Technical Report No. 37, Institute of Sound and Vibration Research, University of Southampton. Vibration analysis of doubly curved honeycomb sandwich plates by the finite element method. 7. T. P. KHATUA and Y. K. CHEUNG 1971 Research Report No. CETZ-13, Department of Civil Engineering, The University of Calgary. Bending and vibration multilayer sandwich beams and plates. 8. H. C. CHAN and Y. K. CHEUNG 1972 Znternational Journal of Mechanical Science, 14,399-406. Static and dynamic analysis of multilayer sandwich plates. 9. W. L. KWOK 1972 M.A.Sc. Thesis, University of Ottawa. Static analysis of skew sandwich plates by the method of finite element. 10. G. R. MONFORTON and M. G. MICHAIL 1972 Journal of the Engineering Mechdnics Division, Proceedings of the American Society of Civil Engineers 98, 763-769. Finite element analysis of skew sandwich plates. 11. KWAN
YUEN LAM 1982 Report Presented to the University of Ottawa for partial fulfillment of the requirements for the degree of Master of Engineering. Static and dynamic analysis of skew
sandwich plates by the finite element method. 12. L. S. D. MORLEY 1963 Skew Plates and Structures. New York: Pergamon Press, The MacMillan Company. 13. G. M. FOLIE 1970 UNZCZV Report No. R56, University of New South Wales, Kensington, Australia. The behaviour and analysis of orthotropic sandwich plates. 14. F. J. PLANTEMA 1966 Sandwich Construction. New York: John Wiley and Sons. 15. C. E. S. UENG 1966 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 33, 683-684. Natural frequencies of vibration of an all-clamped rectangular sandwich panel. 16. C. E. S. UENG and M. E. RAVILLE 1967 Paper No. 1175 Presented at the 1967 SESA Spring Meeting, Ottawa, Ontario. Determination of natural frequencies of vibration of a sandwich plate.
DYNAMIC
AND
STATIC
SANDWICH S. S.
Department
F. NC
of Civil Engineering,
(Received
AND
ANALYSIS
OF SKEW
PLATES D. K. Y. LAM
University of Ottawa, Ottawa, Ontario, Canada
6 July 1983, and in revised form 3 July 1984)
A finite element displacement model is presented for the dynamic and static analysis of clamped and simply supported skew sandwich plates. The geometric admissibility conditions of the principle of minimum total potential energy are satisfied by representing the assumed displacement pattern by a polynomial function. For static analysis, the sandwich plate is assumed to be uniformly loaded although point loads can be handled with very little modification in the computer program. For the dynamic analysis, results are presented for free flexural vibrations of skew sandwich plates with different plate aspect ratios, angles of skew and core rigidities.
1.
INTRODUCTION
Static and dynamic analysis of rectangular sandwich plates have been performed by a number of research workers [ 1,2] in the past using the variational approach and obtaining numerical results by a series types solution. A series solution was also presented by Kennedy [3] for static deformations of parallelogrammic clamped sandwich panels. More recently, the finite element method has been applied to single core or multilayered sandwich plates by several research workers [4-lo]. However scant data is available on the dynamic analysis of skew sandwich plates. Although in general the series solution method can yield accurate results, successful application is limited to sandwich structures with very simple boundary conditions. For solutions involving more general types of boundary conditions, it has been necessary to resort to numerical methods such as finite difference or finite elements. In the present paper, the finite element method is used for the static and dynamic analyses of skew sandwich plates incorporating a parallelogrammic element having five degrees of freedom per node. The parallelogrammic sandwich plate under consideration consists of a relatively thick homogeneous orthotropic core sandwiched between two thin homogeneous isotropic faces of equal thickness. The other assumptions used in this investigation, which are standard for sandwich plates, can be obtained elsewhere [ 111. The stiffness matrix is represented in terms of the element geometry, the stiffness of the faces (membrane, bending) and the transverse shear stiffness of the orthotropic core. The consistent mass matrix includes both rotary inertia and translatory inertia.
2. DERIVATION
OF THE STIFFNESS
MATRIX
Because of the unique behaviour of sandwich plates, it is convenient to build up each individual sandwich plate element by using a set of sub-elements, two skin layers and one core layer (see Figure 1). The stiffness matrices of the skin layers and the core layer are developed separately and then combined to form the total elemental stiffness matrix 393 0022-460X/85/070393 + 09 %03.OQ/O @ 1985 Academic Press Inc (London) Limited
394
S. S. F. NC
AND
D. K. Y. LAM
Figure 1. A typical skew plate.
for the sandwich plate element. In dealing with skew plates, a parallelogrammic-shaped element has been found to be very useful, as it has one additional node over a triangular element which permits more accurate representation of the displacement function and the varying stress field. Use of these elements is obviously more restricted than that of triangular elements in other types of problems.
Figure 2. A typical parallelogrammic element.
A typical parallelogrammic sandwich element is shown in Figure 2 and the displacement vector for this element consists of the in-plane displacements u, u and the lateral displacement w. Although a parallelogrammic element with three degrees of freedom per node has been found to yield sufficiently accurate results for static analysis, experience with this element on dynamic analysis has demonstrated the need for a more refined element. In the present study, a parallelogrammic element with five degrees of freedom per node (u, v, w, WC,wq) was used. The assumed displacement function after introducing the dimensionless
skew co-ordinates
(3 = v/b,
c= [/a),
is -
v =
u=a,+a2~+a,ij+a&,
a5 +
ah<+
a,ij
+
as&j,
~=a,+a,,~+a,,7i+a,,F’+a,,577+a,,iS~+a,,~~+a,6~4 +
aIT&?+
a18fj3+
a19i?ii
+
a205;;3
(1)
ANALYSIS
OF SKEW
SANDWICH
395
PLATES
It can be seen that continuity of element displacements is maintained but continuity of the normal slopes is violated for this element. However this displacement function can still represent rigid body modes for frequency analysis. Using the principle of virtual work gives the stiffness matrix as
[KS1= w-q
[ [~lT[‘hllT[~IIW[~l dfd +XL-‘I,
where [L] is the matrix relating the generalized co-ordinates to the nodal displacements, [B] is the matrix relating the strain in the skew co-ordinate system to the generalized co-ordinates, [H] is the matrix relating the strain in the skew co-ordinate system to the strain in the rectangular co-ordinate system, [O] is the stress-strain matrix, and J is the Jacobian for transforming the integration to the dimensionless skew co-ordinates. The skew co-ordinates for the elements are shown in Figure 2. For the sake of brevity, the matrices [L], [B], [H] and [O] are not shown here but can be obtained elsewhere [ 111. 3. THE CONSISTENT LOAD VECTOR The load vector, which is obtained by calculating the virtual work done by the lateral uniformly applied load of intensity q, can be written as
[SJ = ahin 444 0, f, Zi,A, O,O,$, -&ii, O,O,$, &, -&GO, O,& -4, -&}T. TABLE
(3)
1
Comparison of centre deflection of a clamped rhombic sandwich plate with various shear rigidities; ES = 10’psi, v, = vc = 0.32, t, = 1.0 in, t, = 0.25 in, a = b = 40 in, G,, = Gyz = G,_., q = 1 psi
Centre deflection (in) Kwok 191 Skew angle
,3D.0.F. (psi)
\
Monforton 1101 (16X16)
D.O.F. (16X16)
0.248 3 0.023 48
0.241 7 0.033 7 0.023 02
0.245 8 0.032 1 0.020 98
0.233 6 0.031 2 0.020 9
0.227 3 0.030 57 0.020 4
0.2007 0.0223 -
0.203 1 0.023 3 0.014 54
0.192 5 0.023 1 0.014 46
0.187 2 0.022 5 1 0.014 24
0.1324 0.0126 -
0.1402 0.0130 -
0.142 3 0.013 8 0.007 33
0.134 3 0.013 3 0.007 28
0.130 6 0.01296 0.007 2
0.0711 0.0052 -
0.0753 ox@55 -
0.076 7 0.005 8 0.002 3 1
0.072 2 oxlO 5 0.002 24
0.070 2 0.005 4 0.002 2
5D.0.F.
(8x8)
t
$
(12x12)
4
G
90
500 10000 100000
-
-
-
75
500 10 000 100000
0.2452 0.0302 -
0.2301 0.0295 -
0.2433 0.0302 -
60
500 1000 10000
0.2020 0.0223 -
0.1897 0.0217 -
45
500 10000 100000
0.1406 0.0129 -
30
500 10000 100000
0.0749 0.0054 -
t Fully clamped edge. $ Reduced clamped edge.
Kennedy 131 -
5
396
S. S. F. NG AND D. K. Y. LAM TABLE 2
Natural frequencies of a clamped skew orthotropic sandwich plate; Es = lo7 psi, vS = O-34, t, = 0.25 in, t, = 0.016 in, a = 67.75 in, b = 43.5 in, G,, = 19500 psi, G,,, = 7500 psi, pC= 4.02 X lop6 lb s2/in4, pS = 248 x 10P6 lb s2/in4 Skew angle 4
Element with five degrees of freedom m
n
4x4
6X6
75
1 2 1 3 2
1 1 2 1 2
76 143 209 284 539
66 104 166 177 205
63 96 155 156 188
62 92 145 149 178
60
1 2 1 3 2
1 1 2 1 2
91 164 247 320 574
79 121 199 200 249
76 112 173 189 229
75 107 160 181 216
45
1 2 1 3 2
1 1 2 1 2
127 215 338 401 642
113 163 253 283 348
109 150 219 266 304
106 143 200 253 273
30
1 2 1 3 2
1 1 2 1 2
226 345 563 583 815
204 272 392 489 541
197 250 339 439 478
192 236 306 384 485
TABLE
8x8
12 x 12
3
Natural frequencies of a simply supported skew sandwich plate; Es = lo7 psi, v = O-3, t, = 0.25 in, t, = 0.016 in, a = 72 in, b =48 in, G,, = 19500 psi, G,,== 7500 psi, PC= 11.4 X low6 lb s2/in4, pS = 259 x 10e6 lb s2/in4 Element with five degrees of freedom Skew angle
4x4
6X6
8X8
12x12
1 1 2 1 2
28.42 62.06 87.51 124.22 141-84
26.45 52.08 79.59 94.19 109.49
25.94 49.87 77.65 87.66 104.05
25.23 48.29 76.32 83-98 101.09
1 2 1 3 2
1 1 2 1 2
37.79 82.15 110.63 149.75 192.75
33.94 65.01 99.77 109.68 141.52
32.82 60.80 96.83 99.16 133.49
31.69 57.81 93.29 94.71 127.33
1 2 1 3 2
1 1 2 1 2
54.49 111.56 161.89 193.72 271.01
48.92 87.87 140.46 147.49 207.24
47.00 81.01 124.36 142.18 178.42
45.50 76.34 114.79 138.05 161.06
4
m
n
75
1 2 1 3 2
60
45
ANALYSIS
4. THE
OF SKEW
SANDWICH
CONSISTENT
The mass matrix can be computed structure and is given by
MASS
by considering
[M] = Jp[ L-‘]* ”
397
PLATES
MATRIX
the kinetic energy of the sandwich
[CITITIT[~IICl d i!d ii d z[L-‘I,
(4)
Gc =
100000 psi
G,= 10 000 psi
G,= 5000 psi
Cc= 1000 psi
G,= 500 psi
0 9o”
I
75O
I
60”
I
45O
1
3o”
’
Skew angle $ Figure 3. Lowest natural frequencies of a clamped skew sandwich plate ratio = 1.5. E,_ = IO’ psi, V, = 0.34, !, =0.25 in, 1, =O.O16in, (I =60in. 4.02 x 10m6 lb ?/in”, pS = 248 x 10e6 lb s2/in4.
with various skew angle. Aspect h =4Oin, G,, = G,., = G, pe =
398
S. S. F. NC
AND
k
b
Q
aa
II
‘j II
D. K. Y. LAM
B II
m
0
C’
399
ANALYSIS OF SKEW SANDWICH PLATES
where [IL] is the matrix relating the generalized co-ordinates to the nodal displacements, [ Tj is the matrix relating the displacement of the individual layers to the global displacements, [C J is the matrix relating the assumed displacement field to the matrix of undetermined constants in the interpolation function, and p is the mass density of the layer. Again, the mass matrix of each individual layer is formulated separately and then combined to form the elemental mass matrix [ 111. 5. BOUNDARY
CONDITIONS
The boundary conditions for the obique boundary edge of the skew sandwich plates are applied after transforming the global co-ordinate system into an orthogonal system so that the generalized displacements become components normal and tangential to the
Figure 4. Lowest natural frequencies of a clamped skew sandwich plate with shear rigidity. various Aspect ratio = 1.5. E, = 1, = V, = 0.34, 10’psi, 0.25 6. ,nini = f;o=,,“.Oz i:G a 1 G, ’ ’ I&4$x 10e6 lb s2/in4, Ps = 248 x 10e6 lb sz/in4.
1
1.0
I
1.5
I
2.0
Aspect ratio
Figure 5. Lowest natural frequencies of a clamped skew sandwich plate with various as ct ratios. (A,). p = 4.02 x 10-G s2/ in4 Gxy = G,,= = G, ?, = 0.016 in, r, = 0.25 in, u, = 0.34, 24gx10-61bs2/in4. a=4Oin, b=4Oin, A,=l.O; a=60in, b=4Oin, A,=1.51 a=80in, b=4Oin, ;\,=?z G,=1ooOOOOpsi,-, G,=XIOpsi. --1 Es = 10’psi,
400
S. S. F. NG
AND
D. K. Y.
LAM
oblique boundary. The symmetries in the structural matrices are maintained. When particular natural or kinematic boundary conditions are known, the boundary conditions are enforced by zeroing the corresponding row and column and placing unity in the leading diagonal of the structural matrices. In this way shifting of the resulting matrix is avoided.
6. NUMERICAL
RESULTS
AND DISCUSSIONS
A comparison of the centre deflection of clamped rhombic sandwich plates obtained from the five degrees of freedom model for various skey angles and shear rigidities is shown in Table 1, and it can be seen from the table that excellent agreement with the results of previous investigators is obtained. Also it can be observed that as the skew angle is increased the centre deflection decreases; similar decrease in deflection is also observed as the rigidities of the core are increased. As there are no results available in the technical literature for the vibration of skew sandwich plates, no comparisons can be made. Some numerical results are presented in Tables 2 and 3 for vibration of skew sandwich plates with various skew angles. It can be observed that the convergence of the solutions for high skew angles is as good as for low skew angles although it is again noted that the rate of convergence is slightly slower for high angles of skew. This may be in part due to the presence of unbounded bending stresses at the obtuse corners whose edges are not completely clamped or free (12). As the skew angle increases this singularity is strengthened and the accuracy of the results usually decreases proportionately. However, in this analysis meaningful results are also obtained for plates with relatively high skew angles. In Figures 3-5, lowest natural frequencies of clamped skew sandwich plates with various skew angle, shear rigidities and aspect ratios are presented, respectively. As can be observed the lowest natural frequency of skew sandwich plates increases with increase in the skew angles or the shear rigidity of the core layer. It should also be noted that as the aspect ratio of the skew sandwich plate increases the lowest natural frequency decreases proportionately.
7. CONCLUSIONS
A finite element method for computing the displacements and natural frequencies of clamped and simply supported skew sandwich plates has been presented. Stiffness and mass matrices for the skew sandwich plate element have been derived. The numerical examples indicate that the method presented gives an excellent approximation for the displacements and natural frequencies of skew sandwich plates. By examination of the analytical results obtained, the following conclusions can be drawn. (a) E#ect of the angle of skew. Under constant aspect ratio, and identical physical properties and geometrical dimensions, and identical boundary conditions, the higher the skew angle the higher is the lowest natural frequency and the less is the deflection. (b) E@ct of shear rigidity. Under identical properties and dimensions, and identical boundary conditions and skew angle, the greater the shear rigidity the higher is the lowest natural frequency and the less is the deflection. (c) Eflect of aspect ratio. For a given skew angle, for sandwich plates with identical properties .and thickness, the greater the aspect ratio the smaller is the lowest natural frequency and the larger is the deflection.
ANALYSIS
OF SKEW SANDWICH
PLATES
401
REFERENCES
1. K. T. YEN, S. GENTURKUM and V. P. POHLE 1951 NACA TN 2581. Deflections of a simply supported rectangular sandwich plate subjected to transverse loads. 2. B. D. LIAW and R. W. LITTLE 1967 American Institute ofAeronautics and Astronautics Journal, 5, 301-304. Bending of multilayer sandwich plates. 3. J. B. KENNEDY 1970 7&e Aeronautical Journal of the Royal Aeronautical Society 74, 496-501. On the deformation of parallelogrammic sandwich panels. 4. G. R. MONFORTON and L. A. SCHMIT, JR 1968 Proceedings of the Conference on Matrix Method in Structural Mechanics, Wright-Patterson Air Force Base, Ohio, AFFDL-TR-68-150, 573-616. Finite element analysis of sandwich plates and cylindrical shells with laminated faces. 5. J. F. AEEL and E. P. POPOV, 1968 Proceedings of the Conference on Matrix Method in Structural Mechanics Wright-Patterson Air Force Base AFFDL-TR-68-150, 213-243. Static and dynamic
finite element analysis of sandwich structures. 6. K. M. AHMED 1970 Technical Report No. 37, Institute of Sound and Vibration Research, University of Southampton. Vibration analysis of doubly curved honeycomb sandwich plates by the finite element method. 7. T. P. KHATUA and Y. K. CHEUNG 1971 Research Report No. CETZ-13, Department of Civil Engineering, The University of Calgary. Bending and vibration multilayer sandwich beams and plates. 8. H. C. CHAN and Y. K. CHEUNG 1972 Znternational Journal of Mechanical Science, 14,399-406. Static and dynamic analysis of multilayer sandwich plates. 9. W. L. KWOK 1972 M.A.Sc. Thesis, University of Ottawa. Static analysis of skew sandwich plates by the method of finite element. 10. G. R. MONFORTON and M. G. MICHAIL 1972 Journal of the Engineering Mechdnics Division, Proceedings of the American Society of Civil Engineers 98, 763-769. Finite element analysis of skew sandwich plates. 11. KWAN
YUEN LAM 1982 Report Presented to the University of Ottawa for partial fulfillment of the requirements for the degree of Master of Engineering. Static and dynamic analysis of skew
sandwich plates by the finite element method. 12. L. S. D. MORLEY 1963 Skew Plates and Structures. New York: Pergamon Press, The MacMillan Company. 13. G. M. FOLIE 1970 UNZCZV Report No. R56, University of New South Wales, Kensington, Australia. The behaviour and analysis of orthotropic sandwich plates. 14. F. J. PLANTEMA 1966 Sandwich Construction. New York: John Wiley and Sons. 15. C. E. S. UENG 1966 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 33, 683-684. Natural frequencies of vibration of an all-clamped rectangular sandwich panel. 16. C. E. S. UENG and M. E. RAVILLE 1967 Paper No. 1175 Presented at the 1967 SESA Spring Meeting, Ottawa, Ontario. Determination of natural frequencies of vibration of a sandwich plate.