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Some practical considerations on vibrating timoshenko beams
'r,'.:"
Applied Acoustics 34 (1991) 261-266
~
Technical Note
Some Practical Considerations on Vibrating Timoshenko Beams J. A. Reyes, R. E. Rossi Department of Engineering, Universidad Nacional del Sur, Av. Alem 1253, 800(O-Bahia Blanca, Argentina
P. A. A. Laura,* J. L. P o m b o & D. Pasqua Institute of Applied Mechanics, CONICET-SENID-ACCE, Gorriti 43, 800(0--Bahia Blanca, Argentina (Received 21 January 1991; revised version received 12 May 1991: accepted 17 May 1991)
ABSTRACT The present study deals with two types of considerations which are of practical importance when dealing with vibrating, simply supported Timoshenko beams: (1) range of validity of the theory, taking as a basic geometric-mechanical parameter the ratio: radius of gyration of the cross-section/beam length; and (2) the position of the supports at the beam ends.
INTRODUCTION It is common practice i to determine natural frequency coefficients of a beam
obtaining, by analytical or numerical techniques, the eigenvalues of the * To whom correspondence should be addressed. 261 Applied Acoustics 0003-682X/91/$03.50 © 1991 Elsevier Science Publishers Ltd, England. Printed in Great Britain
262
J..4. Reyeset al.
governing mathematical Timoshenko model covering a range of ~alues of the parameter (radius of gyration of the cross-section/beam length, r/L) which goes from 0 (the classical Bernouilli-Euler beam) to 0"10. Certainly, the only possible way to determine the validity of the range is by determining the eigenvalues of a vibrating solid using the three-dimensional theory of elasticity or, if the transversely vibrating solid can be considered as a "beam' or "plate', by means of a dynamic plane stress analysis {where a rectangular cross-section is assumed). The second approach is given in the present investigation, where the lower eigenvalues are determined using plane stress theory (where a standard finite element code is used) and are compared with the values determined by means of the Timoshenko theory. Another simple, basic question is raised when a simply supported beam is considered: in which position of the beam ends are the supports placed? At the edges? At the centre line of the cross-section? Certainly, this question does not arise if the parameter r/L is negligible but it makes sense if r/L is of considerable magnitude and such that the Bernouilli-Euler theory cannot be rationally employed. This study constitutes a modest attempt to partially answer the questions posed.
RANGE OF VALIDITY OF THE T I M O S H E N K O THEORY Table 1 depicts the first three natural frequency coefficients of a simply supported beam of rectangular cross-section determined for L/h -- 5, 4 and 3, which correspond to r/L---0-057 735, 0-072 169 and 0"096225, respectively. The supports are assumed to be placed at the centroids of the beam ends. The eigenvalues are determined using: (a) Dynamic, plane stress theory; and (b) Timoshenko's formulation. Three different nets were used when obtaining the eigenvalues corresponding to point (a), in order to investigate the convergence of the method. It is observed that for L/h < 5 (or r/L > 0-057 735) the results obtained using plane stress theory deviate considerably from the eigenvalues predicted by the Timoshenko model. On the other hand, considering a situation where r/L < 0"05, for instance taking r/L = 0"0372, very good agreement is achieved between the results predicted by the Timoshenko theory and the eigenvalues obtained using the plane stress formulation, at least for the first three modes of vibration (see Table 2).
263
Practical considerations on vibrating Timoshenko beams TABLE 1
Values of f~ = c,J~L',cpA. EI: Comparison Between Rest, Its Predicted by Dynamic Plane Stress Theory and Values Obtained Using Timoshenko's Theory of Vibrating Beams L h
rL
Metho&dogy
5
0-057 735
Finite element (plane stress)
5
0"057735
Timoshenko theory:
4
0"072 169
Finite element (plane stress)
4
0.072 169
Timoshenko theory:
3
0 " 0 9 6 2 2 5 Finite element (plane stress)
3
0.096225
J
f~ i
f2,_
f23
9.259 7 9'1929 9"142 I 9"2740
31.8265 31'2779 30-7828 32"1665
60.1436 58-4269 56-7640 61-458 1
8.936 1 8.837 6 8.754 2
28-9915 28.2073 27-4671
52.1967 49'873 9 47-620 1
8.991 2
29.6509
54.5193
8-383 2 8"158 1 8-008 1 8.471 5
24.5888 23.4032 22.2770 25-8806
40.896 1 37.9728 35.4005 45.3363
• No of nodes
] ~1,
105 369 1 377
No. o f nodes
85 297 1 105
No. ok/nodes 65 225 833
Timoshenko theory: TABLE 2
Lower Natural Frequency Coefficients of a Simply Supported Beam (r/L = 0-037 2): Comparison of Results Determined Using Different Models T3T e o f mathematical model
f2t
02
f~3
Bernouilli-Euler theory Timoshenko's model Dynamic plane stress model (finite element results)
9.87 9'61 9.62
39"5 35"8 35"8
88"8 73.1 72.8
A c c o r d i n g l y , it d o e s n o t s e e m a d v i s a b l e to use the T i m o s h e n k o v i b r a t i n g b e a m t h e o r y f o r r / L > 0"05. A d m i t t e d l y this c o n s i d e r a t i o n has been o b t a i n e d in the case o f s i m p l y s u p p o r t e d b e a m s b u t it m a y p o s s i b l y be used for o t h e r types o f s u p p o r t s .
LOCATION
OF THE SUPPORTS
AT THE BEAM ENDS
C o n s i d e r the b e a m s h o w n in Fig. l(a) a n d the finite e l e m e n t nets s h o w n in Figs l ( b ) a n d l(c).
264
J. A. Reyes et al.
J.85 ¢m
.i_ -.
29.85 cm --
(a)
10
13
16
lg
6,
7
A12
5
8
11
14
17
20
B1 3
6
ig
12
iI~
16
21
Z3
20
7
5zn~^m
(b) Y
T CI gl
""
~ 14 ~ 2, '~
~o 15 ;co 25 30
35
~o 45 50
s~ ~o 6~ 70
n
ao ~
i i ~
i:::A'
~s ~so ~ss ~ o ~,s
C z
X
~----~
(c) Fig. 1. Investigation dealing with the effect of the position and nature of the end supports of a beam. (a) Steel beam studied in the present investigation (r/L = 0-0372);(b) finite element net No. 1; (c) finite element net No. 2. (A~, Az), (Bt, B2), (Cl, C2): locations of the end supports.
Table 3 depicts the values of fundamental frequency coefficients obtained using both nets, assuming: (a) (b)
fixed supports placed at (A 1, Az), (Bt, B2) or (C1, Cz); and fixed supports at A 1, B t or C 1 and sliding supports at Az, Bz or C2.
It is observed that for this particular value ofr/L (0"0372) the fundamental frequency is the same, regardless of the nature of the supports, when they are placed at the centroids of the beam ends. If the supports are placed at (Bz, B2) or (C~,C2) the fundamental frequencies are practically the same as when the supports are placed at (A1,A2), as long as the supports are of a sliding nature. However, drastic increases in the fundamental frequencies are observed when dealing with fixed supports placed at (B 1, B2) o r ( C t , C2). Apparently the fixity of the support coupled with a positional effect traduces into an increase of the
265
Practical considerations on vibrating Timoshenko beams TABLE 3
Fundamental Frequency (Hz) of the Steel Beam Shown in Fig. l(a) as a Function of (i) Position of the Supports and (ii) Nature of the Boundary Supports; Obtained Using Two Different Nets, Figs l(b)and l(c) Finite element
Position 0[" supports
Fundamental /?equency ( Hz) Both supports fixed
One support fixed, other allows motion in x-direction
(B t. B2)
989 1490
989 983
(A,, A,) (C,, C_,) (B t, B2)
978 1 156 1432
978 977 970
net
(A t, A,)
stiffness o f the beam. ( T h e fixity a n d p o s i t i o n i n g o f the s u p p o r t s i n d u c e a tensile force c o m b i n e d with a m o d i f i c a t i o n o f the l o c a t i o n o f the neutral plane o f bending.) It is i m p o r t a n t to p o i n t o u t t h a t for this b e a m c o n f i g u r a t i o n , the T i m o s h e n k o m o d e l yields F 1 = 978 Hz. This value coincides with the one o b t a i n e d by m e a n s o f plane stress t h e o r y (see T a b l e 2). A n e x p e r i m e n t a l p r o g r a m m e was r u n in o r d e r to c h e c k the validity o f the
Fig. 2. Steel beam with supports placed at the centroids of the beam ends; points (A t, A2) are not allowed to slide.
266
J. 4. Reves et al.
TABLE 4 Fundamental Frequency of the System Shown in Fig. lla): Comparison Bet,,~,een Theoretical and Experimental Results P, .~H{ /i )tl
A,,~- . . . . . . . . . 8:
C~-4
f,,cqz,en, c ,,
,:!? .~up/),,r:.~
-~~2
a~2 L,Cz
975
(H_-)
¢32{) 20
1156
090
previous observations. The experimental set-up is shov,'n in Fig. 2 and a comparison with the numerical predictions is shown in Table 4. Obviously there is a very poor agreement between the theoretical and experimental values. (The experimental possibilities of the laboratory are very limited from the point of view of exciting properly a Timoshenko beam and providing for proper boundary conditions.) However, there is a marked difference between the values of fundamental frequencies as the position of the supports is changed and when they are not allo~ved to slide.
ACKNOWLEDGEMENT The present study has been sponsored by C O N I C E T Research and
Development Program (PID 300050088).
REFERENCE 1. Rossi, R. E. & Laura, P. A. A., Vibration ofa Timoshenko beam clamped at one end carrying a concentrated mass at the other. Appl. Acoust., 30 (1990) 293-302.
Applied Acoustics 34 (1991) 261-266
~
Technical Note
Some Practical Considerations on Vibrating Timoshenko Beams J. A. Reyes, R. E. Rossi Department of Engineering, Universidad Nacional del Sur, Av. Alem 1253, 800(O-Bahia Blanca, Argentina
P. A. A. Laura,* J. L. P o m b o & D. Pasqua Institute of Applied Mechanics, CONICET-SENID-ACCE, Gorriti 43, 800(0--Bahia Blanca, Argentina (Received 21 January 1991; revised version received 12 May 1991: accepted 17 May 1991)
ABSTRACT The present study deals with two types of considerations which are of practical importance when dealing with vibrating, simply supported Timoshenko beams: (1) range of validity of the theory, taking as a basic geometric-mechanical parameter the ratio: radius of gyration of the cross-section/beam length; and (2) the position of the supports at the beam ends.
INTRODUCTION It is common practice i to determine natural frequency coefficients of a beam
obtaining, by analytical or numerical techniques, the eigenvalues of the * To whom correspondence should be addressed. 261 Applied Acoustics 0003-682X/91/$03.50 © 1991 Elsevier Science Publishers Ltd, England. Printed in Great Britain
262
J..4. Reyeset al.
governing mathematical Timoshenko model covering a range of ~alues of the parameter (radius of gyration of the cross-section/beam length, r/L) which goes from 0 (the classical Bernouilli-Euler beam) to 0"10. Certainly, the only possible way to determine the validity of the range is by determining the eigenvalues of a vibrating solid using the three-dimensional theory of elasticity or, if the transversely vibrating solid can be considered as a "beam' or "plate', by means of a dynamic plane stress analysis {where a rectangular cross-section is assumed). The second approach is given in the present investigation, where the lower eigenvalues are determined using plane stress theory (where a standard finite element code is used) and are compared with the values determined by means of the Timoshenko theory. Another simple, basic question is raised when a simply supported beam is considered: in which position of the beam ends are the supports placed? At the edges? At the centre line of the cross-section? Certainly, this question does not arise if the parameter r/L is negligible but it makes sense if r/L is of considerable magnitude and such that the Bernouilli-Euler theory cannot be rationally employed. This study constitutes a modest attempt to partially answer the questions posed.
RANGE OF VALIDITY OF THE T I M O S H E N K O THEORY Table 1 depicts the first three natural frequency coefficients of a simply supported beam of rectangular cross-section determined for L/h -- 5, 4 and 3, which correspond to r/L---0-057 735, 0-072 169 and 0"096225, respectively. The supports are assumed to be placed at the centroids of the beam ends. The eigenvalues are determined using: (a) Dynamic, plane stress theory; and (b) Timoshenko's formulation. Three different nets were used when obtaining the eigenvalues corresponding to point (a), in order to investigate the convergence of the method. It is observed that for L/h < 5 (or r/L > 0-057 735) the results obtained using plane stress theory deviate considerably from the eigenvalues predicted by the Timoshenko model. On the other hand, considering a situation where r/L < 0"05, for instance taking r/L = 0"0372, very good agreement is achieved between the results predicted by the Timoshenko theory and the eigenvalues obtained using the plane stress formulation, at least for the first three modes of vibration (see Table 2).
263
Practical considerations on vibrating Timoshenko beams TABLE 1
Values of f~ = c,J~L',cpA. EI: Comparison Between Rest, Its Predicted by Dynamic Plane Stress Theory and Values Obtained Using Timoshenko's Theory of Vibrating Beams L h
rL
Metho&dogy
5
0-057 735
Finite element (plane stress)
5
0"057735
Timoshenko theory:
4
0"072 169
Finite element (plane stress)
4
0.072 169
Timoshenko theory:
3
0 " 0 9 6 2 2 5 Finite element (plane stress)
3
0.096225
J
f~ i
f2,_
f23
9.259 7 9'1929 9"142 I 9"2740
31.8265 31'2779 30-7828 32"1665
60.1436 58-4269 56-7640 61-458 1
8.936 1 8.837 6 8.754 2
28-9915 28.2073 27-4671
52.1967 49'873 9 47-620 1
8.991 2
29.6509
54.5193
8-383 2 8"158 1 8-008 1 8.471 5
24.5888 23.4032 22.2770 25-8806
40.896 1 37.9728 35.4005 45.3363
• No of nodes
] ~1,
105 369 1 377
No. o f nodes
85 297 1 105
No. ok/nodes 65 225 833
Timoshenko theory: TABLE 2
Lower Natural Frequency Coefficients of a Simply Supported Beam (r/L = 0-037 2): Comparison of Results Determined Using Different Models T3T e o f mathematical model
f2t
02
f~3
Bernouilli-Euler theory Timoshenko's model Dynamic plane stress model (finite element results)
9.87 9'61 9.62
39"5 35"8 35"8
88"8 73.1 72.8
A c c o r d i n g l y , it d o e s n o t s e e m a d v i s a b l e to use the T i m o s h e n k o v i b r a t i n g b e a m t h e o r y f o r r / L > 0"05. A d m i t t e d l y this c o n s i d e r a t i o n has been o b t a i n e d in the case o f s i m p l y s u p p o r t e d b e a m s b u t it m a y p o s s i b l y be used for o t h e r types o f s u p p o r t s .
LOCATION
OF THE SUPPORTS
AT THE BEAM ENDS
C o n s i d e r the b e a m s h o w n in Fig. l(a) a n d the finite e l e m e n t nets s h o w n in Figs l ( b ) a n d l(c).
264
J. A. Reyes et al.
J.85 ¢m
.i_ -.
29.85 cm --
(a)
10
13
16
lg
6,
7
A12
5
8
11
14
17
20
B1 3
6
ig
12
iI~
16
21
Z3
20
7
5zn~^m
(b) Y
T CI gl
""
~ 14 ~ 2, '~
~o 15 ;co 25 30
35
~o 45 50
s~ ~o 6~ 70
n
ao ~
i i ~
i:::A'
~s ~so ~ss ~ o ~,s
C z
X
~----~
(c) Fig. 1. Investigation dealing with the effect of the position and nature of the end supports of a beam. (a) Steel beam studied in the present investigation (r/L = 0-0372);(b) finite element net No. 1; (c) finite element net No. 2. (A~, Az), (Bt, B2), (Cl, C2): locations of the end supports.
Table 3 depicts the values of fundamental frequency coefficients obtained using both nets, assuming: (a) (b)
fixed supports placed at (A 1, Az), (Bt, B2) or (C1, Cz); and fixed supports at A 1, B t or C 1 and sliding supports at Az, Bz or C2.
It is observed that for this particular value ofr/L (0"0372) the fundamental frequency is the same, regardless of the nature of the supports, when they are placed at the centroids of the beam ends. If the supports are placed at (Bz, B2) or (C~,C2) the fundamental frequencies are practically the same as when the supports are placed at (A1,A2), as long as the supports are of a sliding nature. However, drastic increases in the fundamental frequencies are observed when dealing with fixed supports placed at (B 1, B2) o r ( C t , C2). Apparently the fixity of the support coupled with a positional effect traduces into an increase of the
265
Practical considerations on vibrating Timoshenko beams TABLE 3
Fundamental Frequency (Hz) of the Steel Beam Shown in Fig. l(a) as a Function of (i) Position of the Supports and (ii) Nature of the Boundary Supports; Obtained Using Two Different Nets, Figs l(b)and l(c) Finite element
Position 0[" supports
Fundamental /?equency ( Hz) Both supports fixed
One support fixed, other allows motion in x-direction
(B t. B2)
989 1490
989 983
(A,, A,) (C,, C_,) (B t, B2)
978 1 156 1432
978 977 970
net
(A t, A,)
stiffness o f the beam. ( T h e fixity a n d p o s i t i o n i n g o f the s u p p o r t s i n d u c e a tensile force c o m b i n e d with a m o d i f i c a t i o n o f the l o c a t i o n o f the neutral plane o f bending.) It is i m p o r t a n t to p o i n t o u t t h a t for this b e a m c o n f i g u r a t i o n , the T i m o s h e n k o m o d e l yields F 1 = 978 Hz. This value coincides with the one o b t a i n e d by m e a n s o f plane stress t h e o r y (see T a b l e 2). A n e x p e r i m e n t a l p r o g r a m m e was r u n in o r d e r to c h e c k the validity o f the
Fig. 2. Steel beam with supports placed at the centroids of the beam ends; points (A t, A2) are not allowed to slide.
266
J. 4. Reves et al.
TABLE 4 Fundamental Frequency of the System Shown in Fig. lla): Comparison Bet,,~,een Theoretical and Experimental Results P, .~H{ /i )tl
A,,~- . . . . . . . . . 8:
C~-4
f,,cqz,en, c ,,
,:!? .~up/),,r:.~
-~~2
a~2 L,Cz
975
(H_-)
¢32{) 20
1156
090
previous observations. The experimental set-up is shov,'n in Fig. 2 and a comparison with the numerical predictions is shown in Table 4. Obviously there is a very poor agreement between the theoretical and experimental values. (The experimental possibilities of the laboratory are very limited from the point of view of exciting properly a Timoshenko beam and providing for proper boundary conditions.) However, there is a marked difference between the values of fundamental frequencies as the position of the supports is changed and when they are not allo~ved to slide.
ACKNOWLEDGEMENT The present study has been sponsored by C O N I C E T Research and
Development Program (PID 300050088).
REFERENCE 1. Rossi, R. E. & Laura, P. A. A., Vibration ofa Timoshenko beam clamped at one end carrying a concentrated mass at the other. Appl. Acoust., 30 (1990) 293-302.