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i (Siilitmlüa ttnturmitii in Ihr Clüu of ü\"rui t'nrh

DKPARTMhNT Ot ( IV11. LNl.lM.l.RIM

TIMOSHENKO'S SHEAR COEFFICIENT

FOR FLEXURAL VIBRATIONS OF BEAMS

hy

R D MINDLIN and H 1)1 RI.MIiWK /

Office nt Naval Research Prnj<tt NROol SHS I

Contract Nonr-*66((W)

Technical Report No 10

CD !1 -MONR 2M>(09) ( I

June 19 V*

1 I*

JJ

TP1POTWS 9HEAP KP* \k VIBiUTKliS OF

Introduction

rj r

; I

Ihe range of applicability of the rmp-d linens lot a 1 theory of flex-

ural vibrations of beams was extended to higher frequencies by Tlmoshenko

when he took into account the effect of transverse shear deformation. He

arrived at the free-vibration equations (l)#

m

governing the transverse deflection) y , and the slope» <f> , of the deflec-

tion curve when the shear is neglected. The coefficient n. is defined as

the ratio of the average shear stress on a section to the product of the

shear modulus and the angle of ahear at the neutral axis. This ratio de-

pends upon the distribution of shear stress on the section and, hence, -fc,

depends upon the <<hape of the section, as Tlmosnenko observed. However,

the distribution of shear stress on a section depends also on the mode of

motion of the beam. For example, for low modes of motion of a slender

beam, the shear stress has a maximum at the neutral axis, while, for very

high modes, th« shear stress has a minimum at the same place. Thus, j|,

depends both on the shape cf the section and the frequency of vibration.

Numbers in parenthesis refer to Bibliography at the end of the paper.

Ul

r jf-H ■r I

» ' '

In the solution of Equations [1], the simplest interpretation of

A ia that it ia a constant. For a beam of given cross-sectional shape«

then, A. oan have the correct value for only one frequency. Tn the past,

several calculations of A have been wade, for various cross-sectional

shapes, on the basis of statical considerations, that is, at aero fre-

quency. These values are satisfactory for the low modes of notion of

slender beans, where, in fact, the influence of transverse shear deforma-

tion is small in comparison with its influence at high frequencies.

As the frequency is increased, a point is reached at which a dras-

tic change occurs in the spectrum. This is the frequency, u) , of the

first thickness-shear mode of a bean of infinite length, i.e., the lowest

frequency at which th> infinite beam can vibrate with no transverse de-

flection, the displacement being entirely parallel to the axis of the

bean.

The reason for the change in character of the frequency spectrum

is that, at the frequencies of the first thiclcness-shear mode and its

overtones, there is strong coupling between the flexural and thiclcness-

shear modes of motion. Examples of this phenomenon have been described

in detail tn previous papers (2, 3) for special esses of crystal plates

for which the equations of motion are the sar*e AS those for a Tinoshenko

beam.

It is desirable, then, that the thickness-shear frequency calculated

from Equations [1] should be the same as that calculated from the equations

to which [1] are an approximation, namely, the three-dimensional equations

of small vibrations of an elastic body. In this connection it must be recog-

nized that, whereas Timoshenko interpreted ^ as the slope of the deflection

curve when the shear deformation in neglected, the product of ^ and the depth

of the beam may be interpreted a? the maximum axial displacement of a transverse

u -2-

r section. Thus, ul' is calculated from [1] by setting j/ = 0 and ^

proportional to €#f(<.cit't) , following which a/ is set equal to the cor-

responding frequency calculated from the three-dimensional equations.

Since a/ depends upon \ , the result is a formula for 4. •

It has been shown U) that, for a rectangular section, this pro-

cedure leads to n.« try/2 & 0.822 . Comparison «ith experiments

(2, 3) shows that, when 4. is calculated in this manner, Tlmoshenko's

equations give good results for both low and high frequencler.

In the present paper, A. i<* calculated in the same manner for a

variety of cross-aectional shapes. To do this, it is necessary to solve

the three-dimensional equations of elasticity for the appropriate fre-

quency and equate it with the value of 0> obtained from Equations [1].

In the case of motion parallel to the axis of the bar, the three-dimen-

sional equations ax 1 boundary conditions reduce to equations governing

a familiar hydrodynamical problem for which many solutions are known.

The determination of n. 1» thus reduced to an Interpretation of these

solutions and some additional computations. Results are given for the

following sections: circle, ellipse, orthogonal parabolas and a variety

of ovalolds. The values of It computed for these sections all lie within

about lOJl of that for the rsctangular section.

fit

\

Thlt.knes8-ai*ar Hotlont TiatoabeqltQ Theory

If u is set equal to zero in Equations [1], the second of the

equations gives "b^föt = 0 , so that the first of [1] reduces to

%>m-° [2]

-3-

r

i

whem c • (<£*/*) y it«»» tha velocity of shear waves in an infinite,

Isotropie, elastic medium, and f ■ (I/A ) , i.e., the radiue of gyra-

tion of the oross-seotion. Hence, the frequency of pure thlckness-shsar

vibration of a Timoahanko bean is

«•= c*'y> [3]

This la the frequency which is to be equated to the one calculated fron

the solution of the three-dimensional equations for each section.

IMflhrift-fr—1 Motion i Three-Dlmenalonal Theory

Let the nvutral axis of the beam he the z -axis of a rectangular

coordinate system and let u . v »nd ** ^e components of displacement in

the & , u and e directions, respectively. For pure thickness-shear

motion,

u s \r * 0

K]

Then the thre: -^ilmcnsional equations of motion (!>) reduce to

where «^= %*»/: and V'» "t*/^*1 * * /^jf * . The condition that the

traction vanish on the cylindrical surface of the b?am reduces to

l//3r7 -- 0 [6]

on the boundary, where n in the normal to the boundary.

The differential equation [5] and the boundary condition [6] are

the same as those governing the small oscillations of a fluid in a basin

-L- J

a-

of uniform depth (6) and the small vibration« of a pas in a ripid cylin-

drical container (7). Solutions are available for a variety of boundaries.

Rectangle

In the case of a rectaninilar section, the frequency is independent

of the width. If the depth 1 s 2a , the frequency is (6)

CO « TTc/lA [7]

Equa" ing [7] and [)], and noting that rls cff3 » the shear constant, A. ,

is found to be jrX//Z , or, approximately, 0 8tt ■

Circle

For a circular section of railius a , the lowest antisymmetric mode

has a frequency (6)

at = I.B+I C/A rg-.

The radius of pyration of a circle about a diameter is a/t . Hence,

equating [8] and [?], £ r 0 $¥7

The elliptic section was treated Ly Jeffreys (8), who computed

frequencies for two values of the eccentiicity

where a and ^ are the semi-major and semi-minor axes, respectively.

-5- [

J

1t ■

! Additional values wer« computed and are listed In the columns headed u>a/c

and cuh/c In Table I. The corresponding values of CO were then equated

to that In Equation [l] to obtain the values of A listed in Table I.

It is interesting to notice that the frequency ( / S8& c/« ) of

the first antisymmetric mode of a very narrow ellipse, about Its minor

axis, is greater than that {iSllc/m ) of a rectangle of depth equal to

the major diameter of the ellips*. »affrays observed that this is due to

the concentration of motion near the center of the ellipse. On the other

hand, the corresponding frequency for a vary wide ellipse appears to ap-

proach that of the rectangle. However, the corresponding values of -4-

are not the same because of the difference« in radii of gyration.

The hyirodynamic problem for a symmetric section bounded by a pair

of orthogonal parabolas was studied by Hidaka (9). Such a section has

the property a/h ■ 2 , where 0 and b are the semi-major and semi-minor

axes, respectively. Foi motion antisymmetric with respect to the minor

axis, Hidaka found a secular equation which can be reduced to

I l

where S(t) is the Bessel function of the "irst kind. The lowest root

of Equation [10] is (10)

fa/l = I-05SS [n]

so that

u> = l.m c/a [12]

-6- 1

[~r~ ++%0Stt*- n ■ i

Equating [12] and [3] and noting that r1« a*/s , the result la A « 0 S96

This completes the calculations for sections for which exact solu-

tions are available. The rrsuits are assembled in Table I and the sec-

tion are illustrated in Fig. 1.

-

Qvalolds

Approximate frequencies, good for narrow sections, may be obtainea

by neglecting the variation of displacement acrofls the width. Hidaka (9)

did this for sections oounded by *he curves

l/tm

[131

f

I

i - - - L

A set of '.hese curves, for the case a*Zb , is shown in Fig, 2.

Hitlaka's results for the lowest mode, antisymmetric about the hor-

i'^ntal axiJ of Kip. ?, are piven in Table II, alone with the correspond-

ing values of 4. . In finding 4. » the radius of gyration (r ), defined by

rL * a _

/y *

[U]

must te computed. Inserting y from Equation [13] into Equation [H], and

making the substitution i * a sin 6 n the inteprals? we find

r

a

J OS 0 J6

COS 0 dO 6 I

[15]

-7- * J

4 I By aid of the formula (11)

r/t

f a****** „ H /r r(*&) 2 r(l+,)

n > -I

-f*--

and the recursion formula for the Genua, function, Equation [15] reduces to

r*. al/(3 ♦ tm) [16]

The . Jsuits in Table II are the same as in Table I for the rectangle

{ji ■ «e ) for any b/a since the assurapticn of uniform displacement across

the width is exact in pure thickness-shear notion. For the ellipse ( /t */ ),

Table II give« the same value as Table I only for b/* -O (i.e., e-/ ),

since here, too, there is no variation of displacement -».cross the width.

As the section becomes wider, there is some discrepancy, but it is not

great. Even for * -b (the circle), Table I gives 4L - 0.847, while the

approximation yields 0.889. Th'1 approxiration Is not intended, of course,

to apply to b 7 a . Another comparison may be made for the parabola with

o« tb . The value of $. In Table I is 0.896, while the approximation

gives 0.924. Thus it may le expected that the approximate solution of

Equation [5] will give results good to within a few percant for sections

at least twice as deep as they are wide.

,.JL~L.

1 -8- f

i

?AB1Z I

■ Jeffreys "(8) ** Convergence clow

rt

i

4

1, it

-9-

%

c Motion antiayiuwetric

about minor axis Motion antisynnetric

about major a: is

A/A/C A C*t&/c 4

1.571 ,822 1.571 .822

j EIHruM

! e - IcO 1.886« .889 ••

e = 0.9 1.878 .882 #*

e = 0.8 1.87 • .87 1.78 * .79

e = 0.7 1.858 .863 1.814 .323

e = 0.6 1.856 .861 1.823 .831

e s 0.3 1.845 .851 1.837 • 8U

e = 0 (circle) 1.841 -8^7 1.841 .847

Orthogonal r>raboja3

i 2.L17 .896

r ir « I

TABLE II

Ji *?': u/a/c r»/** —l

4

0 1 * 2.150 V5 .924

1 1/? •• 1.886 V4 .889

2 V4 1.717 2/7 .862

1 V8 1.655 V13 .&i3

/. V16 1.616 8/25 .836

5 1/32 1.591 16/49 .826

oo 0 •** 1.571 VI .822

parabolas; ** ellipse; *** reetanple.

-10-

I 1 T—T

! h

(l) S. Timoahenko, "Vibration Problem« in Engineering," D. Van Noatrand

Company, Nev York, 2nd Edition, 1957, p. 338.

It is of some historical interest that both the rotatory iner-

tia correction, usually attributed to Rayleigh ("Theory of Sound,"

Cambridge, England, first edition, 1877V current edition, reference (7),

art. 162),and the transverse shear correction, usually attributed to

Tiaoshenko (FhllMoptai^l M»g-zine, Ser. 6, Vol. 41» 1921, pp. 744-746,

and Vol. 43, 1922, pp. 125-Hl),are given by M. Dresse in his "Cours

ie Mecanique Appllquee," "fcllet-Bachalier, Paris, 1859, p. 126.

(2) R. D. "indlin, "Thickness-Shear and Flexural Vibrations of Crystal

Plates," J. Appl. Phys., Vol. 22, 1951» op. 316-323.

(3) R. D. MinriHn, "Forced Thickness-Shear and Flexural Vibrations of

Piezoelectric Crystal Plates," J. Appl. Phys., Vol. 23, i<-52, pp. 83-88.

(4) R. D. Mindlin, "Influence of Rotatory Inertia and Shear on Flexural

Motions of Isotropie, Elastic Plates," J. Appl. Mach., Vol. 18, 1951,

pp. U-*8.

(5) S. Timoshenko and J. N. Goodier, "Theory of Elasticity," McGraw-

Hill Book Co., New York, 2nd Edition, 1951, p. 452.

■ l 1 i

(6) H. Lamb, "Hydrodynamics," Dover Publications, New York, 6th Edition,

1945, pp. 282-290.

(7) Lorti Rayleigh, "Theory of Sound," Dover Publications, New York, 2nd

Edition, 1945, art. 339.

I_L

I ±.

.11-

; 1 :

i ,

i

* (8) H. Jeffrey«, "The Free Oeoillaticna of Water in an Elliptic lake,"

Proo. Lordon Math. Soo., Ser. 2, Vol. 23, 1924, pp. 455-476.

(9) K. Hidaka, "The Oscillations of Water in Spindle-Siaped and Elliptic

Basins as wall as the Associated Problems," Mas. Imp. Marine Obeorr.,

Kobe, Japan, Vol. 4» 1931, pp. 99-219.

r *-

(10) "Tables of Besael Functions of Fractional Order," Columbia Unirersity

Press, New York, 1948, Vol. 1, p. 384.

(11) B. 0. Peiree, "A Short Table of Integrals," Ginn and Company, Boston,

3rd reTised edition, 1929, No. 483.

-12-

V !

am ~ r-n-i i 4-j—*

A»

i

I !

II

Fig. Is Elliptic and parabolic sections for which thiclcnesfl-shear frequency (a/) and Timoshenko's con- stant (•*€) are piven in Table I. DusheH line is para- bolic section. Thickness-shear motion is antisymmetric about axis A-A ■

\ i

LL_

r 1—(T

-'ip. 2: Ovaloid sections for which approx- imate values of w and 4 are given In Table II,

_i L

[j I

T — i

i i

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Institut de Mathematlquea Universite post. fal> 55 Skoplje, Yugoslavia (1)

Professor L. S. Jacobsen Dept. of Mechanical hngineerlng Stanford University Stanford, Califomlü 11)

Professor Sruce G. Johnston Jnlv"rsltj of Michit?an Ann Arbor, Michigan (1)

Prof«39or K. Klotter Stanford University Stanford, California (l)

Professor W, J. Krefeld Jept. of Civil ünflneering Columbia Unii'ersity New York 27, New York (1)

Professor B. J. Lauer. Dept. of Materials u.^inee ring University of Minnesota Minneapolis, Minnesota (1)

Jr. E. H. Lee Division of Applied Mathemtics Brown University irovidence, Rhode Island (1)

Professor George Lee Rensselaer Polytechnic Institute Trey, New York (1)

Professor J. M. Leasells Massachusetts Institute of Technology Cambridge *9, Massachusetts (1)

Library, Engineering i >nd>.tion 29 West 19th Street New York, I4*w York (1)

Professoi p l Lieber Oept. o<* Kjigi leering Rensselaer Polytechnic Institute Troy, New York (1)

Or. H>m Lo Pordue University !afayefUt, Indiana U)

Professor C. T. G. Looney Dept. of Civil Engineering Yale University New Haven, Connecticut (1)

Dr. J. L. Lubkln Midwest Research Institute 4(U° Pennsylvania Avenue Kansas City 2, Missouri (1)

Professor J. F. Ludloff School of Aeronautics New York University New York 53, New York (1)

Professor J. N. Menduff Rensselaer Polytedmic Institute Troy, New York (1)

Professor C. W. MaoGrego" UVversity of Pennirylvania Philadelphia, Pennsylvania (1)

Professor Lawrence E. Malvern Ospt. of Mathematics Carnegie Institute of Technology Pittsburgh l'<, Pennsylvania (1)

Dr. J. H. Merchant Brown University Providence, hhode Island (1)

Professor J. Marln Pennrylvania State College State College, Pennsylvania (1)

Jr. W. P. Mason Bell Telephone Laboratories Murray Hill , New Jersey (1)

Professor R. D. Mlndlin Dept. of Civil Engineering Columbia University 612 West 125th Street Ni.w York 27, New York (15)

;*-. A. Nadai 116 Cherry Valley Koad Pittsburgh 21, Pennsylvania (1)

R .fessor Paul M. Naghdl Dspt. of En Sneering Mechanics University of Michigan Ann Arbor, Michigan Ü)

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Contractors and Other IflTectimto« Actively BBOaUd *° RaLated Research (cent.)

Professor N. M. Newmark Dapt. of Civil Engineering University of Illinois Urban«, 111 in MB

Professor Jess« Ormomlroyd University of Michigan Ann Arbor, Michigan

Dr. W. Oagood Illinois Institute of Technology Technology Center Chicago 16, Illinois

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Dr. George B. Pegraa Conaittee on Government Aided Research 313 Low Memorial Library Columbia University New York 27, New York

Or. R. P. Peteraen Director, Applied Physics Division Sandia Laboratory Albuquerque, New Mexico

Mr. R. E. Peterson Westlnghouse Research Laboratories East Pittsburgh, Pennsylvania

Dr. A. Pnilllps School of Engineering Stanford University Stanford, California

Professor Gerald Pickett Dept. of Mechanics University of Wisconsin Madison 6, Wisconsin

Dr. H. Poritzky General Electric Research Labs. i'jhenectady, New York

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Dr. W. Prager Graduate Division of Applied Mathematics Brown University Providence, Rhode Island (1)

RAND Corporation 1500 4th Street Santa Monica, California attnt Dr. D. L. Judd (1)

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Dr. S. Kaynor Armour F*search Foundation Illinois Institute of Technology Chicago 16, Illinois (i)

Professor E. Retssner Dept. of Ma the*« tics Massachusetts Institute of Technology Cambridge 39, Massachusetts (1)

Professor H. Reissner Polytechnic Institute of Brooklyn 99 Livingston Street Brooklyn 2, New York (1)

Dr. Kenneth Rchinson National Bureau of Siandarda Washington, D.C. (1)

Professor M. A. Sadcwsky Illinois Institute of Technolog} Technology Center Chicago 16, Illinois (1)

Professor M. G. Salvador! Dept,, of Cl\'il engineering Colusjbia University New York 27, New York (1)

Dr. F. S. Shaw Polytechnic Institute of Brooklyn 99 Livingston Street Brooklyn 2, New York (1)

Dr. Daniel T. Sigley Applied Physics Laboratory The Johns Hopkins University 0621 Georgia Avenue Silver Spring, Maryland (1)

Dr. C. B. Sarith Department of Mathematics Walker Hall University of Florida Gainesville, Florida (1)

Professor C. R. Soderbezg Massachusetts Institut», of Technology Cambridge )9, Massachusetts 7*1)

Professor R. V. Southwell Imperial College of Science and

Technology South Kensington London S.W. /, England (1)

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Professor E. dternberg Illinois Institute of Technology Technology Center Chicago 16, Illinois

Professor J. J. Stoker New York Unlvers' ty Washington Square i >w lork, New York

Mr. K. A. Sykes Bell Telephone Laboratories Murray Hill, New Jersey

Professor P. S. Symonds Brown Jni*rereity Providence, Rhode Island

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Professor J. L. Syr^e Dublin Institute for Advanced Studies School of Theoretical Physics 64-65 Merrlon Square Dublin, Ireland

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Professor F. K. Telchaann Dept. of Aeronautical Engineering New York University university Heights, Bronx New York, New York

Professor S. P. Tinoahenko School of engineering Stanford University Stanford, California

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(1) {*• r" A. Truesdell

Hloo«ington, Indiana

Professor Karl S. Van Dyke Department of Hiysics Scott Laboratory Wssleyea University Middletown, Connecticut

Dr. I. Vlgness Naval Research Lahor«t; Anacostia Station Washington, D.C.

Dr. Leonarde Villen» Gran Via J. Antonio 6 Madrid, Spain

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iQegtirch (fiaa^ Professor E. Volterra Rensselaer Polytechnic Institute Troy, New York (1)

Mr. A. M. Wahl Uestlnghouse Research Laboratories East Pittsburgh, Pennsylvania (1)

Professor C. T. Wang Dept. of Aeronautical haglneerirg New York University University Heights, Bronx New York, N-w York (1)

Dr. R. L. w>pel RFD 2 reeksklll, New York

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Professor E. E. w»ibel University of Colorado Boulder, Colorado

Dr. Alexander »toinstein Institute of Applied Mathematics University of Maryland College Park, Maryland (1)

Professor Dana Young Dept. of Mechanical Engineering University of Minnesota Minneapolis L£, Minnesota (1)

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