VIBRATION OF TAPERED BEAMS

By Arvind K. Gupta , 1 M. ASCE

ABSTRACT: Stiffness and consistent mass matrices for linearly tapered beam element of any cross-sectional shape are derived in explicit form. Exact expres­sions for the required displacement functions are used in the derivation of the matrices. Variation of area and moment of inertia of the cross section along the axis of the element is exactly represented by simple functions involving shape factors. Numerical results of vibration of some tapered beams are obtained us­ing the derived matrices and compared with the analytical solutions and the solutions based upon stepped representation of the beams using uniform beam elements. The significance of the severity of taper within beams upon solution accuracy and convergence characteristics is examined.

INTRODUCTION

Nonprismatic members are increasingly being used in structures for economic, aesthetic, and other considerations. Design of such structures to resist dynamic forces, such as wind and earthquakes, requires a knowledge of their natural frequencies and the mode shapes of vibra­tion. There is an ample bibliography on the subject of transverse vibra­tion of beams of variable cross sections (2). Prior published analytical results for tapered beam elements have been limited to a few cross-sec­tional shapes and simplest sets of end conditions. Approximate methods such as mass lumping techniques or modeling the tapered beams as sub-assemblages of uniform beam elements have been utilized in the past.

The computational efforts involved in computing the dynamic re­sponse of beams to arbitrary forcing functions, when required, can be minimized if stiffness and mass matrices of good accuracy are utilized. With the availability of the digital computers, stiffness matrices, appli­cable to a certain class of tapered beam elements, have been developed within the last two decades. This paper presents stiffness and consistent mass matrices for tapered beam elements of any cross-sectional shape. First, expressions for the displacement functions are derived from the equation of motion for the free undamped vibration of a beam of vari­able cross section. In this equation, the variable cross-sectional area and moment of inertia are represented by exact expressions using cross-sec­tional dimensions of the beam. The multiples of these displacement functions and their derivatives are integrated and the stiffness and con­sistent mass matrices are derived.

Vibration analysis results for some typical tapered beams are pre­sented using the derived matrices. These results are then compared with the available analytical solutions and the results based upon stepped representation of the beams using uniform beam elements. The cost and advantages and disadvantages of the tapered and uniform beam element

'Consultant Engr., Quadrex Corp., 1700 Dell Ave., Campbell, Calif. 95008. Note.—Discussion open until June 1,1985. To extend the closing date one month,

a written request must be filed with the ASCE Manager of Journals. The manu­script for this paper was submitted for review and possible publication on Jan­uary 20, 1984. This paper is part of the Journal of Structural Engineering, Vol. Ill , No. 1, January, 1985. ©ASCE, ISSN 0733-9445/85/0001-0019/$01.00. Paper No. 19426.

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formulations are discussed. The results showing significance of the se­verity of taper within beams upon solution accuracy are also presented.

LINEARLY TAPERED BEAM ELEMENT

The beam element is assumed to be associated with two degrees of freedom, one rotation and one translation at each end. The location and positive directions of these displacements in a typical linearly tapered beam element are shown in Fig. 1. Some commonly used cross-sectional shapes of beams are shown in Table 1 (1). The smaller end of the beam is denoted as end A and the larger as end B. The depth of the cross sections at ends A and B are da and db, respectively. The length of the element is /. The axis about which bending is assumed to.take place is indicated by a dashed line in the cross section of a beam as shown in Table 1. The only exceptions are those cross-sections in which the mo­ment of inertia is the same for every centroidal axis.

For most beam shapes the variation in cross-sectional area along the length may be represented as

Ay — Aa 1 + r-l

(1)

and the variation in the cross-sectional moment of inertia about the axis of bending as

A a (1 + r 1 >

U ^ q ^ ' " " »UV

t£

U 2 = q 2 e i u , t U 4 - q 4 e

i ii f t

FIG. 1.—Geometry and Sign Convention of Tapered Beam Element

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TABLE 1.—Tapered Beam Gross-Seettona! Shapes and Shape Factors

Shape (1)

Shape Factors

n (2) (3)

Jlf

U«ul~™

Wide-flange or I-section Constant dimensions b, tw, t/ Varying depth d Bending about horizontal axis

ilf -«

-L b t

—̂4

1«

Closed box section Constant dimensions b, tw, tf Varying depth d Bending about horizontal axis

Solid, rectangular section d Constant width b ,, Varying depth d

Bending about horizontal axis

i

! V ,

Solid, rectangular section d Constant width b

Varying depth d Bending about vertical axis

1 = T T Open-web section 1

fe=U L-S-J " f

Constant dimensions b, tf Varying depth d Bending about horizontal axis

FT

2.1 to 2.6

2.1 to 2.6

Tower section Constant areas concentrated near corners Varying dimension d

CD Solid, circular section Varying diameter d

i

I, !

d

d Solid, square section Varying dimension d

X 1 + r -

/

in which r = 1

(2)

(3)

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Ax and Ix are the area of cross section and moment of inertia at distance x from the small end A; Aa and Ia are the area of cross section and mo­ment of inertia at end A; Ab and Ib are the corresponding parameters at end B, and m and n refer to the shape factors that depend on the cross-sectional shape and dimensions of the beam. The shape factors m and n may be evaluated by observing that Eqs. 1 and 2 should give Ax = Ab and Ix = Ib when x = I. This condition gives

1 O S(A m = 7 — (4)

<db\ log

• * * and n = (5)

(db\ log

Thus, the shape factors can be found readily using the dimensions of the cross sections at the two ends.

Calculation of values of m and n from Eqs. 4 and 5 means that the expressions for Ax and Ix are exact at both ends of the beam. For beams of closed box or I-section (Table 1), it has been found that, at interme­diate points along the beam, Ax and Ix will deviate slightly from true values. The amount of this deviation is very small and for beams of all usual proportions, Eqs. 1 and 2 give values at every section along the beam within one percent of the exact values. The shape factors n and m are dimensionless quantities; n varies between the limits 2.1-2.6 for all practical shapes and tapers of beam elements of the closed box or I-section. For beam elements of other cross-sectional shapes shown in Ta­ble 1, Eqs. 1 and 2 are exact representations of the area of cross section and moment of inertia along their lengths.

EQUATION OF MOTION

According to the simple beam theory, the equation of motion for the free undamped vibration of a beam of variable cross section has the fol­lowing form (Fig. 1)

in which E = modulus of elasticity, Ax and Ix are given by Eqs. 1 and 2, p = mass per unit volume, uy represents lateral deflections of the vi­brating beam and t is the time. Small deformations of the center line of the beam are assumed in the derivation of this equation. Translation of the beam element is taken into account, assuming that, before defor­mation, the center line is straight and coincident with the x-axis of the coordinate system. Moreover, linear stress distributions and small ratio of height of cross section to span are assumed, thus allowing the influ-

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ence of shear forces on the vibrations of the beam element to be disre­garded.

The equation of motion of free undamped vibrations of a structural system in matrix form may be written as

M.U + KU = Q

in which, K = Stiffness matrix = I bjdb dv .

and, M = Consistent mass matrix = I qT p a dv

(7)

(8)

(9)

Here & and b are element deflection distribution and element strain dis­tribution respectively, due to unit values of nodal displacements.

The derivation of the matrices a and b is given in Appendix I. Using these matrices the stiffness and consistent mass matrices M and K for the tapered beam element are derived in Appendices II and III, respec­tively.

NUMERICAL EXAMPLES

Use of the stiffness and mass matrices derived above is illustrated in the vibration analysis of linearly tapered cantilever and simply-sup­ported beams of solid rectangular cross-section as shown in Fig. 2.

A finite element computer program HIGHER (Ref. 2) was written for vibration analysis of these beams. The modal frequencies computed by

h5:,

l"<2»54 CM)

2.25 ,'(5.7is CM)

L

1"(2.54 CM)

2.25"(5.715 CM)

L

FIG. 2.—Tapered Beams of Rectangular Section

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TABLE 2.—Frequencies of Linearly Tapered Cantilever Beam [Fig. 2(a)] Using Ta­pered Beam Elements

Num­ber of

ments

(D 1

2

3

4

5

Natural Frequencies, in Cycles per Second

1

(2)

154.81 (95.676) 152.55

(133.38) 152.29

(143.33) 152.24

(147.09) 152.23

(148.90)

2

(3)

1,316.2 (942.67) 682.13

(578.03) 679.77

(630.01) 677.93

(649.58) 677.21

(658.82)

3

(4)

2,095.2 (2,075.1) 1,725.5

(1,585.3) 1,715.4

(1,636.6) 1,710.3

(1,661.0)

4

(5)

5,574.2 (4,759.9) 3,746.8

(3,774.1) 3,301.8

(3,133.9) 3,263.1

(3,159.7)

5

(6)

7,178.7 (6,994.4) 5,954.8

(6,013.6) 5,433.4

(5,242.8)

6

(7)

11,981 (11,069)

9,695.7 (9,636.3) 8,685.3

(8,769.2)

7

(8)

15,728 (15,218) 12,945

(12,924)

8

(9)

20,548 (19,801) 18,939

(18,794)

9 (10)

' 27,294 (26,148)

10

(11)

31,743 (31,516)

Note: Numbers in parenthesis represent frequencies obtained using stepped representation.

this program for the beams are compared with the analytical solutions obtained by other investigators. The number of elements in the cantil­ever beam [Fig. 2(a)] was varied to study convergence of the modal fre­quency solutions. The simply-supported beam [Fig. 2(b)] was considered to study the severity of taper within the beam upon solution accuracy and convergence characteristics. Modulus of elasticity and mass density of these beams are assumed to be 30 x 106 lb/in.2 (2.109 x 106 kg/cm2) and 0.00073386 lb sec2/in.4 (0.0000079974 kg-s2/cm4), respectively.

Modal frequencies of the cantilever beam [Fig. 2(a)] corresponding to the 1, 2, 3, 4, and 5 tapered element idealizations, are given in Table 2, This table also contains the values obtained using stepped representa­tion. The basis of comparison of the solutions for the first two modal frequencies thus obtained is drawn from the results given by Wang (4) who employed a hypergeometric series procedure to solve the govern­ing differential equation of motion. The first two frequency values of the beam given by Wang are 152.0 and 681.0 which have been assumed to be the "exact" values for the present study. In Fig. 3 the percentage errors in the first and second mode frequencies (Table 2) are plotted versus the number of elements employed in the idealization in order to show more clearly the general trends in solution convergence.

A perusal of Table 2 and Fig. 3 reveals that, except for the one element idealization case in which error in the first mode frequency is about 1.6%, the error is nearly zero for 2, 3, 4, and 5 element idealizations of the beam employing tapered elements. When stepped representation de­fines the beam model, error in the first mode frequency varies from 37% for the one element idealization to about 2% for the five element ideal­ization of the beam. This shows that the rate of decrease in percentage error in the first mode frequency, when the number of elements is in­creased, is much greater when the tapered beam elements are used in­stead of the uniform elements to idealize the beam. In the second mode frequency, except for the one and five element idealizations of the beam, considerable improvement in accuracy is indicated from Fig. 3 when ta­pered beam elements are used instead of the uniform elements.

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or o a. LU

ui to < t-z LLl O CC Id

TAPERED BEAM ELEMENT

UNIFORM BEAM ELEMENT

-40

FIG. 3.—Percentage Error in Frequencies of Beam [Fig. 2(a)] versus Number of Elements

Two-element idealization of the simply-supported beam [Fig. 2(b)] is considered here to study the significance of the severity of taper within the beam upon the convergence characteristics of the solutions obtained using the tapered beam elements. Frequencies in the first three modes corresponding to the different tapers are given in Table 3. These values were obtained by employing the tapered and uniform beam elements. The basis of comparison of these solutions is drawn from the fifth order approximation of the first three modal frequencies given by Heidebrecht (3) who used Fourier expansions of the area and moment of inertia of cross section functions (Eqs. 1 and 2) for simply-support beams of rec­tangular cross section.

These "exact" values are also given in Table 3. Fig. 4 describes the percentage error in the predicted values of. frequencies (Table 3) in the three modes of vibration as a function of the taper.

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TABLI 3.—First Three Modal Frequencies of Simply-Supported Beam [Fig. 2(d)] for Various Tapers

Taper

(db l)

U v d) 0.1 0.2 0.3 0.5 0.7 0.9

NATURAL FREQUENCIES, IN CYCLES PER SECOND

USING TWO ELEMENT IDEALIZATION

Tapered Elements

Mode 1

(2)

168.22 175.35 189.12 202.33 215.09

Mode 2

(3)

745.21 778.27 843.64 908.14 971.95

Mode 3 (4)

1,880.9 1,955.6 2,116.5 2,275.3 2,431.7

Uniform Elements

Mode 1 (5)

160.77

174.37 186.69 198.08 208.76

Mode 2

(6)

712.36

782.52 854.31 927.13

1,000.5

Mode 3 (7)

1,787.5

1,942.4 2,086.4 2,223.4 2,356.0

"Exact" Values Given by Heidebrecht

Mode 1

(8)

160.32 167.62 174.73 188.45 201.62 214.34

Mode 2

(9)

641.49 671.26 700.57 758.08 814.33 869.58

Mode 3 (10)

1,443.29 1,510.1 1,575.9 1,704.9 1,831.2 1,955.6

It is observed from Fig. 4 that the solution error in all the three modal frequencies remains virtually constant, i.e., remains unaffected by the amount of taper when the beam is idealized by tapered elements. In contrast, the solution error varies considerably with the amount of taper when stepped representation of the beam is considered. It appears that the variation is linear for all the three modal frequencies, sloping in the

IE O C C LI

III

< H Z 111 O C LU

a.

.<£ THIRD MODE FREQUENCY

SECOND MODE FREQUENCY

FIRST MODE FREQUENCY

r___ 0.2 0.4 O.B TAPER

TAPERED BEAM ELEMENT

UNIFORM BEAM ELEMENT

FIG. 4.—Percentage Error in Frequencies Obtained from Two Element Solution of Beam [Fig. 2(b)] versus Taper

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TABLE 4,—-Computer Time to Calculate Frequencies and Mode Shapes of a Ta­pered Cantilever Beam [Fig. 2(a)]

Number of elements

(1)

1 2 3 4 5

Central Processer Time, in Seconds

Tapered elements (2)

0.46 0.54 0.78 0.99 1.39

Uniform elements (3)

0.41 0.52 0.69 0.88 1.22

same direction for alternate frequencies. Computer Costs.—The computer time required to calculate the fre­

quencies and mode shapes by the compiled computer program HIGHER and the related subroutines for the tapered cantilever beam of rectan­gular section [Fig. 2(a)] with varying degree of grid refinement is given in Table 4.

It is noted from Table 4 that the computer time needed in the vibration analysis of the beam idealized by the tapered elements is slightly higher than when it is idealized by the uniform elements. However, since the results obtained using tapered elements are considerably better than those obtained using uniform elements, the use of the former type of elements is expected to be more economical in the vibration analysis of beams and frames.

CONCLUSIONS

Stiffness and consistent mass matrices for linearly tapered beam ele­ment of practically any cross-sectional shape are presented here in ex­plicit form. The variation of the area and moment of inertia of cross section along the axis of the element is exactly represented by simple functions involving shape factors. Vibration analysis results obtained us­ing the derived matrices are compared with the available analytical so­lutions and the approximate solutions based upon stepped representa­tions of the beams using uniform beam elements.

The results of the investigation show that the convergence of the fre­quencies to their appropriate values, at the same level of grid refine­ment, is faster when the tapered beam elements are used instead of the uniform elements. It was further observed that the severity of taper within the beams has significant effect on the accuracy and convergence char­acteristics of the solutions obtained from the stepped representation of the beams and practically no effect on the solutions obtained from their tapered element idealization. This improvement in accuracy of the vi­bration analysis results of tapered beams using tapered beam elements may not be significant in analysis of beams where refined grids are usu­ally employed; however, it is of much greater importance in frame­works, composed of tapered members, where each member is usually represented by only one or two elements. This improvement in accuracy is achieved, of course, at the expense of more computation. This dis-

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advantage is largely offset by the reduced efforts required in data prep­aration and reduced number of specified displacements where tapered element idealization is used.

ACKNOWLEDGMENT

This paper is based on the author's Ph.D. thesis (Ref. 2). The financial assistance and computer funds for the graduate study, provided by the College of Engineering, Utah State University, are gratefully acknowl­edged. Dr. Vance T. Christiansen was the graduate adviser for this study.

APPENDIX I.—DISPLACEMENT FUNCTIONS

The displacement function for the tapered beam element are derived as follows.

Expansion of Eq. 6 yields

d% d2uy dlx S3uu d4w„ d2uv

dx2 dx2 dx dx3 dx4 dt2

Substituting the Eqs. 1 and 2 for Ax and Ixr respectively, in Eq. 10 leads to

n(n - 1) P2Zn ̂ + 2nPZ"+1 ^ + Z"+2 ^ = ~ Zm+2 ^ (11) ' dZ2 dZ3 az4 c 4 dt2 v '

in which P =-, Z = 1 + Px, (12a)

and C4 = — (12b) PA

The displacements ux, which occur along with the transverse displace­ments uy (Fig. 1) are given by

Buv duv y ux = y = -I —I n, in which r\ = ^- (13)

dx dx I

Now the solution for displacement u is assumed to be given by

u = qU = qq eM (14)

in which, assuming simple harmonic motion (Fig. 1)

U = {U1U2U3U4} = {q1q2q3q4} e'w (15)

q is the vector of nodal amplitudes in ith mode of vibration. Thus

ux = axq e'w (16)

and uy = ayq e"°f (17)

Substituting the expression for uy from Eq. 16 into the equation of mo­tion (Eq. 11) yields

Z"+2g7+ 2nP Z"+1q';+ n(n - 1) P2Z" a'y = 0 (18)

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. d4fl„ . , d3a„ , d2a., or, Z4 -T=% + 2nZ3 - f f + «(« - 1)Z2 - = * = 0 dz4 ' " '" rfz3 ' "v" *'" dz2'"v (19)

This is the Euler equation of the fourth order and its general solution is

C, + C2Z + C3Z (-"+3) + C4Z<~"+2) (20)

in which Cx, C2, C3, and C4 are arbitrary constants. The matrix a is now used to satisfy the following boundary conditions

Ay = \1\ at x = 0, a'y=U2 at x = 0

qy= U3 at x = Z, fly = IZ4 at x = Z

Successive application of these boundary conditions yields

Ci + C2 + Q + Q = Zii

C2 + ( - n + 3) C3 + ( - « + 2) Q = —2

(21a)

(21b)

(22«)

(22b)

d + (1 + PI) C2 + (1 + Pl)<-~"+3)C3 + ( 1 + PZ)(""+2)C4 = U3 (22c)

C2 + ( - n + 3)(1 + Piy~"+2)C3 + (-n + 2)(1 + PZ)(~"+1)C4 = — (22d)

Solution of these four simultaneous equations gives

gy = {[B18 + BI5Z + BUZB' - B10ZB>][B19 + B16Z - B13Z

B> + B9ZB>]

[1 - Bw - B15Z - B\2ZBi + B1QZB*][B20 + B17Z - BUZB> + BnZ

B>]} (23)

Substituting Eq. 23 into Eq. 13 yields

ax = {{-B15 - BuBiZ*-1 + B10B2ZB^] [-B16 + B13B1Z

B^1 - B9B2ZB^]

[Bis + BizBiZ81"1 - B10B2ZB'-1][-B„ + B iAZ 8 ' " 1 - BllB2Z

B^1]}r^ .... (24)

Here

Bt = -n + 3, B2 = -n + 2, B3 = l + Pl

B4 = Bf1,

B8 =

B5 = BB3\ B6 = Bf

1

B'j — Bo • B (-«+!)

B2 — B2B3 — 1 + B5 B2 — By

B, B, = -

B„ =

Bi — B1B3 — 1 + B4 Bi — B6

I - B 3 1

Bi — BjB3 — 1 + B4 Bi — B6

Bs

Bin — ' Bs

B] — 6163 — 1 + B4

•B12 -

B13 —

[P-M-BM Bio(B2 - B7)

B,-B6

B9(B2 - B7)

B i - B 6 P(B1-B6)'

1 Bu — + •

1 Bu(B2 - B7

Bi - B6 P ^ - B6)

(25a)

(25b)

(25c)

(25d)

(25c)

(25/)

(25g)

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Bis - —Bi• B\2 + B2• Bw, B16 — — + Bi• B13 — B2- B9 (25n)

B17 = Bj • B14 - B2 • B n , B18 = 1 - B15 - B12 + B10 (25r)

B19 = ~B16 + B13 — B 9 , B20 = - B i 7 + Bu - Bn, r = — - 1 (25/)

This completes the derivation of the displacement matrix a. With dis­placements known at all points within the beam element, the strains at any point can be determined as

dux d\ d\ e = — =--iy=-l—i^\ = bu (26)

to to to

Substituting Eq. 24 into Eq. 26 gives

b = {[B12B1(B1 - 1)ZB>-2 + BWB2(B2 - 1)ZB2"2]

[B^iBr - l)ZB l"2 - B9B2(B2 - 1)ZB2"2]

[BnB^ - l)ZB l"2 - BWB2(B2 - 1)ZB2~2]

[BUB1(B1 - l)ZB l"2 - BnB2(B2 - 1)ZB2-2]} r T, P (27)

The matrix b is utilized to determine the stiffness matrix of the beam element.

APPENDIX II.—CONSISTENT MASS MATRIX

The consistent mass matrix is obtained by substituting the transverse displacement ay (Eq. 23) into Eq. 9 as

M = I p aryay d Adx = p q^ayAxdx (28)

Jo JA Jo

in which Ax is given by Eq. 1. The coefficients of the matrix M in explicit form are given.

Mil = Bio5[Bi8 • B106 + B]5 • Bi07 + Bn • Bi0S + Bi0 • B109] (29a)

Mi 2 = Bi05[Bi9 • Bi06 + Bj6 • B107 - B i 3 • Bi08 - B9 • BW9] = M 2 1 (296)

M13 = Bi05[(l - B i8) • Bi06 - Bis • Bi07 - B n • B108 - B10 • B109] = M31.... (29c)

M1 4 = Bi05[B20 • Bi06 + B„ • B1Q7 - Bu • Blm - B„ • BW9] = Mu (29rf)

M 2 2 = Bi05[Bi9-Bii0 + B 1 6 - B m + Bi3-Bn2 + B9-Bn3] (29e)

M ^ = Bi05f(l - Bi8) • B 1 M - Bi5 • Bm + B12 • B„ 2 + B w • B113] = M 3 2 . . . . (29/ )

M2 4 = Bi05[B20 • B n o + B17 • Bm + Bu • B m + B„ • B n 3 ] = M 4 2 (29g)

M 3 3 = B105[(l - B18) • B U 8 + B15 • B„ 9 + B u • B120 + B10 • B n l ] (29/?)

M M = B105[B20 • B „ 8 - B17 • B119 + B M • Bi20 + B„ • B12i] = M 4 3 (29/)

M44 = BW5[B20 • B „ 4 + B17 • Bus + B14 • B „ 6 + B„ • B117] (29;)

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in which

(1 + f)'"+1 - 1 (1 + r)",+2 - 1 (1 + r)m+Bl+1 - 1 B61 = - '— , B62 = — — , B63 = l — ^ - - — - (30a)

m + 1 m + 2 (m + Bi + 1)

(1 + r)m+Bl+1 - 1 (1 + r)'"+3 - 1 (1 + r )

m + B l + 2 - 1 B« = , B69 = , B70 = (30b)

m + B2 + l m + 3 (m + Bl + 2) v '

(1 + jAm+B2+2 — 1 (1 + r\2Bi+m+l _ j B7i = , B76 = , (30c)

m + B2 + 2 2B1 + m + l K '

(1 + r)"'+Bi+B2+1 - 1 (1 + r)mi+2Bl+1 - 1 pA, B77 = —' B«2 = > Bi0S = —-• • • • • (30d)

m + Bi + B2 + 1 m + 2B2 + 1 P v ;

BW6 = BX8 • B6i + Bis • B62 + Bn • B63 - Bio • B64 (30c)

B107 = Bis • B62 + Bis • B69 + B12 • B70 - Bio • B7i (30/)

Bios = Bi8-B63 + Bis-B7o + Bi2-B76 - B10-B77 (30g)

B109 = _ Bi 8 • B64 - B15 • B7i - B12 • B77 + Bio • B82 (30ft)

B„o = B19 • B61 + Bi6 • B62 - B13 • B63 + B9 • B^ (30/)

Bui = B w • B62 + BX6 • B69 - B13 • B70 + B9 • B7i (30;)

B112 = _Bi9 • B63 — B16 • B70 + B i 3 • B76 - B9 • B77 (30fc)

B„3 = Bi9 'B6 4 + B16-Bn - Bi3'B77 + B9-B82 (30Z)

B114 = B20 • B6i + B17 • B62 - B14 • B63 + Bn • B64 (30m)

Bus = B20 • B62 + Bi7 • B69 — Bu • Bm + Bn • Bn (30«)

B116 = _ B 2 0 • B63 - B17 • B70 + B i4 • B76 - Bu • B77 (30o)

B117 = B20 • B64 + B17 • B71 — BJ4 • B77 + Bn • B82 (30p)

Bus = (1 _ Bis)' B6i - B15 • B62 _ Bi2B63 + Bio • B64 (30g)

B119 = _ ( 1 _ Bis) • B62 + B15 • B69 + B n • B70 - Bio • B71 (30r)

B120 = - ( 1 _ Bis) • B63 + B15 • B70 + B12 • B76 - Bio • B77 (30s)

B121 = (1 ~~ B18) • BM - Bi5 • B71 - Bi2 • B77 + Bio • B82 (30f)

APPENDIX III.—STIFFNESS MATRIX

Substituting matrix b (Eq. 27) into Eq. 8 and performing integration yields the coefficients of the stiffness matrix as given.

-K11 = Bi6o[B]2 • Bi6i + Bio • Bi62 — 2Bio • B12 • Bi63] = K33 = — K31 = — K]3 (31«)

^12 = B]60[—B12 • B13 • BWi + (Bio" B13 + B9 • B12)Bi63 — B9 • Bxo • Bi62]

= ^21 = _^32 = ~K23 (316)

^14 = B160[—B12 • BM • Bi6i + (Bio" BJ4 + Bn • Bi2)B163 — Bi0 • Bn • B162]

= Kn = -Ki3 = -KM (31c)

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^22 — Bi6o[Bi3 • B16i + B9 • Bx62 — 2 • B9 • B13 • B163]

^24 = Bi60[B13 • B14 • B16i — (B9 • B14 + Bn • Bi3)B163 + B9 • B n ' B162]

K42

K44 - B160[Bi4 • B161 + Bn • B162 - 2 • B14 • B n • Bi63]

E-r1

in which Bi, /2 •P-h

Bi6i » B ^ - l)2(B3Bl - 1)

o2B2-Bi B162 - ^ ^ ( B T - - 1)

Blfi3 -

(2B2 - BO

B.jB, - 1)B2(B2 - 1)

B, (B?2 - 1)

(31d)

(31c)

(31/)

(32a)

(32b)

(33c)

(32d)

It is to be noted that no numerical value is assigned to the shape factors m and n in the derivation of the stiffness and mass matrices given pre­viously. It is found that, w h e n an integer value is assigned to the shape factor n, as for categories 2 to 5 (Table 1), the equation of motion (Eq. 6) yields a different ay for each of these categories (Ref. 2).Because of ay being different, the constants B8 through B2 0 , B61 through B ^ , B69

through B7 1 , B7 6 , B7 7 , B8 2 , and Bm through B163 are also different for these categories. The expressions for these constants for the four cate­gories are given below. Corresponding expressions given previously can be used for the beams with noninteger value of the shape factor n (Cat­egory 1). By utilizing proper expressions for the previous constants, given in this paper, the stiffness and mass matrices, derived previously, can be used for the tapered beams of any cross-sectional shape given in Ta­ble 1. Following the procedure given in Ref. 2, the constants B8 through B2 0 , B61 through B ^ , B69 through B7 1 , B7 6 , 677, B82 and Bm through B163

can easily be derived for the tapered beam element of any other cross-sectional shape.

For category 2:

B8 + -1 + B3 2 — B3 — B3

B3 1 - B3 + In B3

B„ = :

1 + • l n B 3

1-B:

B 3 - l / Bio -

Bs

(1 - B3 + In B3)

(33a)

(33b)

BB-B3 _ (1 + B3

B11 ~ 77777 7T/ B12 — — Bw

B13 — —B 1 / B3

P KB, - 1 (33c)

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/ I + B3\ 2 / B3 \

Bu = - + B9 + B13, Bi7 = B11 + Bli, Bw = 1 - B15 + B10

-Bw - B9, B20 B17 - B n , B6i - - (B3 - 1).

1 D2 n2 1

Bffl = - ( B i - l ) , B63 = Y l n B 3 - ^ + 4 ' 6^ = 6 3 - 1 ,

1 o3 1 1 1 B69 = - ( B | - 1 ) , B70 = T

3 l n B 3 - - B l + - , B71 = -(B32-1)

4 j y y 2

B76 -Biln2B3 BilnB3 . B§ 1

+ , Br, = B3 In B3 - B3 + 1 2 4 4

(33d)

(33e)

(33/)

(33g)

(33/;)

(330

B82 = In B3, B161 = In B3, B162 = -2(B3"2 - 1), B163 = 2(B3-

J - 1) (33;)

For Category 3:

2(1 - B3) 1 - B3 =T, B9 =

B8

InB, B3 In B3 1 +•'

I - B 3

lnB3 B3 In B, ! + •

1 - B ,

1 _ B8 _ 2B10(1 - B3) \1 - B3 + B3 In B3J P In B3 In B3

B13 — 2B9(1 - B3) , 1

InB. + • PlnB "/ Bu ~ —

2B„(1 - B3) 1

lnB3 P l n B ,

(34a)

(34b)

(34c)

B15 - -Bn + 2B10, B16 - — + B13 - 2B9, B17 - B14 - 2Bn

Bis - 1 — B15 + B10, Bw - — B16 — B9, B20 - —Bn - Bn

B«°plf , n B >^K B« = — ( B J - 1 ) , B69 = B64, B70 = - - I n B 3 - - +

4P

1

PV4 16

B71 = — (B| - 1), B76 = — (B| In2 B3) (B|( In B3 - - J + -5P 4PV ' 8 P \ \ 4/ 4. B3/ 1\ 1 1 « B , = - l n B 3 - i U - , B82 = - ( B ! - 1 ) . . -

(34d)

(34e)

(34/)

(34$)

(34ft)

(34f)

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B161 = l n B 3 / B162 = 2(B32 - 1), B163 = 2(B3 - 1)

For Category 4:

Bs=r 1

1 + • l n B 3

1 - B 3

B -* - 8

l n B 3 B3 In B3 1 +

B3 In B3 In B3 03 i i ' 03 + 1 - B3 J

Bin — B> 1 - B3 + B3 In B

I - B 3 J

/ B u

B8 _ B ^ l - B J 1 ) '/ B12 - —

B,(l - B3-1) , 1 Dl3 - : Z r

InB. P In B3

B14 — —

P l n B 3

Bn(l - Br1)

l n B 3

InB , P l n B 3

B15 - —B12 + Bio / B16 - — + B13 — B 9 , B17 - BM — B n

Bis - 1 _ B15, B19 - —B16, B2o — ~ B 1 7 , B6j — — (B3 — 1 ) . .

B62 = -(B^-1), B- = p l ! ( l n B 3 4 ) + I 1

B64 = - ( B 3 l n B 3 - B 3 + l)

1 1 B69 = — (B3 _ 1)/ B70 = -

B*=Kf H 4 +i>

rs / — In B3 -L3 V

l \ 1 - - + -

3/ 9

B7fi — B| In2 B3 2

3P 9P Bi In B3 - - + -

B77 - • B?ln2B, 1 /Bi

2P

1\ 1 - p l y | I n B 3 - r l +

4 / '

(34;)

(35a)

(35b)

(35c)

(35d)

(35e)

(35/)

(35*)

(35h)

B82 = - (B3 In2 B3 - 2B3 In B3 + 2B3 - 2) .

Bwi = \ (B3 - 1), B162 = - 1 (B,-1 - 1), B163. = ~ (In B3)

For Category 5:

(350

(35;)

B8 = - 2 + 2B3"

3 3 - 2B3 - BJ2'

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1

1 - B 3 ~ 2 ~

1

1 - B31

1 + -I - B 3 J

^ " p T T n ^ ~ — j - T ^ r r (36a)

1 \ „ Bs B10(2 - 2B3~"3) Bio = Bs ? / B„ = -, B12 = — ~ (36b) 10 8 V2-B 3 -B 3 -V P ( - l + B3-

2)' 1-B3-2 ^ '

B9(2 - 2B3'3) 1 B„(2 - 2BJ3)

B l 3 ~ 1 - B 32 P(l-B3"2) ' "U~ I -B3- 2

1

P(-i + B;1)

Bt

(36c)

BM = " V " ' B<a = "^T~' Bs3 = ~^T~' 6̂4 = B3 - 1 (36rf) 3 4 z

_ Bj - 1 _ B J - 1 B69 - —- , B70 - - , B71 - B63, B76 - BM, B77 - In B3 . . . (36e)

D O

BB = 1 - B3-1, B161 = 4(1 - S3"1), B162 = 12(1 - B;%

B163 = 6(1 - B3'2) (36/)

APPENDIX IV.—REFERENCES

1. Gere, J. M., and Carter, W. O., "Critical Buckling Loads for Tapered Col­umns," Journal of the Structural Division, ASCE, Vol. 88, No. ST1, Paper 3045, Feb., 1962, pp. 1-11.

2. Gupta, A. K., "Vibration Analysis of Linearly Tapered Beams Using Fre­quency-Dependent Stiffness and Mass Matrices," thesis presented to Utah State University, Logan, Utah, in 1975, in partial fulfillment of the require­ments for the degree of Doctor of Philosophy.

3. Heidebrecht, A. C., "Vibration of Non-Uniform Simply-Supported Beams," Journal of the Engineering Mechanics Division, ASCE, Vol. 93, No. EM2, Apr., 1967, pp. 1-15.

4. Wang, H. C, "Generalized Hypergeometric Function Solutions on the Trans­verse Vibration of a Class of Non-Uniform Beams," Journal of Applied Mechan­ics, Vol. 36, 1967, pp. 702-708.

APPENDIX V.-—NOTATION

The following symbols are used in this paper:

A„ = area of cross section at small end A of beam element; At, = area of cross section at large end B of beam element; A* = area of cross section of beam element at distance x from

small end A; a. = element deflection distribution due to unit values of no­

dal displacements; qx = rotational deflection distribution due to unit values of

nodal displacements; gy = transverse deflection distribution due to unit values of

nodal displacements;

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array of constants; element strain distribution due to unit values of nodal displacements = -I d2ay/dx21\; arbitrary constants evaluated from boundary conditions; elasticity matrix containing appropriate material prop­erties; total depth of beam element at small end A; total depth of beam element at large end B; Young's modulus of elasticity; strains at any point within the beam element; moment of inertia of cross section at small end A of beam element; moment of inertia of cross section at large,end B of beam element; moment of inertia of cross section of beam element at distance x from small end A; subscript; stiffness matrix; length of beam element; consistent mass matrix; exponent in function for Ax, called shape factor; exponent in function for Ix, called shape factor; r/l; vector of nodal amplitudes; dimensionless ratio = db/da — 1; superscript indicating the transposition; time; nodal displacements; second derivative of nodal displacements with respect to time; internal displacements in the element; rotational internal displacements in element; transverse internal displacements in element; volume of element; distance from small end A along length of element; distance along y-axis from center line of element; dimensionless ratio = 1 + Px; y/b mass per unit volume; natural circular frequency of vibration; and subscript denotes matrices.

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