Journal of Sound and Vibration (1995) 187(5), 737756DYNAMIC
RESPONSE OF TAPERED COMPOSITEBEAMSUSING HIGHER ORDER
SHEARDEFORMATION THEORYS. R:x:iiNcrsv:r: R:o :Nb N. G:Nrs:NMachine
Dynamics Laboratory, Department of Applied Mechanics, Indian
Institute ofTechnology, Madras 600 036, India(Received 6 May 1994,
and in nal form 5 September 1994)Harmonic response of tapered
composite beams is investiaged by using a nite elementmodel based
on a higher order shear deformation theory. Only uniaxial bending
is consideredand the interlaminar shear stresses are neglected. The
Poisson ratio eect is incorporated
intheformulationofbeamconstitutiveequations.Theeectofin-planeinertiaandrotaryinertiaisalsoconsideredintheformulationof
themassmatrix. Aparametricstudyisdone of the inuence of anisotropy,
taper prole andtaper parameter. Linearlyandparabolically varying
thickness variations of increasing, decreasing,
decreasingincreasingand increasingdecreasing type are used.7 1995
Academic Press Limited1. INTRODUCTIONConventional metals are being
replaced by bre-reinforced composite materials in a
varietyofstructural
componentsowingtotheirhighstrength-to-weightandstiness-to-weightratios.
Non-uniformbeams, tapered and stepped, can be used to achieve a
better distributionof strength and weight and sometimes to satisfy
architectural and functional
requirements.Interestinlaminatedbeams,uniformandnon-uniform,isgrowingastheyarendinganumber
of applications in turbine blades, helicopter blades and robot
arms. Therefore, thereis a need for a better understanding of the
vibration characteristics of laminated beams,and far laminated
composite beams, especially non-uniform beams, have not been
studiedextensively.ThepaperbyKapaniaandRaciti [1]
givestherecentdevelopmentsinthevibrationanalysis of laminated
composite beams. In recent years several authors have tried to
predictthenaturalfrequenciesoflaminatedbeamsofuniformthickness.MillerandAdams[2]studied
the vibration characteristics of orthotropic clampedfree beams
using the classicallamination theory. Vinson and Sierakowiski [3]
have given exact solutions based on classicallamination theory.
Chen and Yang [4] and Chandrashekhara et al. [5] have undertaken
thefree vibration analysis of composite beams based on rst order
shear deformation theory.Recently Chandrashekhara and Bangera [6]
have studied the free vibration characteristicsof laminated
composite beams using a nite element model based on a higher order
platetheory[7]. Theyhave correctedgeneralizedforce
andgeneralizedstrainrelations, toconsider the Poisson eect, by
ignoring the forces in the y-direction. This operation
involvesinversionof certainmatrices andis limitedtothetypeof
beamtheoryoneis using.7370022460X/95/450737+20 $12.00/0 7 1995
Academic Press Limiteds. r:x:iiNcrsv:r: r:o :Nb N. c:Nrs:N 738It
wouldbe appropriate tocorrect stressstrainrelations rather
thancorrecting thegeneralized force and generalized strain
relations [8]. Oral [9] has formulated a shear exiblenite element
for linearly tapered symmetrically laminated composite beams using
rst ordersheardeformationtheory.
TheelementhasthreenodesandtherearesixDOFateachnodethree
displacements and three independent rotations. The author [9] has
used thiselement to nd the static response of tapered beams.In the
present work the dynamical behaviour of tapered composite beams
subjected toa point harmonic excitation is studied by using a nite
element method. Higher order sheardeformation theory based on a
higher order plate theory [7] is used in developing the niteelement
model for tapered beams as it does not require shear correction
factors and becauseresultspredictedbyit
areaccuratecomparedtothoseobtainedwithrst
ordersheardeformationtheory.
ThePoissonratioeectisincorporatedbycorrectingstressstrainrelations.
The eect of in-plane inertia androtary inertia is
alsoconsideredinthe formulationof the mass matrix. Only uniaxial
bending is considered and the eect of interlaminar
shearstressesisneglected.
Avarietyofparametricstudiesareconductedtodemonstratetheinuence of
taper and material anisotropy on the dynamic response.2.
FORMULATIONThe displacement equations [7]
usedinobtainingone-dimensional laminatedbeamequations (see Figure
1) areU(X, z, t)=u(x, t)+z[cx(x, t)(4/3)(z/h)2(cx(x, t)+1w(x,
t)/1x)],W(x, z, t)=w(x, t), (1)where u and w are the in-plane and
lateral displacements of the middle surface,cx is therotation of
the normal to the middle plane about the y-axis, and h is the
thickness of thebeam. The strains associated with the displacements
in equation (1) areex=e0x+z[k1x+z2k2x], gxz=g*xz+z2k*xz,
(2)wheree0x=1u/1x, g*xz=cx+1w/1x, k1x=1gx/1x,
k*xz=(4/h2)(cx+1w/1x),k2x=(4/3h2)(1gx/1x+12w/1x2).Figure 1. A
straight uniform composite beam.1:irrrb coxiosi1r nr:x b.N:xic
rrsioNsr 739Figure 2. A linearly tapered composite beam of constant
width.The stresses in the nth layer, for layers stacked in the
y-direction as shown in Figure 2, theprincipal material axis of
whichis orientedat anangle u tothe x-axis, are relatedtothe
strainsby the relationsxxC11C12C130 0 C16exx F J K L F JG G G G G
GsyyC12C22C230 0 C26eyyG G G G G Gszz=C13C23C330 0 C36ezz. (3)j f G
G j fJ F G G J F tyz0 0 0 C44C450 gyzG G G G G Ggxy0 0 0 C45C550
gxyG G G G G GtxzC16C26C360 0 C66gxz f j k l f jIf the beam is
undergoing uniaxial bending and if there is no torsional loading,
one can takesyy=szz=tyz=txy=0, and upon substitution of this
equation (3) one arrives at the followingrelation which accounts
for Poissons ratio eect [8]:6sxxtxz 7=$Q11Q16Q16Q66%6exxgxz7.
(4)Here Q11=C*11, Q16=C*16 and Q66=C*66.For layers stacked in the
z-direction as shown in Figure 1 one can obtain, using the
sameprocedure and with appropriate stressstrain relations [8], the
following relations among theQij and C*ijconstants:
Q11=C*11C*16C*16/C*66, Q16=0 and Q66=C*44. The relations for
C*ijinterms of Cij are given in reference [8].2.1. s1r:iN rNrrc.The
stiness matrix is formulated by using the strain energy expressions
for a beam givenbyU=12gvol(sxxexx+txzgxz) d (vol), U=12gl0{e0}T{N0}
dx, (5)where {e0}T=[e0x k1x k2x g*xz k*xz] and {N0}T=[Nx Mx Px Qxz
Rxz]. Upon expressing thegeneralized stress vector as {N0}=[D]{e0}
and substituting this in equation (5), theexpression for the strain
energy becomesU=12gl0{e0}T[D]{e0} dx, (6)s. r:x:iiNcrsv:r: r:o :Nb
N. c:Nrs:N
740whereA11B11E11A16D16B11D11F11B16E16[D]=GGGGGKkE11F11H11E16G16GGGGGLl.A16B16E16A66D66D16E16G16D66F66For
the conguration shown in Figure 1(Aij, Bij, Dij, Eij, Fij, Hij)=b
snlk=1 ghkhk1Qij(1, z, z2, z3, z4, z5) dzand for the conguration
shown in Figure 2(Aij, Bij, Dij, Eij, Fij, Hij)=b snlk=1 ghxhx
gbkbk1Qij(1, z, z2, z3, z4, z5) dy dz.The layer conguration shown
in Figure 2 is adopted for the tapered beams considered inthe
present parametric study. Taper is considered for the thickness
with the width of
thebeamkeptconstant.Onlysymmetricallytaperedbeamsofrectangularcross-sectionareconsidered.
Dierent
taperedbeamsareobtainedfromauniformbeambyalteringitsthickness along
the length with its length, width and volume kept constant. For a
taperedbeam dened by a function f(x) and for any particular taper
parameter b, the half-thicknesshx at any distance x can be
calculated by using the relationhx=h1{1bf(x)},
(7)wherebisthetaperparameter=(1h2/h1),
h1isthemaximumhalf-thickness, h2istheminimum half-thickness and
f(x) is a function dening the taper prole (see the Appendix).In
equation (7) the only unknown is h1, which can be calculated by
equating the volume ofthe tapered beam to that of uniform beam.2.2.
iiNr1ic rNrrc.The expression for the kinetic energy may be written
asT=12 gvolr(U2+W2) d (vol). (8)2.3. rirxrN1 x:1ricrsFor the nite
element formulation a two-node beamelement with four degrees of
freedom[uj wj (1w/1x)j cxj] per node is used. For this conguration
the generalized displacements areinterpolated by using expressions
of the formu(x, t)=s2j=1uj(t)Nj(x), cx(x, t)=s2j=1cxj(t)Nj(x),w(x,
t)s2j=1{wj(t)zj(x)+(1w(t)/1x)jjj(x)}, (9)1:irrrb coxiosi1r nr:x
b.N:xic rrsioNsr 741where Nj are the Lagrange linear interpolation
functions andzj(x) andjj(x) are Hermitecubic interpolation
functions. By using equation (9) the generalized strains can be
expressedas{e0}=[B]{de}, (10)where[B] isthematrixof
shapefunctionsandtheirderivativesand{de}isthenodaldisplacement
vector. The element stiness matrix can be obtained by substituting
equation(10) in equation (6) and the element mass matrix can be
obtained by substituting equation(9) in equation (8):
[K]e=fl0[B]T[D][B] dx; [M]e=fl0[N]T[r][N] dx. Here [N] is the
matrix ofshape functions and its derivatives.2.4. s1r:b. s1:1r
rrsioNsrThe natural frequencies are obtained by solving the
eigenvalue problem[K]{d}=v2[M]{d}, (11)where [K] and [M] are the
global stiness and the global mass matrices, respectively,d isthe
corresponding eigenvector andv is the natural frequency.For a small
amount of structural damping, the steady state response of a beam
subjectedto harmonic excitation is given by[Kd]{U}={F}, (12)where
[Kd] is the dynamic stiness matrix [Kd]=[K](1+hi)V2[M], in which
{F} is the forcevector consisting of amplitudes of nodal forces,
{U} is the displacement vector, V is thefrequency of excitation and
h is the structural damping factor. The dynamic displacements{U}
obtained fromequation (12) are then used to obtain dynamic stresses
by using equations(2), (4) and (9).3. RESULTS AND DISCUSSIONA
number of examples have been considered. Unless mentioned
otherwise, the followingAS4/3051-6 graphite/epoxy material
properties are used: E1=14480 GPa, E2=965 GPa,G23=345 GPa,
G12=G13=414 GPa, n12=03, r=138923 kg/m3. The
non-dimensionalparametersusedinpresentingresultsarethefollowing:
v=vL2zr/E1h2; V*=(naturalfrequencyof
ataperedbeam)/(correspondingnatural frequencyof
auniformbeam);w*=(maximum dynamic displacement w of a uniform or
tapered beam)/(maximum staticdisplacement w of a uniform beam);
s*=(maximum dynamic normal stress sx of a uniformor tapered
beam)/(maximum static normal stresssx of a uniform
beam);t*=(maximumdynamicshearstresstxzofauniformortaperedbeam)/(maximumstaticshearstresstxzof
a uniformbeam); wn=(maximumdisplacement wof a
uniformbeam)/(maximumdisplacement wof a uniformorthotropic (0)
beam); s1n=(maximumprincipal normal stresss1 of a uniform
beam)/(maximum principal stresss1 of a uniform orthotropic (0)
beam);s2n=(maximum principal normal stresss2 of a uniform
beam)/(maximum principal stresss1 of a uniform orthotropic (0)
beam);t12n=(maximum shear stress in principal
planet12ofauniformbeam)/(maximumprincipal
stresss1ofauniformorthotropic(0)beam).For Tables 13 the laminated
plate strip (i.e., plies stacked in the z-direction as
showninFigure 1) is considered. InTable 1the natural frequencies of
asimplysupportedorthotropic (0) graphite/epoxy beam are compared
with existing results. The comparisonis quite good. It is well
known that the classical theory over predicts the natural
frequenciesin case of thick beams.s. r:x:iiNcrsv:r: r:o :Nb N.
c:Nrs:N 742In Table 2 the non-dimensional natural frequencies of
symmetrically laminated cross-plyand angle-ply beams of uniform
thickness for various boundary conditions are
compared.Onlytherstnatural frequencyisconsideredforcomparison.
InTable3isshowntheT:nir 1Comparison of natural frequencies (kHz) of
a simply supported orthotropic (0) graphite/epoxybeamMode CLT FSDT
HSDTL/h number [3] [5] [6] Present120 1 0051 0051 0051 0051(L=762
mm) 2 0203 0203 0202 02023 0457 0454 0453 04544 0812 0804 0799
08045 1269 1262 1238 125215 1 0813 0755 0756 0754(L=381 mm) 2 3250
2548 2554 25553 7314 4716 4742 47534 13002 6960 7032 70525 20316
9194 9355 9382T:nir 2Comparison of non-dimensional natural
frequencies (v ) of symmetrically laminated cross-ply[0/90/90/0]
and angle-ply [45/45/45/45] beams under various boundary
conditionsBoundary[0/90/90/0] [45/45/45/45]ZXXXXXXCXXXXXXV
ZXXXXXXCXXXXXXVconditions HSDT [6] Present HSDT [6] PresentSS 25023
24980 08295 07966CC 45940 46440 18472 17611CF 09241 09231 02965
02849T:nir 3Comparisonof non-dimensional frequencies(v ) of
asimplysupported[45/45/45/45]antisymmetric angle ply beamMode
HSDTnumber [6] Present1 08278 07972 32334 31203 70148 67914 107449
118625 119145 13030T:nir 4Comparison of tip displacements of a
tapered cantilever isotropic beamBending StretchingD/d Reference
[9] Present Reference [9] Present2 002729 002782 023077E03
02310E035 004789 004856 032609E03 03350E031:irrrb coxiosi1r nr:x
b.N:xic rrsioNsr 743T:nir 5Comparison of non-dimensional dynamic
displacements of a simply supported isotropic beamof uniform
thicknessxL/4 xL/2Mode ZXXXXXXCXXXXXXV ZXXXXXXCXXXXXXVnumber
Reference [10] Present Reference [10] Present1 10000 9747 14140
137842 1250 1258 011913 12350 12608 1746 17821Figure 3. Dierent
types of thickness variation: (a) linear; (b) parabolic.T:nir 6Eect
of anisotropy on the maximum static response of a simply supported
composite beamof uniform thicknessType beam wn s1n s2n =t12n =
R.F.Orthotropic 1000 10000 00000 00331 100[0/90/90/0] 1679 17410
00000 00332 263[15/15/15/15] 1040 07490 00355 01666
606[30/30/30/30] 4024 05873 01546 03034 2879[45/45/45/45] 15170
04042 03300 03671 7889[60/60/60/60] 26070 02157 05184 03365
15507[75/75/75/75] 28810 00676 06687 02157 23239[90/90/90/90] 29030
00000 07373 00363 27277s. r:x:iiNcrsv:r: r:o :Nb N. c:Nrs:N
744T:nir 7Eect of anisotropy on the maximum dynamic response of a
simply supported composite beamof uniform thicknessType of beam wn
s1n s2n =t12n = R.F.Orthotropic 1000 1000 0000 00493 100[0/90/90/0]
1655 1843 00000 00479 253[15/15/15/15] 1048 0896 00382 01915
642[30/30/30/30] 4146 0754 01926 03845 3676[45/45/45/45] 1563 0510
04059 04581 9712[60/60/60/60] 2609 0273 06446 04229
19381[75/75/75/75] 2874 0086 08291 02721 28843[90/90/90/90] 2894
0000 09154 00497 33885T:nir
8Maximumdynamicresponsevaluesofsimplysupportedcross-plyandangle-plybeamsofuniform
thicknessType of Res. vbeam no. (rad/s) w* s* t*Cross-ply 1 3 571
4993 3292029920[0/90/90/0] 2195902992902 12 896 541
1211021020087902102903 25 410 099 40409853002790985390Angle-ply 1 1
633 5104 3896 3023[45/45/45/45] 2 6 463 449 1350 20763 14 303 065
425 767 Superscripts over the numerical values in cross-ply cases
denote the ber direction in the layer.T:nir 9Maximumresponse values
of a simply supported cross-ply beam[0/90/90/0] subjected to a
pointharmonic load for linearly varying thickness variation:
increasing typeLayer 1 Layer 2Res. ZXXXXCXXXXV ZXXXXCXXXXVno. b V*
w* s* t* s* t* 1 025 09933 5218 31 3446 28 3564 1 230 28 3564 1050
09629 6144 28 4045 22 4931 1 270 22 4931 1075 08700 8938 23 5973 15
9244 1 398 15 9244 12 025 09971 609 16 1335 13 2412 1 089 13 2412
1050 09833 738 13 1592 13 2984 1 106 13 2984 1075 09401 1073 11
2235 9 4399 1 149 9 4399 13 025 09972 102 10 419 10 1018 1 028 10
1018 1050 09846 108 10 424 10 1049 1 028 10 1049 1075 09447 106 8
419 6 984 1 028 6 984 1 Node at which maximum displacement is
obtained. Element in which maximum normal stress is obtained.
Element in which maximum shear stress is obtained.1:irrrb coxiosi1r
nr:x b.N:xic rrsioNsr 745T:nir 10Maximumresponse values of a simply
supported cross-ply beam[0/90/90/0] subjected to a pointharmonic
load for linear thickness variation: decreasing typeLayer 1 Layer
2Res. ZXXXXCXXXXV ZXXXXCXXXXVno. b V* w* s* t* s* t* 1 025 09933
4719 31 3118 34 3221 60 208 34 3221 60050 09629 4789 34 3144 37
3842 60 210 37 3842 60075 08700 5591 37 3736 46 5783 60 249 46 5783
602 025 09971 543 46 1193 46 2148 60 079 46 2148 60050 09833 563 49
1221 49 2277 60 081 49 2277 60075 09401 657 49 1369 52 2693 60 091
52 2693 603 025 09972 108 52 438 52 1070 60 029 52 1070 60050 09846
124 52 502 52 1208 60 033 52 1208 60075 09447 160 52 647 55 1535 60
043 55 1535 60 Node at which maximum displacement is obtained.
Element in which maximum normal stress is obtained. Element in
which maximum shear stress is obtained.T:nir 11Maximumresponse
values of a simply supported cross-ply beam[0/90/90/0] subjected to
a pointharmonic load for linear thickness variation:
decreasingincreasing typeLayer 1 Layer 2Res. ZXXXXCXXXXV
ZXXXXCXXXXVno. b V* w* s* t* s* t* 1 025 09393 5888 31 42.16 31
2573 4 281 31 2573 4050 08252 8426 31 6598 31 2277 55 440 31 2277
55075 05914 19670 31 16550 31 2027 7 1104 31 2027 72 025 09929 548
16 1223 43 2416 31 082 43 2416 31050 09630 580 19 1311 19 3034 31
087 19 3034 31075 08731 727 19 1652 37 4817 31 110 37 4817 313 025
09969 111 31 447 31 1015 40 030 31 1015 40050 09894 132 31 519 31
1049 22 035 31 1049 22075 09768 178 31 656 31 1082 22 044 31 1082
22 Node at which maximum displacement is obtained. Element in which
maximum normal stress is obtained. Element in which maximum shear
stress is obtained.T:nir 12Maximumresponse values of a simply
supported cross-ply beam[0/90/90/0] subjected to a pointharmonic
load for linear thickness variation: increasingdecreasing typeLayer
1 Layer 2Res. ZXXXXCXXXXV ZXXXXCXXXXVno. b V* w* s* t* s* t* 1 025
10405 4290 31 2676 25 3439 1 178 25 3439 1050 10633 3825 31 2395 43
4410 1 160 43 4410 1075 10279 3783 31 2681 10 7384 1 179 10 7384 12
025 09945 545 16 1218 13 2410 1 081 13 2410 1050 09667 573 13 1299
49 3019 1 087 49 3019 1075 08791 710 13 1621 7 4766 1 108 7 4766 13
025 10003 101 10 405 10 1057 1 027 10 1057 1050 09952 105 10 409 52
1177 1 027 52 1177 1075 09719 113 10 448 55 1451 1 030 55 1451 1
Node at which maximum displacement is obtained. Element in which
maximum normal stress is obtained. Element in which maximum shear
stress is obtained.s. r:x:iiNcrsv:r: r:o :Nb N. c:Nrs:N 746T:nir
13Maximumresponse values of a simply supported cross-ply
beam[0/90/90/0] subjected to a pointharmonic load for parabolic
thickness variation: increasing typeLayer 1 Layer 2Res. ZXXXXCXXXXV
ZXXXXCXXXXVno. b V* w* s* t* s* t* 1 025 09796 5482 30 3625 27 3414
1 242 27 3414 1050 09236 6952 27 4577 23 4384 1 305 23 4384 1075
07651 13350 24 8470 17 7673 1 564 17 7673 12 025 09944 621 15 1368
15 2320 1 091 15 2320 1050 09735 803 15 1710 13 2743 1 114 13 2743
1075 09038 1435 13 2806 11 3870 1 187 11 3870 13 025 09966 103 11
424 10 990 1 028 10 990 1050 09803 111 10 452 9 982 1 030 9 982
1075 09201 114 9 445 8 805 1 030 8 805 1 Node at which maximum
displacement is obtained. Element in which maximum normal stress is
obtained. Element in which maximum shear stress is obtained.T:nir
14Maximumresponse values of a simply supported cross-ply
beam[0/90/90/0] subjected to a pointharmonic load for parabolic
thickness variation: decreasing typeLayer 1 Layer 2Res. ZXXXXCXXXXV
ZXXXXCXXXXVno. b V* w* s* t* s* t* 1 025 09796 4923 32 3254 34 3063
60 217 34 3063 60050 09236 5323 34 3505 38 3355 60 234 38 3355
60075 07651 7993 38 5066 44 4592 60 338 44 4592 602 025 09944 550
47 1209 46 2051 60 081 46 2051 60050 09735 586 47 1247 48 2002 60
083 48 2002 60075 09038 734 49 1435 50 1980 60 096 50 1980 603 025
09966 108 51 444 51 1037 60 030 51 1037 60050 09803 125 52 509 52
1108 60 034 52 1108 60075 09201 175 53 686 53 1239 60 046 53 1239
60 Node at which maximum displacement is obtained. Element in which
maximum normal stress is obtained. Element in which maximum shear
stress is obtained.T:nir 15Maximumresponse values of a simply
supported cross-ply beam[0/90/90/0] subjected to a pointharmonic
load for parabolic thickness variation: decreasingincreasing
typeLayer 1 Layer 2Res. ZXXXXCXXXXV ZXXXXCXXXXVno. b V* w* s* t* s*
t* 1 025 09413 5888 31 3910 30 2543 5 261 30 2543 5050 08308 8483
31 5521 30 2292 9 368 30 2292 9075 05951 20840 31 12030 30 2137 14
802 30 2137 142 025 09848 570 17 1292 17 2327 30 086 17 2327 30050
09368 655 18 1499 19 2764 30 100 19 2764 30075 07898 1005 19 2279
22 4032 30 152 22 4032 303 025 09956 111 31 437 30 1057 21 029 30
1057 21050 09780 136 31 503 30 1167 21 034 30 1167 21075 09207 211
31 682 30 1388 22 045 30 1388 22 Node at which maximum displacement
is obtained. Element in which maximum normal stress is obtained.
Element in which maximum shear stress is obtained.1:irrrb coxiosi1r
nr:x b.N:xic rrsioNsr 747T:nir 16Maximumresponse values of a simply
supported cross-ply beam[0/90/90/0] subjected to a pointharmonic
load for parabolic thickness variation: increasingdecreasing
typeLayer 1 Layer 2ZXXXXCXXXXV ZXXXXCXXXXVno. b V* w* s* t* s* t* 1
025 10410 4297 31 2773 30 3564 1 185 30 3564 1050 10711 3825 31
2349 30 4806 1 157 29 4806 1075 10659 3629 31 2213 7 8452 1 148 7
8452 12 025 10009 529 15 1172 14 2503 1 078 14 2503 1050 09820 538
14 1180 11 3283 1 079 11 3283 1075 09159 607 13 1401 6 5496 1 093 6
5496 13 025 09982 099 11 397 9 1100 1 026 9 1100 1050 09868 101 10
397 8 1297 1 026 8 1297 1075 09514 107 9 430 5 1759 1 029 5 1759 1
Node at which maximum displacement is obtained. Element in which
maximum normal stress is obtained. Element in which maximum shear
stress is obtained.comparisonof the non-dimensional natural
frequencies of a simply supportedantisymmetricangle-ply beamof
uniformthickness. For all these beams the L/h ratio is taken as 15,
h beingequal to 254 cm. The present results in both Table 2 and
Table 3 are in good agreementwith those in reference [6]. A slight
deviation of the present results from those in reference[6] may be
due to the procedure employed in considering the Poisson ratio
eect. In reference[6] this eect is considered after the evaluation
of stress resultants, whereas in the presentcase it is considered
before the evaluation of stress resultants.For checking the
accuracy of the present formulation for tapered beams, linearly
taperedclampedfree isotropic beams withdecreasing thickness
variationandof rectangularcross-section have been analyzed for
stretching and bending modes. Two beams, one withh1/h2=2 and the
other with h1h2=5, were taken for analysis. The following
properties wereused: E=30106lb/in2, n=0, L=10 in, b=1 in. The force
is applied at the tip and is equalto 1000 lb. The tip displacements
are compared in Table 4 with those in reference [9]. Thecomparison
is good.To validate the present formulation for dynamic analysis
the non-dimensional dynamicdisplacements of a simply supported
isotropic beam of uniform thickness, subjected to
apointharmonicexcitationattherstthreenatural frequencies,
arecomparedwiththeanalytical solutions giveninreference[10].
Theforceis appliedat quarter span. Thecomparison is shown in Table
5. The non-dimensional displacement is given byw(x)p4EI/PL3, where
Eis the Youngs modulus, I is the moment of inertia, Pis the
magnitudeof the harmonic force and L is the length of the beam. The
hysteretic damping constant istaken as 0.01. The results obtained
by the present formulation are in good agreement withthose in
reference [10].3.1. rrsioNsr or 1:irrrb nr:xsThe
followingdiscussiondeals withthe steadystate response of
simplysupportedcomposite beams, with plies stacked in the
y-direction as shown in Figure 2, when
subjectedtoharmonicexcitation.Threetypesofharmonicloadingwiththeexcitationfrequenciesequal
to the rst three natural frequencies are considered. The amplitude
of the load is keptconstant in all three types of harmonic
excitation. The point harmonic load is applied atquarter span from
the left end of the beam. For harmonic response the hysteretic
dampings. r:x:iiNcrsv:r: r:o :Nb N. c:Nrs:N 748constant is taken as
002. For tapered beams two types of proles, linear and parabolic,
areconsideredforthicknessvariation.
ThedierenttaperprolesconsideredareshowninFigure 3.The eect of
anisotropy on the response, static and harmonic, is shown in Tables
6 and 7.The maximumharmonic response values of simply supported
cross-ply and angle-ply beamsof uniformthicknessareshowninTable8.
MaximumresponsevaluesalongwiththeFigure 4. The distributionof
displacements andstresses for a SS(45/45/45/45) beam; increasing
typelinear,w, hx=0; W, hx=025; Q, hx=050; R, hx=075.1:irrrb
coxiosi1r nr:x b.N:xic rrsioNsr 749Figure 5. The distribution of
stresses for a SS (45/45/45/45) beam; increasing typelinear. Key as
Figure 4.resonant frequencies of a simply supported symmetric
cross-ply beams of dierent thicknessvariations when subjected to a
point harmonic load are shown in Tables 916. The maximumresponse
values of [45/45/45/45] beams are not given, as a more or less
similar trendas that observed in cross-ply beams is observed. The
variation of displacement and stressesalong the length of the beam
are shown in Figures 411. It is to be noted that the variationsof
lateral displacement and stresses of [45/45/45/45] beams of linear
variation only areshown, because the variation is more or less
similar in all the types of beams considered.From these tables and
gures, the following observations are made.In the case of symmetric
angle-ply beams the magnitude of the stresses developed is thesame
in all the laminae. In the case of cross-ply beams the magnitude of
the normal stressdeveloped in the 0 ply is greater than that
developed in the 90 ply and the ratio betweenthese two stresses is
approximately equal to the ratio of the elastic moduli. However the
sameshear stresses are developed in both the 0 ply and the 90ply.
In all the beams consideredthevariationofnormal
stressalongthelengthofthebeamisthesameasthatofthedisplacement for
any particular mode and for any particular thickness variation
considered.However, the variation of shear stress is opposite to
that of displacement.3.1.1. Eect of anisotropyThe eect of
anisotropy on the response values of simply supported uniform
compositebeams is shown in Tables 6 and 7. For Tables 6 and 7,
graphite/epoxy material with thes. r:x:iiNcrsv:r: r:o :Nb N.
c:Nrs:N 750followingproperties was used: E1=30106psi, E2=075106psi,
G12=0375106psi,n12=025. Only the resonant response values of beam
at the lowest natural frequency arepresented in Table 7. The
stresses presented are those in the principal material directionsof
the rst layer. Asimilar trendis observedinboththe static anddynamic
cases. The laminaearrangement is as shown in Figure 2. In the case
of an angle-ply beam, as the bre orientationangle,u to the x-axis
increases, the normal stress in the bre direction decreases and
thatin the direction perpendicular to the bre increases. The shear
stress, t12n, increases with theFigure 6. As Figure 4, but
decreasing typelinear.1:irrrb coxiosi1r nr:x b.N:xic rrsioNsr
751Figure 7. As Figure 5, but decreasing typelinear.increase in u
up to u=45 and then decreases. The normal stress in the bre
direction of across-ply beam is greater than that in all the other
types of beams considered. In contrast,the shear stresst12n in a
cross-ply beam is less than that obtained for the angle-ply
beams.In the case of an angle-ply beam the beam becomes exible as
the inclination of bre to thex-axis increases. It canbe seenthat
the relative factor of safety (withrespect toanorthotropic(0)beam)
is a minimumin case of an orthotropic (0) beam. In the case of an
angle-ply
beam,therelativefactorofsafetyincreaseswithincreaseinbreorientationangleandisverygreat
compared to that of the orthotropic (0) beam. The relative factor
of safety for
anylaminationschemeiscalculatedbydividingthevalueobtaineduponsubstitutionofitsprincipal
stresses in the TsaiHill failure criterion by the value obtained
upon substitutionof the principal stresses of the orthotropic (0)
beam. It can be seen that, the orthotropic(0) beam is better
insofar as uniaxial bending is considered.3.1.2. Eect of taper
proleIt is shown in Tables 916 that, for increasing and decreasing
thickness variations, themaximum dynamic displacement and the
maximum dynamic normal stress obtained witha parabolic prole are
greater than those obtained with linear prole, whereas the
maximumshear stress obtained with parabolic prole is less. In the
case of a decreasingincreasingthickness variation, the maximum
dynamic displacements obtained with both proles arethe same,
whereas the stresses obtained with a linear variation are greater
due to the presences. r:x:iiNcrsv:r: r:o :Nb N. c:Nrs:N
752ofthestressconcentrationeectasthereisasuddenchangeincross-section.Thestressconcentrationeectbecomespredominantforataperparameter,
bq05.Inthecaseofdecreasingincreasingthicknessvariationthemaximumdisplacementandthemaximumnormal
stress obtained with both proles are more or less the same, but the
maximum shearstress obtained with a parabolic variation is greater.
Among all the variations considered,insofar as the rst resonance is
concerned, the decreasing-increasing thickness variationgiveshigher
maximumdynamic displacement andhigher maximumdynamic normal
stressFigure 8. As Figure 4, but decreasingincreasing
typelinear.1:irrrb coxiosi1r nr:x b.N:xic rrsioNsr 753Figure 9. As
Figure 5, but decreasingincreasing typelinear.whereas the
increasingdecreasing thickness variationgives lower
maximumdynamicdisplacement and maximum dynamic normal stress. Upon
comparison one can concludethat the increasingdecreasing with a
linear prole is better suited for simply supported
endconditions.3.1.3. Eect of taper parameterIt is shown in Tables
916 that, in the cases of both cross-ply and angle-ply beams,
highermodefrequenciesdecreasewiththeincreaseinthetaperparameterforanyparticularthicknessvariationconsidered.
Exceptingtheincreasingdecreasingtapervariation, forall
theotherthicknessvariationsofboththecross-plyandangle-plybeamsconsideredtherstnatural
frequencydecreasesasthetaperparameter bincreases.
Inthecaseofincreasingdecreasing thickness variation, the change in
the rst natural frequencyisminimumfor025Qbq05.
Fromthegraphspresentedonecanobservethat if thethickness of the
beamat any particular section increases the stresses developed at
that sectiondecrease and vice versa. Also, the locations of the
maximumdisplacement and the maximumstresses shift towards the
smaller cross-section end as the taper parameter increases.
Thepeaks of dynamic shear stress occur near points where the
bending moment is zero. In thecase of a uniform beam peaks of
dynamic shear stresses developed near the points wherethe bending
moment is zero are equal in magnitude. In the case of a tapered
beam, amongall the sections where the bending moment is zero
maximum dynamic shear stress would bes. r:x:iiNcrsv:r: r:o :Nb N.
c:Nrs:N 754developed at the smallest cross-section. Usually the
maximumdynamic shear stressdeveloped increases with increase in the
taper parameter.4. CONCLUSIONSAnumber of results
havebeenpresentedtoshowtheeect of taper prole, taperparameter
andmaterial anisotropyontheharmonicresponseof
laminatedcompositeFigure 10. As Figure 4, but increasingdecreasing
typelinear.1:irrrb coxiosi1r nr:x b.N:xic rrsioNsr 755Figure 11. As
Figure 5, but increasingdecreasing typelinear.beams. The validity
of the present results has been established by comparing results
withexisting results in the literature. From the results presented
it can be concluded that, insofaras uniaxial bending is considered,
the orthotropic (0) beam is better than all the other typesof beams
considered. It can also be seen that, among all thickness
variations considered theincreasingdecreasing thickness variation
with taper parameter equal to 05 would be betterfor simply
supported end conditions.REFERENCES1. R. K. K:i:Ni: and S. R:ci1i
1989 American Institute of Aeronautics and Astronautics Journal
27,935946. Recent advances in analysis of laminated beams and
plates, part II: vibrations and wavepropagation.2. A. K. Miiirr and
D. F. Ab:xs 1975 Journal of Sound and Vibration 41, 433449. An
analyticmeansofdeterminingtheexuralandtorsionalresonantfrequenciesofgenerallyorthotropicbeams.3.
J. R. ViNsoNandR. L. Sirr:iovsii 1986Behaviour of Structures
Composedof CompositeMaterials: Martinus Nijho. See pp. 139144.4. A.
T. CnrN and T. Y. Y:Nc 1985 Journal of Composite Materials 19,
459475. Static and dynamicformulation of symmetrically laminated
beam nite element for microcomputer.5. K. Cn:Nbr:snrin:r:, K.
KrisnN:xir1n. and S. Ro. 1990 Composite Structures 14, 269279.Free
vibration of composite beams including rotary inertia and shear
deformation.s. r:x:iiNcrsv:r: r:o :Nb N. c:Nrs:N 7566. K.
Cn:Nbr:snrin:r: and K. M. B:Ncrr: 1992 Computers and Structures
43(4), 719727. Freevibration of composite beams using a rened shear
exible beam element.7. J. N. Rrbb. 1984 Journal of Applied
Mechanics 51, 745752. A simple higher order theory forlaminated
composite plates.8. A. Bnix:r:bbi andK. Cn:Nbr:snrin:r:1991
Composite Structures 19, 371380. Someobservations on the modelling
of laminated composite beams with general lay-ups.9. S. Or:iAshear
exible nite element for non-uniformlaminatedcomposite beams.
1991Computers and Structures 38(3), 353360.10. G. B. W:rnir1oN 1976
The Dynamical Behaviour of Structures. Oxford: Pergamon Press.
Seepp. 131133.APPENDIX: FUNCTIONS FOR VARIOUS THICKNESS VARIATIONS
CONSIDERED1. Linearincreasing:f(x)=1x/L.2.
Lineardecreasing:f(x)=x/L.3.
Linearincreasingdecreasing:f(x)=62x/L2(1x/L)for 0ExEL/2,for
L/2ExEL.7.4. Lineardecreasingincreasing:f(x)=612x/L(12x/L)for
0ExEL/2,for L/2ExEL7.5. Parabolicincreasing:f(x)=1(x/L)2.6.
Parabolicdecreasing:f(x)=1(1x/L)2.7.
Parabolicdecreasingincreasing:f(x)=1((2x/L)1)2.8.
Parabolicincreasingdecreasing:f(x)=(12x/L)2.
