Transverse shear deformation and rotary inertia effects in vibrating beam

Egle, D. M.

English

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N A S A  C O N T R A C T O R  
R E P O R T  
A N  APPROXIMATE THEORY FOR 
TRANSVERSE  SHEAR DEFORMATION 
AND ROTATORY INERTIA EFFECTS 
I N  VIBRATING BEAMS 
by De M ,  Egle 
Prepared by 
UNIVERSITY OF OKLAHOMA 
Norman, Okla. 
for Langley Research Center 
\ 
i 
i Li N A T I O N A L   A E R O N A U T I C S   A N D   S P A C E   A D M I N I S T R A T I O N  W A S H I N G T O N ,  D. C .  M A Y  1 9 6 9  
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r- TECH LIBRARY KAPB, NY 
NASA CR-1317 
AN APPROXIMATE THEORY FOR TRANSVERSE  SHEAR 
DEFORMATION AND ROTATORY INERTIA 
EFFECTS IN VIBRATING  BEAMS 
By D. M. Egle 
Distribution of this  report is provided  in  the  interest of 
information exchange. Responsibility for the contents 
resides  in  the  author or  organization  that  prepared it. 
Prepared  under  Grant No. NGR 37-003-035 by 
UNIVERSITY OF OKLAHOMA 
Norman, Okla. 
for Langley Research  Center 
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 
For sa le  by the Clearinghouse for Federal Scientific and Technical lnformotion 
Springfield, Virginia 22151 - CFSTI price $3.00 
P -  

AN APPROXIMATE  THEORY  FOR  TRANSVERSE SHEAR 
DEFORMATION AND ROTATORY  INERTIA 
EFFECTS IN  VIBRATING  BEAMS' 
D.M.  Egle 
Assistant  Professor 
School  of  Aerospace  and 
Mechanical  Engineering 
University  of  Oklahoma 
Norman,  Oklahoma 
Abstract 
A beam  theory,  which  includes  the 
shear  deformation  and  rotatory  inertia 
eff cts  of  transverse 
and  whose  kinetic  and 
potential  energy  may  be  written  in  terms  of a ingle  depen- 
dent  variable,  is  developed  in  this  paper.  This  simplifica- 
tion  will  reduce  the  computational  effort  required  in  the 
analysis  of  complex  beams  or  structures  composed  of a number
of  beams.  This  is  accamplished  by  neglecting  the  coupling 
between  the  transverse  shear  deformation  and  the  rotatory 
inertia.  Comparisons  of  natural  frequencies  calculated  by 
this  theory  with  those f the  Timoshenko  theory  show  that 
this  coupling  is  indeed  negligible. 
'This work  was  sponsored by NASA  under  Research  Grant  NGR 37- 
003-035, with  Mr. John L.  Sewall,  Dynamic  Loads  Division,  Langley 
Research  Center  acting  as  technical  monitor. 
Introduction 
T h e '  Ti~noshenko theory of vibrating b.eams 111 has been 2. 
thebasis  of a large number'of investigations into the effects 
of transverse  shear deformation and rotatory  inertia on the 
dynamics of beams. I t  has been used extensively- because it is 
simple t o  formulate and its results compare very well with 
those obtainable by l inear elasticity theory [2,3] . In ex- 
change for the accuracy of this  theory as compared to the 
classical Bernoulli-Euler beam theory, it is necessary to  
introduce an additional dependent variable, thus complicating 
the solution. While the addition of this variable is not of 
serious consequence in  many of the simpler beam problems, it 
can be a considerable disadvantage if  an approximate solution 
by the methods of R i t z  o r  Galerkin is being sought for a canplex 
beam or  a structure ccarrpased of a number of beams. 
In this paper, the energy expressions for a beam theory, 
which includes the effects of transverse shear deformation and 
rotatory  inertia and which may be written  in terms of a single 
dependent variable, are developed. To accomplish this, it 
is necessary t o  neglect the coupling bemeen the transverse 
shear deformation and the rotatory inertia. The natural fre- 
quencies of a simply supported and a cantilevered beam calculated 
with. this theory compare very well t o  those determined by- the 
Timoshenko theory, implying that the neglect of this coupling 
h e r s  in  brackets designate references a t  the end of the paper. 
2 
is justified. 
Namemlature. 
A 
E 
G 
h 
I 
k 
R 
M 
n 
t 
T 
v 
W 
"b 
W S 
a 
V 
P 
w 
6 
beam cross-sectional  area 
Young's  .modulus 
shear modulus 
depth of rectangular beam 
moment of inertia of beam cross  section about the y-axis 
transverse shear coefficient 
length of beam 
bending moment 
vibration mode (1, 2,  3, etc.) 
time 
kinetic energy 
shear force 
total  deflection of neutral  axis 
deflection due t o  bending moment 
deflection due t o  shear force 
E1 /kGA 
wavelength of vibration 
density of beam material 
frequency 
; @2
Theory  Development 
Let w be the total  deflection of the beam's neutral axis, 
see Figure 1. 'This deflection may be divided into.tw0 components, 
that  due to  the bending moment and that due to transverse shear 
3 
deformation. Thus 
w = Wb + ws (1) 
The shear deflection, ws, may be related  to  the  shear  strain 
in the beam in several ways (see reference [4] for example) and 
the value of the transverse shear coefficient is dependent on this  
relationship. However, for our purposes, it is only necessary to  
s ta te   that  
where k is the appropriate transverse shear coefficient. 
The bending moment is given by 
The s t ra in  energy due to  the bending moment and the shear 
force is 
u = 1 / 2  1 c, + ml dx R M2 V2 
0 
and the  translational and rotatory  kinetic energy is 
where the dot above a variable denotes differentation with res- 
pect t o  time. 
In the energy formulation of the Timoshenko theory, equations 
(1, 3, 4) are substituted into equations (5, 6) which gives energy 
expressions which are functions of two variables, for example, w 
and wb. 
If it is assumed that, for the purpose of determining the 
shear force and, hence, the shbar deflection, the rotatory 
iner t ia  is negligible, then 
V = - 
4 
Thus, the shear force is related t o  the hending moment just as 
it is in s t a t i c  beam theory. Note that the rotatory inertia 
has not been neglected  altogether  as it still appears in the 
second term of equatzon (6). In fact, only' the coupling .between 
.the  rotatory.inertia and the transverse shear force, and hence, 
tihe transverse shear deflection, has been omitted. 
Ccrmbhiig equations (3, 4, and 7) , 
w = - c r w  
,x b,= 
where a = EI/M;A. 
Equation (8) may be integrated t o  determine the shear 
deflection 
W = - a W  
S b,= + c ( t )  (9) 
C(t) is determined from the end conditions. However, because C(t) 
influences the energy only through the sum w = wb + ws, it will be 
deleted from equation (9) with the understanding that it may be 
added to  wb (or w) t o  satisfy the end conditions. 
The energy expressions (5,6) may now be written in  tenns of 
the bending deflection, wb Thus 
U = (1/2) /: E1 C W ~ , ~  + a w2 1 dx 
b ,= (10) 
T = (1/2). 1: p A I[+,, - b! Gb,- l 2  + A .  1 ;2 c.111 
The end conditions are taken..to be similar t o  those of the 
Timoshenko theory. 'The natural end conditions (those- end condi- 
tions for which the  forces and moments a t  the end do no work) 
are  either 
5 
M = E I w  = o  
b,= 
or 
w = o  
b,x 
and,  either 
V =  - E1 w = o  
b,= 
or 
(12-a) 
(12-b) 
(12-c) 
(12-d) 
Application  to a Uniform  Simply-Supported  Beam 
The  end  conditions  (12-a,d)  are  satisfied  at x = 0, R if 
wb = A sin nrx/a sin wt  (13) 
With  equations  (10, 11, and  13),  the  natural  frequencies 
may  be  calmlated  by  Rayleigh's  method 
where 
wl = 2 ,/$ , natural  frequencies  of  the  simply 
supported  Bernoulli-Euler  beam 
The  ratio a€ bending  deflection  to  total  deflection  may  be 
shown to  be 
Figure 2 ccnnpares  the  frequencies  calculated  with  the 
6 
present  analysis  to  those f the  Timoshenko  theory  (see  Appendix) 
for a rectangular  bar  of  material  with  Poisson's  ratio 0.3. The 
shear  coefficient, k was  taken  to  be  0.85 as recommended  by 
Cowper [4] . 
The  frequencies  are  plotted  over a wide  range  of  the  variable 
3 which  is a measure  of  the  depth  of  the  beam  to  the  wavelength  of 
the  vibration.  For  the  rectangular  beam,  for  example, 
where h = depth  of  the  beam 
v = 2a/n, wavelength 
The  range  of (h/v) in Figures 3, 4, 5 is  from 0.06 to  1.8,-  the 
upper  limit  being  much  higher  than  one  would  expect  to  encounter 
in determining  the  first  few  frequencies  of a beam.  However,  the 
upper  limit  was  selected  to  show  the  wide  range  of  agreement  between 
the  present  analysis  and  the  Timoshenko  theory. 
As is  seen  in  Figure 2,  the  agreement  between  the  two  theories 
is  excellent  up  to 5 = 10.  The  values of ( w / w ~ )  calculated  by 
the  Timoshenko  theory  were  slightly  lower  than  those  of  equation 
(14),  the  maximum  difference  being  about  1.25%  for 5 1. The 
differences  for  values  of 5 < 1 and 5 > 1 are  less  than  1.25%. 
The  ratio  of  deflection  due  to  bending  moment to the  otal 
deflection  for  the  simply  supported  beam  is  shown  in  Figure 3.
The  discrepancy  between  the two theories  is  more  apparent i  the 
deflections  but  is  still  not  excessive  if 3 0.5  which  correspond 
to a depth  to  wavelength  ratio  of  about 0.4. Because  the  present 
7 
analysis  underestimates  the  bending  deflection,  it  will  yield 
values  for  the  shear  deflection  which  are  too  large.  This  is 
verified  in  Figure 4 which  shows  the  shear  force  in a simply- 
supported  beam  normalized  by 
which  is  the  shear  force  corresponding  to  the  Bernoulli-Euler 
theory  and  is  equal  to  the  shear  force  predicted  by  the  present 
theory. As would  be  expected,  the  agreement  between  the  present 
approximation  and  Timoshenko  beam  theory  is  much  worse  for  the 
shear  force  than  the  natural  frequencies,  but  the  error  is  less 
than  10%  if 5 < 0.1  or  if  the  hedght  to  wavelength  ratio  is  less 
than 0.18. 
Application  to a Uniform  Cantilevered  Beam 
The Rayleigh-Ritz  method  is  applied  to  the  energy  expressions, 
equations  (10,  11)  to  obtain n approximate  solution  for  the 
natural  frequencies  of a uniform  beam  of  length R, clamped  at 
x = 0. The  bending  deflection  is  taken in the  form 
wb = 1 An  [Xn(x) + a X: (0) 3 sin ut 
n=l 
where  the  Xn (x) are  the  Bernoulli-Euler  eigenfunctions  for  the 
clamped-free  beam [5]  and  the X; = an/&.  Equation  (16)  satis- 
fies  the  end  conditions  (12-b,  d)  at  x=O  and  (lZ-a,c)  at X=R.
The  maximum  kinetic  and  potential  energies  may  be  calculated 
with  eqs.  (10,  11, 16). 
8 
m m 
‘max = 1 / 2  E1 /’ 0 C ( n=l 1 An X; (X))’ + a ( n=l 1 An X:‘ ( x ) ) ~ }  dx 
(18) 
Applying the Rayleigh-Ritz technique t!o eqs. (17,18) with 
the  aid of the beam eigenfunction integrals  tabulated by Fel- 
gar [6] , resul ts   in  
1 (Kin - x Min) An = 0 i = 1, 2, . . . 
n=l 
9 
Numerical values of an and Bn are given i n  reference [SI. 
Equations (19) may be solved approximately .by considering 
only the first few t e r n  i n  the.series. The problem is then a 
linear eigenvalue problem and may be solved any of the  standard 
numerical techniques. 
Table 1 compares the five lowest frequencies of a clamped- 
free uniform beam for  four  gemetrical  configurations  calculated 
with equation (19) to  values given by the Timoshenko theory i n  
reference [7]. Seven terms were used i n  equation (19), which 
w a s  sufficient for 3-place accuracy in the f i f t h  lowest frequency 
when compared to  a ten-term series. 
If we consider only a single term in (19) and neglect the 
a2 tern in Min, the frequency equation is 
where 
10 
w = B i  E , natural frequency of ~ r n o u l l i - ~ l e r  2 
clamped-free beam 
The term which was neglected in  arriving  at  equation (20) can 
be shown t o  be the contribution of the  shear  deflection  to  the 
translational  kinetic energy and the terms remaining in the 
dencnninator of (20) are the contributions of the bending deflec- 
tion, and the rotatory inertia. Equation (20) w i l l  give values 
of the frequency larger than those calculated with (19) and the 
error w i l l  increase as yn increases. 
The point to be made, however, i n  presenting equation (20) 
is that it suggests the use of yn as a similariw parameter in 
plotting the frequencies. Figure 5 shows the frequencies calcu- 
lated with a ten-tern series in  equation (19) plotted as a 
function of yn for the f i r s t  four modes  of vibration. I t  is 
apparent that yn serves reasonably well as a similarity para- 
meter since the values tend t o  l i e  on a single curve. 
11 
It is evident, from the excellent agreement hetween the 
natural  frequencies  calculated with this analysis and those of 
fhe Tmoshenko theory, that the neglect of the caupling between 
the rotatory  inertia and the transverse shear deformation is 
justified. 'The energy expressions equations (10, 11), in  
which the present theory is embodied, are  slightly more complex 
than those of the Timoshenko theory, but, because they are 
functions of a single dependent variable, a considerable shtpli- 
fication  in  the  solution t o  problems t o  which they are applied 
will result. 
The concept used t o  develop this theory, that is, ignoring 
the coupling between the transverse shear deformation and the 
rotatory  inertia, would seem t o  be applicable t o  plates and 
shells as well as beams. A study has been initiated t o  do this 
and it w i l l  be interesting t o  see i f  this w i l l  agree as well 
with the Mindlin theory, reference (8), as the present analysis 
agrees with the Timoshenko theory. 
12 
. . .. 
REFERENCES 
1.  Timoshenko,  S.P., "On the  Correction  for  Shear  of  the  Differential 
Equation  for  Transverse  Vibrations  of  Prismatic  Bars,"  Philo- 
sophical  Magazine,  vol.  41 , pp.  744-746  (1921) . 
2.  Timoshenko, S.P., "On the  Transverse  Vibrations  of  Bars  of  Uniform 
Cross-Section,"  Philosophical  Magazine,  vol.  43,  pp.  125-131 
(1922). 
3.  Davies,  R.M. , "A Critical  Study  of  the  Hopkinson  Pressure  Bar," 
Philosophical  Transactions  of  the  Royal  Society  (A),  vol.  240, 
pp.  375-457  (1948) . 
4. Cowper,  G.R.,  "The  Shear  Coefficient in Timoshenko's  Beam  Theory," 
Journal  Applied  Mechanics , vol.  33 , pp.  335-340  (1966). 
5.  Young, D. and  Felgar,  R.P.,  Jr.,  "Tables  of  Characteristic  Functions 
Representing  Normal  Modes  of  Vibration  of a Beam,''  Publication 
4913,  University  of  Texas  (July  1949). 
6 .  Felgar,  R.P.,  Jr.,  "Formulas  for  Integrals  Containing  Characteristic 
Functions of a Vibrating  Beam,"  Circular  14,  University  of 
Texas,  Bureau of Engineering  Research  (1950). 
7. hang, T.C.  and  C.S.  Kung,  "New  Tables of Eigenfunctions  Representing 
Normal  Modes  of  Vibration  of  Timoshenko  Beams,"  Developments in 
Theoretical  and  Applied  Mechanics,  vol. 1, pp.  59-71  (1962). 
8.  Mindlin.  R.D..  "Influence  of  Rotatory  Inertia  and  Shear  on  Flexural 
Motions of Isotropic,  Elastic  Piates",  Journal  of  Applied  Mechanics, 
Transactions ASME, vol.  73,  pp.  31-38  (1951). 
13 
APPENDIX 
The  equations  governing  the  motion f a Timoshenko  beam  are 
v = kGA (w,x - w ) 
b ,x 
M = E I w  
V,x = pAG 
b,= 
Mpx + V = PI wb,x (A- 4) 
For the simply  supported  beam,  the  deflections  are of th form 
w = C1 sin nvx/R  sin  wt (A- 5) 
wb = Cz  sin n.rrx/k sin ut (A- 6) 
Substitution of equations  (A-5,-6)  &to  (A-1,-4)  eventually 
leads  to 
where 
V1 = - E1 w 
b,= 
14 
Tab1 .e  1.  Ccanparison  of  the  natural  frequencies of a uniform 
cantilevered  beam  calculated w i t h  the  present  theory 
to  those  of  the  Timoshenko  Theory. 
"" ~ E/kG = 4, I /h2  = 4x10m4 
n 1 2 3 4 5 
(L)2 W 1  a .99080  .93925  .86773  . 824  .696 
b .99081 93881 86758 77710  .69535 
E/kG = 6.25, I / h 2  = 4x10- 4 
~ 
n 1 2 3 4 5 
a .98671  .91421  .81909  , 28.6
(% a1 b .98671  .91398  . 1 01  .71254  .6 175 
E/kG = 1.78, I / h 2  = ~ x I O - ~  
n 1 2 3 4 5 
a .98854  .92 0  . 4144  .7 6165
f L 1  
\U1' b .98854  .92 04  .8 116 .7 514 .6 205 
E/kG = 2.78, I / h 2  = ~ x I O - ~  
n 1 2 3 4 5 
a .98447  .90121  .79640 .68 35 2 
(W) 
W 1  b .98447  .go112  .79609  .68318  .58 30 
Nuation (19) , 7 terms 
bTimoshenko  theory,  reference [ 71
15 
- dx - 
V +  V,x dx 
Figure 1. Coordinates and Notation 
0. I 
0.0 I 
- ($ 
TI MOSHENKO THEORY 
PRESENT THEORY - Eqn. 14 
- 
K G  
E = 3.06 \ 
0. I 1.0 10.0 
Figure 2. Effect of Shear Deformation and Rotatory Inertia 
on the Natural Frequency of a Simply supported 
Beam. 
16 
1.0 
0.1 
7 - - - 
- 
- 
- 
TIMOSHEN KO THEORY - 
- 
PRESENT THEORY 
3 
W - - - - 
- 
- E 
" K G  - 3.06 - 
- 
- 2 
OD I I I ~ I I I I 1 1 1 . 1  ~ ~~ I I I I l l  I 1 1 I 1  I l l  
0.01 0. I LO 10.0 
Figure 3. Effect of Shear Deformation and Rotatory Inertia 
Simply Supported Beam. 
on the Deflection due to Bending  Mment in a 
E =  
K G  3.06 
0.0 I 0.1 1.0 IQO 
Figun 4. Effect of Shear Deformation and Rotatory Inertia 
on the Shear Force in a Simply Supported Beam. 
17 
o n = l  
0 n = 4  
" K G  - 3.06 E 
0.80 - 
a 
0.70 - a0 
a 
a o  
0.60 - a 
0 a 
0.50 - 
T 7, I I I I I I I I I I 
0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0 .20  
Figure 5. Effect  of Shear Defmation and Rotatory Inertia 
on the Natural Frequency of a  Cantilevered Beam. 
18 NASA-Langley, 1969 - 32 CR-1317 


An approximate theory for transverse shear deformation and rotatory inertia effects in vibrating beams

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