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Effect of bending preload on flexural vibration of a bimodular tube
Journal of Sound and Vibration (1989) W(2),
EFFECT
359-363
OF BENDING PRELOAD ON FLEXURAL OF A BIMODULAR TUBE
VIBRATION
1. INTRODUCTION
It is well known that a direct axial preload has a significant effect on the flexural vibration frequencies of a slender beam; cf. the paper by Bokaian [ 11. The frequency is increased if the preload is tensile and decreased if the preload is compressive. However, by using the TrefItz type theory of prestressed elastic solids, cf. the paper by Reismann and Pawlik [2], it can be shown that the familiar bending stress distribution, that is linear in the depth co-ordinate, has no effect on the flexural vibration frequencies. This can be explained on the basis that the increase in frequency due to the tensile portion of the prestress distribution is exactly balanced by the decrease in frequency due to the compressive portion. Nevertheless, most fiber-reinforced composites having elastomeric matrices are highly “bimodular” in their mechanical behavior: i.e., they have a much smaller elastic modulus in compression than that in tension [3]. For such materials, prestressed media theory does predict a slight effect of bending stress distribution on natural frequencies. For instance, Doong and Chen [4] investigated this effect for prestressed, bimodular, thick plates. To the best of the author’s knowledge, there has been no investigation of the effect of bending prestress on the vibration of either solid or hollow circular section beams of bimodular material. However, the static bending of solid and hollow circular section beams was analyzed by Adler [5] and Shlyakhman and Lepetov [6], respectively. In the present analysis, transverse shear deformation is neglected, since the material of physical interest had a shear modulus greater than its elastic modulus. 2.
ANALYSIS
In the absence of body forces, the three-dimensional elastodynamic equations of motion, according to Reismann and Pawlik ([2], p. 145), are uij,j-(a~kui,k),,=Piii
(LJ
k =
(1)
L&3),
where ui is the displacement component in the Xi direction, p is the material density, o” is the prestress tensor, u is the stress tensor due to dynamic motion, ( ),j = a( )/aXj, (“) = a’( )/at*, and t is time. Consider a beam of hollow circular cross section and let x, y and z be the position co-ordinates along the beam axis, in the direction normal to the plane of bending preload, and in the other normal direction (in the plane of bending), respectively. Now consider a varying but uniaxial prestress a:, acting in the axial direction. Then the equations of motion corresponding to the x and z directions are UXX,X +7X,,, -(&&,X),X
= P;i,,
7X,,,+ a,,, -(&%,X),X = PK.
(293)
Note that rZXhas been replaced by T,= due to the symmetry of the stress tensor and that thickness-direction normal stress a,, has been neglected, as is customary in beam theory. Then equations (2) and (3) become ox*,x+ 7XZ.Z - ( UlXux,x),X= P& 9
TX,,, -+2x%,x).*
= Pk.
(495)
359 0022460X/89/200359
+ 05 %03.00/O
@ 1989 Academic
Press Limited
360 The
LETTERS
TO THE
EDITOR
bending moment and transverse shear force are defined as M,=
zaxx dA,
IIA
v, =
IIA
(6)
7x, dA,
where A is the cross-sectional area and z is the distance from the neutral plane (not the midplane). Multiplying equation (4) by z, integrating over A, and assuming that 7,, = 0 at the top and bottom fibers, one obtains Mx,x - v, -
(&Q,xz
IIA
zii, dA.
dA = p
(7)
Similarly, integration of equation (5) gives
(8)
ii, dA.
Now, the Bernoulli-Euler hypothesis is invoked; i.e., transverse shear and transverse normal strains are neglected. Thus, (9)
UZ(X,z, t) = w(x, t),
a,(-% z, t) = -zw,x(x, t),
where w is the normal detlection. Equations (7) and (8) become Mx,, - v, +
IIA
v,,x -
((+:x~xx),xz~ dA = -PI+,,,
IIA
(a!&Jx
dA = pA+. (10,11)
The material is assumed to be bimodular with elastic moduli EC and E, in compression and tension, respectively. Thus, ,J
_
uxxwhere
K’
EcK”Z,
(12)
Er~oz,
is the initial bending curvature. Equations (10) and (11) become M,,, - v, + E, V.,, - E,
IIA‘ IIA‘
(K”W,xx),xz3dA+ E, (K”‘+‘,x),xzdA - E,
IIA,
( K”W,,),,Z3
(K’W,,),,Z
IIA,
dA = -PI@,,, dA = pAti,
(13) (14)
where A, and A, denote the respective portions of the cross-sectional area in compression and tension. A higher-order rigidity K is defined as follows: K=ECj-{z3dA+E,jA,lz3dA. Also, note that the neutral-surface found from
(1%
position (i.e., the origin of the z co-ordinate)
E.j-[zdA+E,j-jzdA=O.
may be
(16)
Thus, equations (13) and (14) reduce to W., - v, + K(~~W,xx),x = +G,xx,
V,, = pAti.
(17,W
LETTERS
TO THE
361
EDITOR
Combining equations (17) and (18) to eliminate V,, one obtains MX,, + K ( K~w,,),,
- pAti = 0.
+ pl@,,
(19)
In view of the definition of M, in equation (6) and the assumed kinematics in equation (9), one can write M, = -Dw,,,
(20)
where D is the flexural rigidity of the bomodular beam which, in view of equations (12), can be expressed as D=E,~~,Ir’dA+E,~~,Ir’dA.
(21)
Thus, equation (19) can be rewritten in terms of w as Dw,,,
-pA~++(~‘w,,),,f~l~,~~~=O.
(22)
For the case of a uniformly distributed applied bending curvature (K’ = constant) and neglecting the rotatory inertia (pl), one can simplify equation (22) to (D+KK~)w,,,,-~AG=O.
(23)
This is of the same mathematical form as the Bernoulli-Euler beam equation here the flexural rigidity (EI) is replaced by D + KK’. If the beam cross section is assumed to be thin-walled, it can be shown that position (CX)of the neutral plane (a = arc sin z,,/r), where r is the radius of shaft and z, is the z co-ordinate of the neutral plane, can be determined by following implicit equation: (E,-E,)(cos
(Y+a sin a)-(n/2)(E,+E,)
Once LYis known, the flexural and higher-order D/2tr),
= E,
($+sin2cw)-isin2a
K/2tr~=(E,-E,)[(1/6)(4+11
except that the angular the tubular solving the
sin (Y=O.
]+E,[(f+a)(f+sin2a)+fsin2a],
(25)
sin2 (Y)cos (Y
+ (3/4)(2cz - W) sin (Y+ (Ysin3 (Y]-(E, + EE)( r/2) sin3 (Y. 3.
(24)
rigidities can be calculated from
NUMERICAL
(26)
RESULTS
Numerical results have been obtained by solving equation (24) iteratively to obtain the value of LYcorresponding to the input value of E,/ EC. Then equations (25) and (26) were and used to obtain the dimensionless flexural and higher-order rigidities, D/2E,ri K/2E,trz. The relative increase in effective flexural rigidity is (K/D)KO=(K/DL)B.
(27)
Since the natural frequency is proportional to the square root of the effective flexural rigidity, the dimensionless frequency ratio is w/w,=,~+(K/DL)B,
(28)
where w. denotes the frequency of an unprestressed beam. Numerical results are tabulated in Table 1, for a slender shaft with a length/mean radius of 50.91 and prebent through an angle of 15 degrees. It can be seen that there is a very slight increase in natural frequency for E,/E, < 4.60 and a small decrease for E,/ EC > 4.60. However, from a practical engineering standpoint, the effect turns out to be negligible.
LETTERS
362
TO THE EDITOR TABLE
Dimensionless
1
sti$nesses and natural frequency KI2E
1.00 4.00
4.50 4.60 5.00 10.0 20.0 100 125 150 200 250 300
l-571 2.718
0 0.07742
2.812 2.829 2.894 3.390 3.790 4.350 4.399 4.433 4.48 1 4.511 4.532
0.01331 -0.00197 -0.5101 -0.5894 -1.193 -2.285 -2.397 -2.475 -2.579 -2.649 -2.707
(K/DL)Bt 0 0~0001465
0*00002434 -0~OOOOO3581 -0*0009064 -0.0008941 -0.001619 -0.002701 -0+02802 -0.002961 -0XMI2961 -0.003020 -0~00307 1
@/WI 1 *OOoo
1.0001 1*0000 1.0000 0.9995 0.9995 0.9992 0.9986 0.9986 0.9986 0.9985 0.9985 0.9985
t Based on L/r,,, = 50.91 and 0 = 0.2618 radians (15 degrees). The effects of cross-sectional ovalization, i.e., the Brazier effect, shell-type modes, and beam-type geometric non-linearity have not been considered. However, the early experiments and an approximate analysis by Weingarten [7] and a more rigorous numerical investigation by Dawe and Morris [8] indicate that shell action does not significantly modify the natural frequencies of the modes associated with a low circumferential wavenumber. Further, the Brazier effect would change the bending action to a softening one which would tend to lower the frequency relative to the present linear analysis. If there were any axial restraint, geometric non-linearity would effectively stiffen the beam and increase the natural frequencies. In spite of the present demonstration that four effects mentioned above could not have been the cause of an increase in fundamental flexural frequency with increasing beam bending moment, the author recently witnessed a proprietary test in which a significant increase in this frequency was measured. School of Aerospace and Mechanical Engineering, The University of Oklahoma, Norman, Oklahoma 73019, U.S.A.
C. W.
BERT
(Received 30 March 1989) REFERENCES 1988 Journal of Sound and Vibration 126(l), 49-65. Natural frequencies of beams under compressive axial loads. H. REISMANN and P. PAWLIK 1977 Solid Mechanics Archives 2, 129-185. Dynamics of initially stressed hyperelastic solids. M. KUMAR and C. W. BERT 1982 Tire Science and Technology 10, 37-54. Experimental characterization of mechanical behavior of cord-rubber composites. J. L. DOONG and L. W. CHEN 1985 American Society of Mechanical Engineers Journal of Vibration, Acoustics, Stress, and Reliability in Design 107,92-97. Vibration of a bimodulus thick plate. L. ADLER 1970 International Journal of Rock Mechanics and Mining Science 7,357-370. Double elasticity in drill cores under flexure.
1. A. BOKAIAN
2. 3. 4.
5.
LEITERS TO THE EDITOR
363
6. A.A. SHLYAKHMAN and V. A. LEPETOV~~~~ Sooier Rubber Technology M(8), 50-54.Calculation of hose for bending. 2. Determining the radius of curvature of the longitudinal axis of hose in bending, taking into account the displacement of the neutral surface. 7. V. 1. WEINGARTEN 1965American Institute of Aeronautics and Astronautics Journal 3, 40-44. Free vibration of a thin-walled cylindrical shell subjected to a bending moment. 8. D. J.DAWE and I.R. MORRIS 1982Journal of Sound and Vibration 81, 229-237. Vibration of curved plate assemblies subjected to membrane stresses.
EFFECT
359-363
OF BENDING PRELOAD ON FLEXURAL OF A BIMODULAR TUBE
VIBRATION
1. INTRODUCTION
It is well known that a direct axial preload has a significant effect on the flexural vibration frequencies of a slender beam; cf. the paper by Bokaian [ 11. The frequency is increased if the preload is tensile and decreased if the preload is compressive. However, by using the TrefItz type theory of prestressed elastic solids, cf. the paper by Reismann and Pawlik [2], it can be shown that the familiar bending stress distribution, that is linear in the depth co-ordinate, has no effect on the flexural vibration frequencies. This can be explained on the basis that the increase in frequency due to the tensile portion of the prestress distribution is exactly balanced by the decrease in frequency due to the compressive portion. Nevertheless, most fiber-reinforced composites having elastomeric matrices are highly “bimodular” in their mechanical behavior: i.e., they have a much smaller elastic modulus in compression than that in tension [3]. For such materials, prestressed media theory does predict a slight effect of bending stress distribution on natural frequencies. For instance, Doong and Chen [4] investigated this effect for prestressed, bimodular, thick plates. To the best of the author’s knowledge, there has been no investigation of the effect of bending prestress on the vibration of either solid or hollow circular section beams of bimodular material. However, the static bending of solid and hollow circular section beams was analyzed by Adler [5] and Shlyakhman and Lepetov [6], respectively. In the present analysis, transverse shear deformation is neglected, since the material of physical interest had a shear modulus greater than its elastic modulus. 2.
ANALYSIS
In the absence of body forces, the three-dimensional elastodynamic equations of motion, according to Reismann and Pawlik ([2], p. 145), are uij,j-(a~kui,k),,=Piii
(LJ
k =
(1)
L&3),
where ui is the displacement component in the Xi direction, p is the material density, o” is the prestress tensor, u is the stress tensor due to dynamic motion, ( ),j = a( )/aXj, (“) = a’( )/at*, and t is time. Consider a beam of hollow circular cross section and let x, y and z be the position co-ordinates along the beam axis, in the direction normal to the plane of bending preload, and in the other normal direction (in the plane of bending), respectively. Now consider a varying but uniaxial prestress a:, acting in the axial direction. Then the equations of motion corresponding to the x and z directions are UXX,X +7X,,, -(&&,X),X
= P;i,,
7X,,,+ a,,, -(&%,X),X = PK.
(293)
Note that rZXhas been replaced by T,= due to the symmetry of the stress tensor and that thickness-direction normal stress a,, has been neglected, as is customary in beam theory. Then equations (2) and (3) become ox*,x+ 7XZ.Z - ( UlXux,x),X= P& 9
TX,,, -+2x%,x).*
= Pk.
(495)
359 0022460X/89/200359
+ 05 %03.00/O
@ 1989 Academic
Press Limited
360 The
LETTERS
TO THE
EDITOR
bending moment and transverse shear force are defined as M,=
zaxx dA,
IIA
v, =
IIA
(6)
7x, dA,
where A is the cross-sectional area and z is the distance from the neutral plane (not the midplane). Multiplying equation (4) by z, integrating over A, and assuming that 7,, = 0 at the top and bottom fibers, one obtains Mx,x - v, -
(&Q,xz
IIA
zii, dA.
dA = p
(7)
Similarly, integration of equation (5) gives
(8)
ii, dA.
Now, the Bernoulli-Euler hypothesis is invoked; i.e., transverse shear and transverse normal strains are neglected. Thus, (9)
UZ(X,z, t) = w(x, t),
a,(-% z, t) = -zw,x(x, t),
where w is the normal detlection. Equations (7) and (8) become Mx,, - v, +
IIA
v,,x -
((+:x~xx),xz~ dA = -PI+,,,
IIA
(a!&Jx
dA = pA+. (10,11)
The material is assumed to be bimodular with elastic moduli EC and E, in compression and tension, respectively. Thus, ,J
_
uxxwhere
K’
EcK”Z,
(12)
Er~oz,
is the initial bending curvature. Equations (10) and (11) become M,,, - v, + E, V.,, - E,
IIA‘ IIA‘
(K”W,xx),xz3dA+ E, (K”‘+‘,x),xzdA - E,
IIA,
( K”W,,),,Z3
(K’W,,),,Z
IIA,
dA = -PI@,,, dA = pAti,
(13) (14)
where A, and A, denote the respective portions of the cross-sectional area in compression and tension. A higher-order rigidity K is defined as follows: K=ECj-{z3dA+E,jA,lz3dA. Also, note that the neutral-surface found from
(1%
position (i.e., the origin of the z co-ordinate)
E.j-[zdA+E,j-jzdA=O.
may be
(16)
Thus, equations (13) and (14) reduce to W., - v, + K(~~W,xx),x = +G,xx,
V,, = pAti.
(17,W
LETTERS
TO THE
361
EDITOR
Combining equations (17) and (18) to eliminate V,, one obtains MX,, + K ( K~w,,),,
- pAti = 0.
+ pl@,,
(19)
In view of the definition of M, in equation (6) and the assumed kinematics in equation (9), one can write M, = -Dw,,,
(20)
where D is the flexural rigidity of the bomodular beam which, in view of equations (12), can be expressed as D=E,~~,Ir’dA+E,~~,Ir’dA.
(21)
Thus, equation (19) can be rewritten in terms of w as Dw,,,
-pA~++(~‘w,,),,f~l~,~~~=O.
(22)
For the case of a uniformly distributed applied bending curvature (K’ = constant) and neglecting the rotatory inertia (pl), one can simplify equation (22) to (D+KK~)w,,,,-~AG=O.
(23)
This is of the same mathematical form as the Bernoulli-Euler beam equation here the flexural rigidity (EI) is replaced by D + KK’. If the beam cross section is assumed to be thin-walled, it can be shown that position (CX)of the neutral plane (a = arc sin z,,/r), where r is the radius of shaft and z, is the z co-ordinate of the neutral plane, can be determined by following implicit equation: (E,-E,)(cos
(Y+a sin a)-(n/2)(E,+E,)
Once LYis known, the flexural and higher-order D/2tr),
= E,
($+sin2cw)-isin2a
K/2tr~=(E,-E,)[(1/6)(4+11
except that the angular the tubular solving the
sin (Y=O.
]+E,[(f+a)(f+sin2a)+fsin2a],
(25)
sin2 (Y)cos (Y
+ (3/4)(2cz - W) sin (Y+ (Ysin3 (Y]-(E, + EE)( r/2) sin3 (Y. 3.
(24)
rigidities can be calculated from
NUMERICAL
(26)
RESULTS
Numerical results have been obtained by solving equation (24) iteratively to obtain the value of LYcorresponding to the input value of E,/ EC. Then equations (25) and (26) were and used to obtain the dimensionless flexural and higher-order rigidities, D/2E,ri K/2E,trz. The relative increase in effective flexural rigidity is (K/D)KO=(K/DL)B.
(27)
Since the natural frequency is proportional to the square root of the effective flexural rigidity, the dimensionless frequency ratio is w/w,=,~+(K/DL)B,
(28)
where w. denotes the frequency of an unprestressed beam. Numerical results are tabulated in Table 1, for a slender shaft with a length/mean radius of 50.91 and prebent through an angle of 15 degrees. It can be seen that there is a very slight increase in natural frequency for E,/E, < 4.60 and a small decrease for E,/ EC > 4.60. However, from a practical engineering standpoint, the effect turns out to be negligible.
LETTERS
362
TO THE EDITOR TABLE
Dimensionless
1
sti$nesses and natural frequency KI2E
1.00 4.00
4.50 4.60 5.00 10.0 20.0 100 125 150 200 250 300
l-571 2.718
0 0.07742
2.812 2.829 2.894 3.390 3.790 4.350 4.399 4.433 4.48 1 4.511 4.532
0.01331 -0.00197 -0.5101 -0.5894 -1.193 -2.285 -2.397 -2.475 -2.579 -2.649 -2.707
(K/DL)Bt 0 0~0001465
0*00002434 -0~OOOOO3581 -0*0009064 -0.0008941 -0.001619 -0.002701 -0+02802 -0.002961 -0XMI2961 -0.003020 -0~00307 1
@/WI 1 *OOoo
1.0001 1*0000 1.0000 0.9995 0.9995 0.9992 0.9986 0.9986 0.9986 0.9985 0.9985 0.9985
t Based on L/r,,, = 50.91 and 0 = 0.2618 radians (15 degrees). The effects of cross-sectional ovalization, i.e., the Brazier effect, shell-type modes, and beam-type geometric non-linearity have not been considered. However, the early experiments and an approximate analysis by Weingarten [7] and a more rigorous numerical investigation by Dawe and Morris [8] indicate that shell action does not significantly modify the natural frequencies of the modes associated with a low circumferential wavenumber. Further, the Brazier effect would change the bending action to a softening one which would tend to lower the frequency relative to the present linear analysis. If there were any axial restraint, geometric non-linearity would effectively stiffen the beam and increase the natural frequencies. In spite of the present demonstration that four effects mentioned above could not have been the cause of an increase in fundamental flexural frequency with increasing beam bending moment, the author recently witnessed a proprietary test in which a significant increase in this frequency was measured. School of Aerospace and Mechanical Engineering, The University of Oklahoma, Norman, Oklahoma 73019, U.S.A.
C. W.
BERT
(Received 30 March 1989) REFERENCES 1988 Journal of Sound and Vibration 126(l), 49-65. Natural frequencies of beams under compressive axial loads. H. REISMANN and P. PAWLIK 1977 Solid Mechanics Archives 2, 129-185. Dynamics of initially stressed hyperelastic solids. M. KUMAR and C. W. BERT 1982 Tire Science and Technology 10, 37-54. Experimental characterization of mechanical behavior of cord-rubber composites. J. L. DOONG and L. W. CHEN 1985 American Society of Mechanical Engineers Journal of Vibration, Acoustics, Stress, and Reliability in Design 107,92-97. Vibration of a bimodulus thick plate. L. ADLER 1970 International Journal of Rock Mechanics and Mining Science 7,357-370. Double elasticity in drill cores under flexure.
1. A. BOKAIAN
2. 3. 4.
5.
LEITERS TO THE EDITOR
363
6. A.A. SHLYAKHMAN and V. A. LEPETOV~~~~ Sooier Rubber Technology M(8), 50-54.Calculation of hose for bending. 2. Determining the radius of curvature of the longitudinal axis of hose in bending, taking into account the displacement of the neutral surface. 7. V. 1. WEINGARTEN 1965American Institute of Aeronautics and Astronautics Journal 3, 40-44. Free vibration of a thin-walled cylindrical shell subjected to a bending moment. 8. D. J.DAWE and I.R. MORRIS 1982Journal of Sound and Vibration 81, 229-237. Vibration of curved plate assemblies subjected to membrane stresses.