NASA Contractor Report 4510 Inclusion of Transverse Shear Deformation in the Exact Buckling and Vibration Analysis of Composite Plate Assemblies Melvin S. Anderson and David Kennedy CONTRACT NASI-18584 and COOPERATIVE AGREEMENT NCCW-000002 MAY 1993 (NASA-CP-¢510) INCLUSION UF TRANbV£RSE SHEAR DEFORMATION IN EXACT _UCKLING AND VIBRATION ANALYSIS OF COMPOSITE PLATE ASSEMBLIES (Old Dominion Univ.) 23 p THE ";'_' 5-? 7075 Unc|as HI/3? 0167888 https://ntrs.nasa.gov/search.jsp?R=19930017886 2020-04-25T08:05:20+00:00Z
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Inclusion of Transverse Shear Deformation in the Exact ... · Inclusion of Transverse Shear Deformation in the Exact Buckling and Vibration Analysis of Composite Plate Assemblies
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r_ 3 = r_ 9 ot ot o_2: -b-r45 : b-'rl 09 :--b---h4 5
r_9 = bh5 5
rl2=r 6=-a
#r 16 = -bh3 3
r_9 =-r_6 = -bh35
r21 = r 7 = -°_hl 2
r_7 = -bh22
r210 - r57 = -bh26
b
r34 = r98 = S
ot o_h88
r310 = -r58 = -_r38 = -- S
r410 =-r59 - bh56
13
r51 =r610 =-othl6
r 54 = -r9 10 = ot +_-
ot2h8 8
r510 = -bh66 - bS
r 3 --- 1 =-_-r65 =- b rl01 --
r_2 = -By
°_2N2yh8 8r83 = B z + ot2X 2 +
bS
c_Nxyh78r85 = -rl 03 = -°_bX2 - S
bNyh78
r95 = "r104 =- S
a3h14
b 2
ot2 Nxyh88)r53 = "r8 10 = --_--(h46+ S
h78.
r55 = rl 0 10 = -ot(h46+--_)
oc2h I 1r_l = -B x +
b
+ otX 1
r_4 =-r_2 = bX 1
o_Nx_r84 = r93 = S
oc2 ,,r94 = --b--h44 + bzX2+ S
bNxyh_8rl 05 = -b2X2 + b h77 - S
where
_b
Z.
S = 1 - Nyh 88Nz
X1 =_2(4m 1 co2+
X2 =n2(4 m 2 o_2 _ ct2b + bk2 )- _ -h44
B g = n2b(4 m 0 0)2 +_2 " bKI-t
where _t can be x, y or z.
Equations for the classical case can be obtained by setting the transverse
shear strains, _'x and _/y, equal to zero and noting that _x = W,x. The
partially inverted stress strain relations corresponding to equations (7) are
slightly different in that mxy and _Cxy are interchanged. Only the first four
equilibrium equations from equations (1) are used (the fifth is satisfied by
14
incorporation into the final form of the third). Following the same steps asfor the transverse shear case, the matrix whose eigenvalues are thecharacteristic roots is of order eight. Elements with a superscript #previously given for the transverse shear case apply also to the classicalcase if 1 is subtracted from any index greater than 4. The remainingelements not given above are
r14 =-r85 =-2c_h36
r34 = r87 = b2ot2
r54 =-r81 =- b hI6
2c_3r74 = r83 = c_Nxy- b2h46
r24 =-r86 = -2ah26
r44 = r88 = 2ah56
r73 = Bz + c_2X2
r84 = °_2 h44"4h66b + b2X 2 + bNy
Appendix BNumerical Solution of Eauations
There are several possibilities of numerical problems arising in
determining K given in equation (18) as
K = F E -1 (B1)
First the eigenvectors of the matrix R which form the elements of E and F
must be determined accurately for extreme proportions and a wide range
of scaling. With accurate values of the elements of E and F, the solution
procedure is as follows. Solve the system of equations
E* x = F* (B2)
The solution is
x = (E*)-IF* = K* (B3)
where superscript * indicates Hermitian transpose of the matrix. The
solution is accomplished by Gaussian elimination with pivoting on the
largest current diagonal except as will be discussed subsequently for
transverse shear cases. A typical element in row k in the matrix equation
+il3k
(B3) has a multiplying factor of exp(=-_) where 13k is the k th characteristic
root. If the imaginary part of 15k is too large, numeric overflow will occur.
To prevent this, a real number is subtracted from the argument of each
15
exponential in the kth row such that overflow will not occur for the
particular computer being used. This has the effect of multiplying the kth
equation by a constant which does not change the solution.
An additional problem occurs for transverse shear cases where the
transverse shear effects are small. One pair of characteristic roots becomes
very large and the remainder approach the values for the classical case.
Finite element formulations have had serious problems for this case and
much effort has been spent to avoid errors from too stiff elements
associated with locking that can occur with routine application of theory.
This problem is handled for the present case by placing the terms
involving these large roots in the last two rows of equation B2 and not
including these rows in the logic for pivoting on the largest diagonal. This
approach has proved to be quite successful, stiffnesses and buckling loads
having been accurately calculated for aluminum plates with width-
thickness ratios as high as one thousand.
In some cases, particularly for unloaded isotropic plates, the characteristic
roots 13 from equation 12 occur in repeated pairs and the above procedure
breaks down. For this case small perturbations in the positive and
negative directions are made to all three foundation stiffnesses (K x, Ky, K z
in equation 1) in order to separate the repeated roots. The stiffness matrix
is obtained by interpolation between the two stiffness matrices resulting
from the perturbed foundation stiffnesses.
References
1. Williams, F. W., Kennedy, D., and Anderson, M. S., "Analysis features of
VICONOPT, an exact buckling and vibration program for prismatic
assemblies of anisotropic plates," Proceedings of the 31st AIAA/ASME/
ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference,
Long Beach, CA, pp. 920-929, 1990. AIAA paper 90-0970
2 Williams, F. W., Kennedy, D., Butler, R., and Anderson, M. S., "VICONOPT:
Program for exact vibration and buckling analysis or design of prismatic
3. Cohen, G. A., "FASOR- A second generation shell of revolution code,"Computers & Structures Vol. 10, 1979, pp. 301-309.
4. Wittrick, W. H., and Williams, F. W., "Buckling and vibration ofanisotropic plate assemblies under combined loadings." InternationalJournal Mechanical Sciences, Vol. 16, 1974, pp. 209-239.
5. Cohen, G. A., "Transverse shear stiffness of laminated anisotropic shells,"Computer Methods in Applied Mechanics and Engineering, Vol. 13, 1978,pp. 205-220.
6. Noor, A. K., Burton, W. S., and Peters, J. M., "Predictor-Correctorprocedures for stress and free vibration analyses of muitilayeredcomposite plates and shells," Computer Methods in Applied Mechanics and
Engineering, Vol. 82, 1990, pp. 341-363.
17
Z, W
Z_y,v
/// Nxya¢_ NY
#_2_/ J/ Centroid surfaceX, LI
(a)Prebuckling inplane loads
Z
• ymxy_
/ _ _ "_- Reference surfaceX n x
(b) Buckling forces and moments
Figure 1. Positive direction of forces and moments per unit width acting on
a plate element.
5 x-104
4
3SHEAR
MODULUS, psi2
0 [ I t 1
0 5 10 15 20DENSITY, Ib/cu ft
Figure 2. Shear modulus variation of foam core material with density.
18
Solid
b 2
!
k_- bl-_
Sandwich
Classical Theory
I I
Sandwich
Transverse Shear Theory
Configuration
blSolid 1.12Sandwich 2.43
Designs based on Classical Theoryskin bladc mass
b2 t4 5 t O tc t45 tO tc lb/sq ft1.43 .00518 .0183 .00891 .0526 1.1992.69 .0091 .1710 - .0236 .4488 .606
Configuration
blSolid 1.11Sandwich 3.90
Designs Based on Transverse Shcar Theoryskin blade mass
b2 t4 5 tO t c t45 tO t c lb/sq ft1.41 .00520 .0179 .00868 .0533 1.2053.07 .0131 .3914 .0226 .6606 .737
Figure 3. Geometry of solid composite and sandwich configurations
designed to carry axial loading of 5000 lb/in. Sandwich core density is 6.9
lb/cu ft corresponding to a shear modulus of 7100 psi. Panel length is 30inches. All dimensions are in inches
19
1.5 --
1
PANEL
MASS, Ib/sq ft
0.5
0
0
NO CORE MATERIAL
,,,__AN S VE RS E_ _..-_-_
I- _1_ CLASSICAL THEORY
1 1 1 I
5 10 15 20
CORE DENSITY Ib/cu ft
Figure 4. Minimum panel mass as a function of core density.
_*_J_¢ t_04_li_J 048flirt I_(_P _i_ (:o_l_lOtq of mfCkrmatK)it *$ eStlfft&tl_i tO 8wg¢ll_ t Ptour _rf rlrlIOOrt_l_, ittclbldin_ the tl_ for fll_l_wtl1_J irlStl_'_lOrl$. SeBfchlhcJ exlS_ll_ data soclrcts.
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Olev_ H_hwlw, $_¢e 1204. Arlie_toe,. ¥& 2220.].4]03. and to tr_e Office of M_,_lemen! and $_l_et Pal)ee, vorw Red.on Pro e¢t (0704.0 t IMI), W_m<Jton. OC 2023.
I. AGENCY USE'ONLY (Leave'blank) 2. REPORT DATE ' 3. REPORT TYPE AND DATES COVERED
May 1993 Contractor _eoqrl;4. TITLE AND SUBTITLE S. FUNDING NUMBERS
Inclusion of Transverse Shear Deformation in the
Exact Buckliog and Vibration Analysis of CompositePlate Assembl1_
,.AUTHORS)
Melvin S. Anderson and David Kennedy
7. _0RMING ORGANIZATION NAME(,)AND ADDRESSEES) '
Old Dominion University Research Foundation
Norfolk, VA 23508 and
University of Nales College of CardiffP. O. Box 925, Cardiff CF2 1YF, United Kingdom
9. SPONSORING/MONITORING AGENCY NAME(S) ANO ADORE,SEES)
National Aeronautics and Space Administration
Langley Research Center
Hampton, VA 23681-0001
n
C NASI-18584TA 05CA NCCW-O00002
WU 505-63-50-07
8. PERFORMING ORGANIZATION
REPORT NUMBER
10. SPONSORING I MONITORINGAGENCY REPORT NUMBER
NASA CR-4510
11. SUPFLEMENTARY NOTES
Langley Technical Monitor: James H. Starnes, Jr.Melvin S. Anderson: Old Dominion University Research Foundation_ Norfolk, VADavid Kennedy: University of Wales College'of Cardiff, United Kingdom
The problem considered is the development of the necessary plate stiffnesses for use in thegeneral purpose program VICONOPT for buckling and vibration of composite plateassemblies. The required stiffnesses include the effects of transverse shear deformation andare for sinusoidal response along the plate length as required in VICONOPT. The method isbased on the exact solution of the plate differential equations for a composite laminate havingfully populated A, B, and D stiffness matrices which leads to an ordinary differential equationof tenth order.