FEM Tapered Twisted Timeshenko Beams

7/28/2019 FEM Tapered Twisted Timeshenko Beams

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Journal of Sound and Vibration (1978) 56(2),187-200

FINITE ELEMENT EIGENVALUE ANALYSIS OF TAPERED

AND TWISTED TIMOSHENKO BEAMS

R. S. GUPTA

Department 0/Mechanical Engineering, Punjab EngineeringCollege, Chandigarh-Ti, India

AND

S. S. RAO

Department 0/Mechanical Engineering, Indian Institute ofTechnology, Kanpur-208016, India

(Received 13January 1977, and in revised form 14 September1977)

The stiffnessand mass matrices of a twisted beam element with linearly varyingbreadth and depth are derived.The angle of twist is assumedto vary linearly alongthelength of thebeam. The effects of shear deformation and rotary inertia are consideredin deriving the elemental matrices. The first four natural frequenciesand mode shapesare calculated for cantilever beams of various depth and breadth taper ratios atdifferentangles of twist. The results are compared with those available in the literature.

1. INTRODUCfION

The analysis of tapered and twisted beams has wide application in many industrial problems.

The vibration and deflection analysis of compressor blades, turbine blades, aircraft pro

peller 6lades, helicopter rotor blades, gear teeth, springs of electromechanical devices,

electrical contact switches, etc., all can be made by using such beam elements.

Tapered .beams have been analyzed by many investigators using different techniques.

Rao [1] used the Galerkin method to calculate the fundamental natural frequencies of

beams tapered in depth. Housner and Keightley [2] applied the Myklestad [3] procedure to

determine the first three modes of a tapered beam. Martin [4] obtained the frequencies of a

tapered beam using the assumption that the eigenvalues and eigenvectors of a tapered beam

can be expanded about those of a beam with zero taper in terms of the taper parameters. Rao

and Carnegie [5] used the finite difference method to obtain the frequencies and mode shapesof tapered blading. Mabie and Rogers [6, 7] solved the differential equation of vibration of

tapered beams with different boundary conditions using Bessel functions and tabulated the

results of the first five vibrational frequencies for different breadth and depth taper ratios.

In analyzing pretwisted beams, different approaches have been used by various investiga

tors. Mendelson and Gendler [8J used station functions while Rosard [9]applied the Mykle

stad method. The Rayleigh-Ritz method was used by Di Prima and Handelman [IOJ, Carnegie

[11] and Dawson [12]. Rao [13] analyzed pretwisted beams using the Galerkin method.

Carnegie and Thomas [14] used a finite difference procedure for the analysis of pretwisted

beams.

The finite element technique has also been applied by many investigators, mostly for thevibration analysis of beams of uniform cross-section. All these investigations differ one

from the other in the nodal degrees of freedom taken for deriving the elemental stiffness and

mass matrices. McCalley [15] derived consistent mass and stiffness matrices by selecting the

187


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188 R. S. GUPTA AND S. S. RAO

total deflection and the total slope as nodal co-ordinates. Archer [16] analyzed various beams

with specific boundary conditions. Kapur [17J took bending deflection, shear deflection,

bending slope and shear slope as nodal degrees of freedom and derived the elemental matrices

of beams with linearly varying inertia. Carnegie et al. [18] analyzed uniform beams by consi

dering few internal nodes in it. Nickel and Secor [19] used total deflection, total slope and

bending slope of the two nodes and the bending slope at the mid-point of the beam as thedegrees of freedom to derive the elemental stiffness and the mass matrices of order seven.

Thomas and Abbas [20Janalyzed uniform Timoshenko beams by taking total deflection, total

slope, bending slope and the derivative of the bending slope as nodal degrees of freedom.

In this work the finite element method is applied for finding the frequencies of natural

vibration of doubly tapered and twisted beams. The stiffness and mass matrices of the beam

element are developed by taking bending deflection, bending slope, shear deflection and

shear slope in two planes as nodal degrees of freedom. The effects of shear deformation and

rotary inertia, which are of significant importance at higher modes of vibration, are consid

ered in the derivation. The elemental matrices of a doubly tapered beam without pretwist

and a pretwisted beam without taper can be derived as special cases of the present matrices.The stiffness and mass matrices of a tapered beam and that of a uniform beam without

shear deformation are also special cases of the general matrices. The first four natural fre

quencies of vibration have been calculated for untwisted and a pretwisted doubly tapered

cantilever beams by using the finite element thus developed. The effects of variation of

depth and breadth taper ratios of the beam have also been studied. The results compare well

with those reported in the literature.

2. ELEMENT STIFFNESS AND MASS MATRICES

2.1. DISPLACEMENT MODEL

Figure I (a) shows a doubly tapered, twisted beam element of length Iwith the nodes as1 and 2. The breadth, depth and the twist of the element are assumed to be linearly

varying along its length. The breadth and depth at the two nodal points are shown as b., hi

and h2' h2' respectively. The pretwist angles at the two nodes are denoted by 01 and O2,

respectively (a list of notation is given in Appendix B). Figure 1(b) shows the nodal degrees

offreedom of the element, with bending deflection, bending slope, shear deflection and shear

slope in the two planes taken as the nodal degrees of freedom.

The total deflections of the element in they and x directions at a distance z from node I,

namely, w(z) and v(z), are taken as

w(z) = Wb(Z)+ w.(z), v(z) = vb(z) + vs(z), (I)

where wb(z) and Vb(Z)are the deflections due to bending in theyz and xzplanes, respectively,and w.(z) and vs(z) are the deflections due to shear in the corresponding planes.

The displacement models for wb(z), w.(z), Vb(Z)and vs(z) are assumed to be polynomials

of third degree. They are similar in nature except for the nodal constants. These expressions

are given by

- (u3/[2) (Z3 - 21Z2 + f2z) - (u4/fl)(Z3 - I

Z2), ws(z) =(u5/13)(2z3 - 31z2 - [3) + (u6/13){3/ Z2 -2z3)-

- (U7/[2)(Z3 - 21z2 + f2z) - (US/f2)(Z3 -lz2),

Vb(Z)= (u9/f3){2z3 - 31z2 _/3) + (ulof'/3)(3Iz2 - 2z3)- (ulllf2)(z3 - 2 Z2 + f2z) - (uul12)(z3 -

Iz2), vs{z) = (u131 3)(2z3 - 3 Z2_/3) + (uI4/[3)(3Iz2 -2z3)

- (ulsIP)(z3 - 21z2 +[2z) - (uI6112)(z3 - Iz2), (2)


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TAPERED AND TWISTED TIMOSHENKO BEAMS l89

(c)

Figure 1. (a) An element of tapered and twisted beam; (b) degrees of freedom of an element; (c) angleof twist O.

where Ill' liz, U3and 114represent the bending degrees of freedom and 115'116'U7and Us arethe shear degrees offreedom in the yz plane, U9, UIo, Uu and U12 represent the bendingdegrees of freedom and U13, 1114, UI5 and 1116the shear degrees of freedom in the xz plane.

2.2. ELEMENT STIFFNESS MATRIX

The total strain energy Uof a beam of length I, due to bending and shear deformation, isgiven by

u-_JI [{E-lx-:"(iF--Wb)2+EI"y--iF--+W-b-,J2Vb si; (0-2- Vb)2)

+f-lA-G{(ow-s-)Z+(O-Vs)2)] dz. (3)

2 OZ2 OZ2 OZ2 2 ozz 2 OZ OZo

As the cross-section of the element changes with zand as the element is twisted, the crosssectional area A, and the moments of inertia I;"",Iyy and I"y will be functions ofz:

A(z) = b(z) h(z) = {bl+ (b1- bl)zll} {hi + (hz - hi) zl/} = (llf2)(cl Z2 + C21z+ c3IZ),(4)

where

In(z) =I". x'cos" 0 +Iy. rsin?0,

Iyy(z) = Iy.r'cos" 0 + Ix'x' sin!

0,


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(5)

Ixiz) = (Ix x' - Iy.y)tsin 20, (6)


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+ + + +

4 3 2 2 3

190 . R. S. GUPTA AND S. S. RAO

wherex' x' andy'y' are the axes inclined at an angle e, the angle of twist, at any point in the

element, to the original axes xx and yy as shown in Figure l(c). The value ofIx' y' = 0 andthe values ofIx' x' and Iy'y' can be computed as

b(z)h3(z) 1 4 3 2 2 3 (Ix'x'(z) = 12 = 1214[alz a2 Iz a31 z a41 z as], 7)

where

Iy'y'(Z)=h(z).b3(z) I

z +d2lz +d31 z +d41 z+d 14]

(8)

(9)

where

=--4{dl 5 ,12 121

dl = (h2 - hI) (b2 - bl)3, d2 = hl(b2 - bl)3 + 3(h2 - hi) (b2 - bl)2b"

a,= 3{hlbl(b2 - bl)2 + (h2 - hI) (b2- b,) bi},(10)

By substituting the expressions forWb, Ws, Vb' vS'A, I;r:xtIx)'and In from equations (2), (4)and (6) in equation (3), the strain energy Ucan be expressed as

(II)

where u is the vector of nodal displacements U" U2, , U16, and [K] is the elemental

stiffness matrix of order 16. In terms of the integrals defined as

(12)

(13)

(14)

and

jsi; (a;;b) (a;;:)dz = [ul U2U3u4f[DK] [U9 UlO Ull

U12],

(15)

the element stiffness matrix can be expressed as

[AK] [0]

[K] = [0] [CK]

l6xl6 [DK] [0][

[DK]

[0]

[BK]

[0] ][0][0] , (16)

[0] [0] [0] [CK]

where [AK], [BK], [CK] and [DK] are symmetric matrices of order 4 and [0] is a null matrixof order 4. The elements of matrices [AK], [BK], [CK] and [DK] are formulated in Appendix

A.


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-

g ozot

l

TAPERED AND TWISTED TlMOSHENKO BEAMS 191

2.3. ELEMENT MASS MATRIX

The kinetic energy of the element T,including the effects of shear deformation and rotaryinertia, is given by

T=

I'[PA

(O-W+

b-

ows

)2+p-

A

(O-Vb-+

OV-

.)2 +pI-

n

(0-2-Vb)2+

2g ot ot 2g ot ot 2g ozoto

+-P1 (0-2-Wb)(0-2V-b)

+pl-xx (0-2W-b)2]dz

(17)

By defining

g xy ozot ozot 2g ozot .

(18)

II-PIn (0-2-Vb

2dz =

[U9 UIOUu u

n]T

[CM]

.[U9 UIOUl1 Ull]'

(19)

(20)

oand

(21)

where U, denotes the time derivative of the nodal displacement u" i = 1,2, ... , 16, the

kinetic energy of the element can be expressed as

where [M] is the mass matrix given by

[AM] + [BM] [AM1 [DM]

[M] = [AM] [AM] [AM]

i6x16 [DM] [AM] [AM] + [eM]

[0]].

[AM] ,

(22)

(23)

[0] [0] [AM] [AM]

and [AM], [BM], [CM] and [DM] are symmetric matrices of order 4 whose elements are

defined in Appendix A.

2.4. BOUNDARY CONDITIONS

The following boundary conditions are to be applied depending on the type of endconditions:

free end: ow./oz =0 and ov./oz =0; (24)

clamped end: w. = 0, Wb= 0, V. = 0, Vb= 0, owb/oz= 0 and ovb/oz= 0;(25)


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hinged end: w. = 0, Wb= 0, Vs= 0 and Vb= O.(26)

Itis to be noted here that all the forced boundary conditions could be satisfied by the presentmodel. Among the natural boundary conditions, if the condition of zero bending moment is

to be enforced at a free end, the element due to Thomas and Abbas [20] is expected to be

better than the present one.


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Number of elements

1

First mode

3048

Second mode

11870

Third mode

2259'3

Fourth mode

45192

2 298'7 1146'8 16853 4046'5

3 2981 1139'2 1652'2 36474

4 297'9 1137'9 1647-3 3593'5

5 2978 1137-5 1646'0 35856

6 2978 1137'4 1645-3 35788

7 297-8 1137-3 1645'1 3578'58 2978 1137'3 1645'0 3578'3

reference [21] 2991 1142'8 1653-3 35957

3. NUMERICAL RESULTS

The element stiffness and mass matrices developed are used for the dynamic analysis of

cantilever beams. By using the standard procedures of structural analysis, the eigenvalue

problem can be stated as

([K] - (1)[2M]) U = 0,

(27)

where [K) and [M] denote the stiffness and mass matrices of the structure, respectively, U

indicates nodal displacement vector of the structure, and (1)is the natural frequency of

vibration.

A study of the convergence properties of the element is made by taking the special case ofauniform beam with a length of 02540 m, breadth of 00762 m, depth of00704 m,E= 207 X1011 N/m2, G = 3E/8, mass density of 800 kg/m", J1 = 2/3 and 0 = 0. For this beam,the first, second, third and fourth natural frequencies obtained by the present method

(with 4

elements) have been found to have 0'00%, 007%, 030% and 060% errors, respectively.The first four natural frequencies obtained by using 8 elements are 845'8, 3989,5, 88368 and

138271 Hz while the exact values are 845,8, 3988'9, 8834'2 and 138181 Hz, respectively

[20]. The convergence of the natural frequencies of a pretwisted doubly tapered cantilever

beam has also been studied and the results are shown in Table I. In this case the natural

frequencies given by the method of reference [21] have been found to be slightly higher than

those predicted by the present method. It can also be seen that reasonably accurate results

can be obtained even by using four finite elements.

TABLE 1

Natural frequencies of a tapered and twisted Timoshenko beam (Hz)

According to the method of

Data: length of beam = 01524 m, breadth at root = 00254 m, depth taper ratio =2'29, breadthtaper ratio = 2'56, twist = 45, shear coefficient=0'833, mass density = 800 kg/ml, E= 207x 1011Nlm",G= (3/8)E.

Table 2 shows the frequency ratios of an untwisted tapered beam for various combinations

of depth and breadth taper ratios. Six finite elements are used to model the beam. It is

observed that for constant depth taper ratio the frequency ratio of all the four modes

increases with breadth taper ratio while for constant breadth taper ratio the frequency ratio

decreases for the first mode and increases for the second, third and fourth modes with an

increase in the depth taper ratios. The shear deformation effects reduce the frequency of

modal vibration. The present results can be seen to compare well with those reported by

Mabie and Rogers [6] which are also indicated in Table 2. Figure 2 shows comparison of

the results given by the finite element method with those reported by Rosard [9) for a

twisted beam of 00254 m x

0-00635 m cross-section and 02794 m length. It can be seen that the two sets of results are


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193 R. S. GUPTA AND S. S. RAOquite comparable.


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TABLE 2

Comparison of 'frequency ratios of untwisted tapered beams

Depth taper ratio ex= 3 Breadth taper ratio P= 3

P=lMethod] P=I P=2 P=3 P=4 P=5 0:=1 ex=2 ex=4 ex=5 0:=1

First (a) 1-1452 13741 1'5144 16124 1'6849 1'3668 1'4556 1'5582 1'5920 1mode (b) 1'1454 1'3743 1'5148 1-6124 16850 1'3668 14556 1'5584 1'5923 1

(c) 1'1369 1'3626 1'5010 1'5971 16680 1'3505 14413 15444 1'5786 09905

Second (a) 4'8104 5'1337 5-3381 5'4905 5'6125 6,9511 5'7789 5-1073 4,9679 62670

mode (b) 48124 5,1361 5,3404 5'4928 5'6]49 6,9528 57805 4,1103 4,9715 6,2685

(c) 4'6650 4'9759 5'1708 5'3156 5'4313 6'5228 5'5543 4'9670 48432 5-8850

Third (a) 11'8752 122260 12'4526 126318 12'7788 18'2701 14'0614 11,5805 110251 17'5484

mode (b) 1I 8957 12,2477 124771 126549 12-8024 18'2998 14'0867 116056 110554 17'5797(c) 11'1177 11,4410 116491 11,8091 119413 159423 12'9227 109327 10,4713 15'3303

Fourth (a) 22-3834 22,7437 229902 23-1798 23'3410 351250 263652 21,1320 19'9375 34'3857mode (b) 22'5186 22-8936 231429 23-3399 23'5060 35'3637 26'5329 21'3052 20'1406 34'6158

(c) 200672 203905 20'6016 20'7669 20'9054 28'1545 229438 19,2481 183606 27'5757

Data: length of beam = 0254 m, breadth at root = 0'0762 m, depth at root = 0,0254 m, shear coefficient =0'833, mass density =800 kg/rn", E= 2'07X

lOllN/ml,

G= 3/8 E.t Method (a): Mabie and Rogers [6].

Method (b): Finite element without shear deformation effect-Method (c): Finite element with shear deformation effect.


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3

2 First mode

o 30

Angle of twist (degrees)

Figure 2. Comparison of results for an uniform twisted beam. ----, Values by Rosard method; --,values by present method.

Figures 3 and 4 show the variation of modal frequencies with breadth taper ratio for beams

having 0,30,60 and 90 twist with constant depth taper ratio while Figures 5 and 6 show

similar variations for beams with constant breadth taper ratio and varying depth taper

ratio. Here the length of the beam is taken as 0254 m and the root cross-section as 0076 x0038 times the length of the beam. Again the effects of breadth and depth tapers are seen to

Breadth toper ratio

Figure 3. Effects of shear deformation and breadth taper ratio on the first and second naturalfrequencies of a twisted beam. e, Method of reference [22]; --, without shear deformation; ----,Timoshenko beam; depth taper ratio ex=3.


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'"

195TAPERED AND TWISTED TIMOSHENKO BEAMS

4LI------2~----~3------74----~5

Breadth toper ratio, f3

Figure 4. Effectsof shear deformation and breadth taper ratio on the third and fourth naturalfrequencies of a twisted beam. --, Without shear deformation; ----, Timoshenko beam; depth taperratio IX=3.

be pronounced at higher modes of vibration. Here also the effect of shear deformation isseen to reduce the modal frequencies at higher rates in higher modes of vibration in all the

cases. The results found by the method of Carnegie and Thomas [22}for the first two natural

frequencies are also indicated in Figures 3 and 5. It can be seen that the present results

compare excellently with those of Carnegie and Thomas.

4,-----,------,------,-----,

3

.Q

~>-uc::>CT

L'Z

"

2

5

Depth toper ratio, a

Figure 5. Effectsof shear deformation and depth taper ratio on the first and second natural frequenciesofa twisted beam. --, Without shear deformation; ----, Timoshenko beam; breadth taper ratio /3= 3.

_, Method of reference [22].


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4~1-----72----~~----74----~5

Depth toper rotio

Figure 6. Effectsof shear deformation and depth taper ratio on the third and fourth natural frequenciesof a twisted beam. --, Without shear deformation; ----, Timoshenko beam; breadth taper ratiop= 3.

4. CONCLUSION

The finite element procedure developed for the eigenvalue analysis of doubly tapered and

twisted Timoshenko beams has been found to give reasonably accurate results even with four

finite elements. The effects of breadth and depth taper ratios, twist angle and shear deforma

tion on the natural frequencies of vibration of cantilever beams have been investigated.

The present results are found to compare very well with those reported in the literature. The

element developed is expected to be useful for the dynamic analysis ofblades of roto-dynamic

machines.

REFERENCES

1. J. S. RAo 1965 Aeronautical Quarterly 16, 139-144. The fundamental flexural vibration ofcantilever beam of rectangular cross-section with uniform taper.

2. G. W. HOUSNERand W. O. KEIGHTLEY1962Proceedings of the American Society of CiuilEngin- eers 88, 95-123. Vibrations of linearly tapered beam. .

3. N. O. MVKLESTAD1944Journal0/Aerospace Science 2,153-162. A new method forcalculating natural modes of uncoupled bending vibrations of aeroplane wings and other typesof beams.

4. A. I.MARTIN1956Aeronautical Quarterly 7, 109-124. Some integrals relating to the vibrationof a cantilever beams and approximations for the effect of taper on overtone frequencies.

5. J. S. RAO and W. CARNEGIE1971Bulletin 0/Mechanical Engineering Education 10, 239-245.

Determination of the frequencies of lateral vibration of tapered cantilever beams by the use ofRitz-Galerkin process.

6. H. H. MABIE,and C. B. ROGERS1972Journal of the Acoustical Society 0/AmericaS1, 1771-1774.

Transverse vibrations of double-tapered cantilever beams.7. H. H. MABIEand C. B. ROGERS1974Journal of the Acoustical Society 0/America 55, 986-

988.Vibration of doubly tapered cantilever beam with end mass and end support.

8. A. MENDELSONand S. GENDLER1949 NACA TN-2185. Analytical determination ofcoupled bending torsion vibrations of cantilever beams by means of station functions.

9. D. D. ROSARD1953Journal0/Applied Mechanics 20, 241-244. Natural frequencies of twistedcan tilevers.

10. R. C. OJ PRIMAand G. H. HANDELMAN1954 Quarterly on Applied Mathematics 12,241-


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195TAPERED AND TWISTED TIMOSHENKO BEAMS259.

Vibration of twisted beams.


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f

TAPERED AND TWISTED T1MOSHENKO BEAMS 197

11. W. CARNEGIE1959 Proceedings of the Institute of Mechanical Engineers 173, 343-346.Vibration of pretwisted cantilever blading.

12. B. DAWSON1968Journal of Mechanical Engineering Science 10, 381-388. Coupled bendingvibrations of pretwisted cantilever blading treated by Rayleigh-Ritz method.

13. J. S. RAo 1971Journal of the Aeronautical Society of India 23, 62-64. Flexural vibration ofpretwisted beams of rectangulc r cross-section.

14. W. CARNEGIEand J. THOMAS1972Journal of Engineering for Industry, Transactions ofthe.

American Society of Mechanical Engineers 94, 255-266. The coupled bending-bending vibrationof pretwisted tapered blading.

15. R. MCCALLEY1963 General Electric Company, Schenectady, New York, Report No. DIG/SA,63-73. Rotary inertia correction for mass matrices. .

16. J. S. ARCHER1965American Institute of Aeronautics and Astronautics Journal 3, 1910-1918.Consistent matrix formulations for structural analysis using finite element techniques.

17. K. K. KAPUR1966Journal of the Acoustical Society of America 40, 1058-1063. Vibrations of aTimoshenko beam, using finite element approach.

18. W. CARNEGIEJ, . THOMASand E. DOCUMAKI1969 Aeronautical Quarterly 20, 321-332.An improved method of matrix displacement analysis in vibration problems.

19. R. NICKELand G. SECOR1972International Journal of Numerical Methods in Engineering5,243-253. Convergence of consistently derived Timoshenko beam finite elements.

20. J. THOMASand B. A. H. ABBAS1975Journal of Sound and Vibration 41,291-299. Finiteelement model for dynamic analysis of Timoshenko beam.

21. J. S. RAo 1972Journal of Engineering for Industry, Transactions of the American Society ofMechanical Engineers 94, 343-346. Flexural vibration of pretwisted tapered cantilever blades.

22. W. CARNEGIEand J. THOMAS1972Journal of Engineering for Industry, Transactions of theAmeri can Society of Mechanical Engineers 94, 367-378. The effects of shear deformation androtary inertia on the lateral frequencies of cantilever beams in bending.

APPENDIX A: EXPRESSIONS FOR [AK), [BK), ... , [DM)

The following notation is used for convenience:

I

U, = Zl-I dz, i= 1,2, ...,n, (AI)o

L,= 1'-1, i = 1,2, ...,n,

v,=j zl-lcos2 [(02 - OI)Y + 01] dz,

s, =jzl-l sin2 [(02 - 01) T + OI]dZ,

(A2)

i= 1,2, ... ,n; (A 3)

i= 1,2, ...,n, (A4)

where 01 and O2denote the values of pretwist at nodes I and 2, respectively, of the element.

As Wb, w., Vb and Vs are all the same in nature except for their positions in the stiffness andmass matrices, one can use IV to denote anyone of the quantities Wb, IV., Vb or Vs and in a

similar manner the set (17.,172,173,174) can be used to represent anyone of the sets (11.,112,113,114),

(lIs,U6,1I7,lIS), (1I9,UlO,Ull,U12) or (1I13,1I14,1IIS,1I16T)'hus

17 U3 a, ii4w(z) =-(2z2 - 31z2 + 13)- - (Z3 - 21z2+ f2z) + -(3/z2 - 2z3) - - (Z3 -

/Z2)

t f2 f3 f2'

(A 5)

-dw=

-

UI (6z

2-6Iz)--(3z

U3 2

-4Iz+1

2) +U-2(61z-6z)--2(3z U4 2

-2Iz)(A6)

dz f3 f2 [3 [2 '

d2w ~ . ~ Ul U4

dz2


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= 13(I2z - 61) - f2(6z - 41) + 13 (61 - I2z) - f2(6z - 2/).

(A7)


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By letting PI.1.k(i = I, _.. , 4;j = i, ... ,4; k = 1, . c ,7) denote the coefficient ofzk-1J1-kfor the 1I1iil term in the expression of Ip2, QI. J. k(l = I, , 4;j =i, , 4; k= 1, _. _,5) the coefficient ofZk-l/5-k for the ii,llJ term in the expression of (dlv/dz)2,R,.l. k(i = I,

_._,4;j =i, , 4; k= I, , 3) the coefficient ofZk-l P:"for the ill LlJ term in the expression

of(d 2Ir'/dz2)2, H,.j(i = I, , 4;j = i, .. _,4) the index coefficient of J to account for thediffer-

ence in index of Idue to multiplication of rotational degrees of freedom !II and ii2 and thedisplacement degrees offreedom Ll3 and ii4, the values ofPIJ.b Q,.J.k, R,.J.k and HIi can

be obtained as shown in Tables Al and A2_

TABLE Al

Values ofHI.}> RIJk,Q,.).k

R,.J.tfork= Q,.).t fork=

j H,.) 2 3 1 2 3 4 5

1 1 0 144-0 -144,0 360 360 -72-0 360 60 001 2 0 -144,0 1440 -360 -36,0 72-0 -36,0 00 001 3 1 -72,0 840 -240 -18,0 420 -30,0 60 001 4 1 -72-0 600 -12,0 -18,0 300 -12,0 0'0 002 2 0 1440 -144,0 360 360 -72-0 36'0 00 002 3 1 720 -84,0 240 18:0 -420 300 -6,0 002 4 1 720 -60,0 120 180 -30,0 120 00 00

3 3 2 360 -48'0 160 90 -240 220 -8'0 1'03 4 2 360 -36'0 80 90 -180 110 -2'0 004 4 2 360 -24,0 40 90 -120 40 00 00

TABLE A2

Values ofP,.l.k

P',J.t fork=

1

j

1 40

2

-120

3

90

4

4'0

5

-6'0

6

00

7

101 2 -40 120 -9,0 -20 30 00 00

1 3 -20 70 -8,0 20 20 -10 00

1 4 -20 50 -3,0 -10 1'0 00 00

2 2 40 -12,0 90 00 00 00 00

2 3 2'0 -70 80 -3,0 00 00 00

2 4 2'0 -5,0 30 00 00 00 00

3 3 10 -4'0 60 -4,0 10 00 00

3 4 10 -30 30 -10 00 00 00

4 4 1'0 -20 10 00 00 00 00

EVALUATION OF [BK]

As the procedure for the derivation of [AK], [BK], ... , [DM] is the same for each, the

expression for [BK] is derived here as an illustration. One has


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199 R. S. GUPTA AND S. S. RAO(A 8)


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+

where W= Vb and

TAPERED AND TWISTED TIMOSHENKO BEAMS 199

Putting the value ofIyy and w in equation (A8) gives

iiI ii3 ii2 a, ]2X J3 (12z - 6/) - P (6z - 4/) 73(61 - 12z) - [2 (6z - 21) dz, (A9)

[

with

BK11=coefficient ofiiI iiI ==coefficient ofU9U9

= 12~10Ill,.} ![((a1Z4 + a21z3 + a3 J2Z2 + a4 J3z+ as 14) +

+ d1z4 + d21z3 + d3 [2Z2 + d4J3 z+ ds) - (alz4 + a21z3 + a3 J2Z2 +

+a4J3 z+ as 14 cos? 02 - 01)]-+ 01)} {R111Z2 + R1. i.2lz + R1.13 F}]dzE S

= 12[10 L(Ht.t+1) 1~1{a,[R11. 1US-I + L'+1 R1.1. 2 U7-1 + LI+2 R1.

1.3 V6_,] +

This relation can be generalized as

1= 1, ... ,4, J=/, ... ,4,

Similarly

1= 1, ... ,4, J= 1,....,4. (All)

1= 1, ... ,4, J= I,...,4, (A12)


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3 7

S s

S 5

5 5

1 = I ... 4. J = 1; ... 4. (AI3)

AMr.J = :8LL'=1 J-I

[C,LCI.J+Ht.JIUOI-I-JIPr.J.Jl.

1= 1... 4. J=I ...4.

(AI4)

I= 1...4. J= 1...4.(AI5)

BMr.J = 12;[10 L L [{d,L(l+J+Ht.JI U(Il-'-JI Qr.J.J} +'_I J=I

1= 1.. 4. J=I ... 4. (AI6)

CMr.J = 12;/10L L [{a,L(l+J+lIt.JI U(Il-'-JI Qr.J.J} +'-I J=1+ (d, - G,HL(l+J+Ht.JI V(II-I-Jl QI. J. j}l. I= I ... 4. J = I...4.(A 17)

DM1,J = 12;/10 L L[(al - d,)L(I+J+Ht.J) S(1I-1-J)Qr.J,Jl.

1=1 J=1

I= 1..... 4. J =I...4.

(AI8)

APPENDIX B: NOMENCLATURE

A area of cross-sectionb breadth of beamE Young's modulusg acceleration due to gravityG shear modulush depth of beam

I"",1",1xy moment of inertia of beam cross-section about xx, yy andxy axis. respectively[K] element stiffnessmatrix

I length of an element

L length of total beam[M] element mass matrix

time parameteru nodal degreesof freedomU strain energyv displacementin xzplanew displacementin yzplane

x.y co-ordinate axesz co-ordinate axis and length parameter

frequencyratio ratio of modal frequency to frequency of fundamental mode of uniform beamwith the same root cross-section and without shear deformation effects

a depth taper ratio. = hl/h2P breadth taper ratio.

=b1/b2

o angle of twistp mass densityu shear coefficient

Subscripts: b,bending;s, shear.

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FEM Tapered Twisted Timeshenko Beams