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Closed-form solutions of static and dynamic problems of long, composite cylindrical shells
ComposiIes En&Gwering, Vol. I, No. 4* pp. 225-233, 1991. Printed in Great Britain.
0961-9$26/91 S3.W+ .OO @ I991 Pergamon Press plc
CLOSED-FORM SOLUTIONS OF STATIC AND DYNAMIC PROBLEMS OF LONG, COMPOSITE CYLINDRICAL SHELLS University of Missouri-Rolla,
VICTOR BIRMAN Engineering Education Center, 8001 Natural Bridge Road, St Louis, MO 63121, U.S.A.
and MARK G. MAGID MICO, St Louis, MO 63130, U.S.A. (Received
21 Mu&
1991)
Abstract-Vlasov’s semi-membrane isotropic shells theory is extended to composite cylindrical shells. The resulting theory can be used to obtain closed-form solutions for the statics and dynamics of composite cylindrical shells. Two versions of the theory are proposed. The range of applicability and the accuracy of the new theory are established via a comparison of natural frequencies calculated by the theory with those obtained by Love’s first approximation theory. It is shown that the semi-membrane theory accurately predicts the lowest frequencies of long, composite cychndrical shells. NOTATION extensional, coupling and bending stiffnesses width of cylindrical panel shell length stress couples integers stress resultants load intensities shell middle surface radius axial, circumferential and radial displacements axial and circumferential coordinates mass per unit area natural frequency.
INTRODUCTION
Long cylindrical shells are used in pipelines, missiles, pressure vessels and other applications. It is often advantageous to manufacture these shells from advanced composite materials. Therefore, knowledge of the static and dynamic behavior of composite cylindrical shells of large length-to-radius ratio is important for their application in engineering. A large number of shell theories have been developed, e.g. the Donnell, Morley, Love’s first approximation, Sanders, Fhigge and Novozhilov theories. Although all these theories were originally developed for isotropic shells, most of them have been extended to composite material structures. A comparison of results obtained based on the first four theories cited above was performed for composite cylindrical shells by Bert and Reddy (1982), who showed that a relatively-simple Donnell’s theory yields an acceptable accuracy if the radius-to-thickness ratio of the shell exceeds 20. This is an important conclusion since an eighth-order differential equation for a potential function was derived based on Donnell’s theory in the monograph of Ambartsumian (1974). This equation can be used to obtain closed-form solutions of static and dynamic problems for arbitrary boundary conditions. 225
226
V. BIRMAN
and M. G. MAGID
In spite of this important development, the creation of closed-form solutions for composite shells based on the potential function is so laborious that the method developed by Ambartsumian has mostly theoretical importance, i.e. no analytical solutions utilizing this approach have appeared in the literature. It should also be mentioned that Donnell’s shell theory is not accurate for long shells. A simplified theory is necessary to develop closed-form analytical solutions for long composite shells and to justify their advantage compared to results available using computer codes. In the present paper such a simplified theory is proposed, based on an extension of Vlasov’s semi-membrane theory (Vlasov, 1944). According to the proposed theory, stress couples MX, MW and, therefore, the transverse shear stress resultant QX are negligible due to a large length of the shell. In addition, the circumference of the shell is inextensible. An additional Vlasov’s assumption that in-surface shearing deformations are negligible is adopted only for the solution obtained using the compatibility equation (Method 2). The governing equations are developed based on Love’s shell theory. The deflection pattern being assumed harmonic in the circumferential direction, these equations are reduced to a single fourth-order ordinary differential equation for radial deflection. The solution of this equation includes four constants of integration which are related via boundary conditions. In this paper, the stability and vibration of shells with arbitrary boundary conditions are considered and closed-form analytical solutions are obtained. The problem of the range of applicability of the new theory is addressed by comparing solutions derived using this theory with results obtained using Love’s first approximation theory of shells for the only type of boundary conditions where a closed-form solution based on Love’s theory is available. The limitations of the new theory being established, it can be used to obtain closed-form solutions for various boundary conditions. Customary notation of the the theories of shells and composite material structures is used throughout the paper. SEMI-MEMBRANE
Method
SHELL
THEORY
FOR
COMPOSITE
CYLINDRICAL
SHELLS
1: Equations in terms of displacements
The semi-membrane theory is developed based on Love’s first approximation theory. The equations of equilibrium of an infinitesimal shell element are
shell
N&X + Ny,y = -a
The positive direction of the radial coordinate is taken toward the center of curvature. The assumption MX = MW = 0 results in a simplified system of equations: Nx,x + Nw,y = -qx M N .vJ + 4.Y - - Y*Y -- -qY R M y,yy + 2
= -q.
The shell is laminated in such a manner that the stiffnesses Al6 = AZ6 = BIG = Bz6 = Dr6 = Dz6 = 0. This situation occurs if the shell is formed ofi -multiple isotropic layers; -antisymmetric cross-ply layers (B,* = Be6 = 0, Bz2 = -Bll); -a large number of symmetric angle-ply layers (Bij = 0); -symmetric cross-ply or specially-orthotropic layers (Bu = 0).
221
Closed-formsolutionsof cylindricalshells
The stress resultants and the stress couple MY are the following displacements:
functions
of the
(3)
My = B12u.x - 42w.m The substitution
- D22(w,w + 3.
of (3) into (2) yields the system of equilibrium
equations:
A closed-form solution of (4) is possible if the external loads and the displacement field can be represented as harmonic functions of the circumferential coordinate:
b4w,qx,d
= iU,W,&,qJsinb
(9
tu9 $4 = iv9 c&Jax AY where A = n/R.
where
k4 =
(B,2 + B66)A2 - 2 L
ks = [(&
k5 I
-$A]-’
k, = A66 - $f
V. BIUAN
228
and M. G. MACXO
The substitution of the first and the third equations (6) into the second equation yields a fourth-order differential equation for W: n Y@Lx + n~bVxx + n:W = Q
@I
where
Q = -k,k,hzqxx
- (kzk, + kcjhg
- gy - k5k,h
no1 = k,k,klo
+ k 3 k7
n: = klk,kll
+ (k2k, + k6)k,o + k4k, + k8
(9)
nt = (k2 k, + k6)kll + kg. Suppose that axial compression of cylindrical shells is considered. Then consideration of an equilibrium of an infinitesimal shell element yields (Vol’mir, 1967)
where N, is an applied load. The substitution
of (10) into (8) and (9) yields
n I Yxxxx + n2yxx + n3W = 0 where n, = ny - N,k,k,z n2 = ni
+
k,k,o
+ WV,
+ V&
- Wzb + W,z + k, k,, + bk,o + kW, + W,&
n3 = n$’ - k2kllNl. The integral of (11) must satisfy the boundary conditions, If the shell is simply supported, one can use the expression for iWX:
Equation (13) incorporates the assumption that the shell is inextensible in the circumferential direction. The requirement 1?4~= 0 in conjunction with the last equations (6) and (10) yields the boundary condition yXX = 0. Note that this also implies that u,~ = NX = 0, i.e. the ends of the shell are not restrained against axial movements. Other boundary conditions such as clamping, elastic support or elastic clamping can be included in the analysis. Four constants of integration in the integral of (11) are related via boundary conditions that yield a system of homogeneous algebraic equations. The nonzero requirement for constants of integration provides the buckling equation. If the shell is simply supported, W = f sin vx,
(14)
Closed-form solutions of cylindrical shells
229
satisfies the boundary conditions. The substitution of (14) into (11) yields an equation for the buckling load that is omitted here for brevity. The present method can be also used if vibrations of the shell are considered. In vibration problems of long shells that satisfy Vlasov’s theory assumptions regarding inextensibility of the middle surface, circumferential inertia cannot be disregarded as compared to radial inertia. The requirement &y = v,y - !I! = 0 R
WI
indicated that if the number of deformation halfwaves in the circumferential small, I’ is of the same order as W. The substitution of q = pw2W, q,, = -pw2 W/AR into the expression for Q given by (9) yields
1
Q = -pw2 Therefore, the differential of the shell is
direction is & = 0 and
k,k,k12FVxx - (k2k, + k6)k,2W - g . Ufd L equation that can be used to evaluate the natural frequencies rl; yxxxx + n;bvxx + n;w = 0
(17)
where ?I; = rly ni = r$ + pwzkl k, klZ
w
n; = ni + pw2 k12(k2kT + k6) + & . L I If the shell is simply supported, the natural frequencies are given by -1
pw2 = (nyv4
k12(kl k,v2 - k2k, - k6) - &
. I
Method 2; Utilization
of compatibility
equation
This method follows an approach illustrated by Vol’mir Equations (2) are transformed to
(1967) for isotropic shells.
WV
RMy.yyyy + ;My ,yy + Nx,xx = FJ where
(21)
IJ = -qx,x + qy,y - Rqyy.
The assumptions that the shell circumference is inextensible and in-surface shearing deformations are negligible yield v,~ = w/R and u,~ = -v,~. These relations, substituted into expressions for the axial strain .sX= u,~ and for the change of curvature in the crosssections perpendicular to the shell axis JC~= -(w,~,, + w/R), yield the compatibility conditions: &,yyyy
+ ; %yy =
KYJX
Gw
*
Introducing nondimensional coordinates (CY,/?) = (x, y)/R and the differential Q = ( ~m3L3 + ( ).Lw~ one can transform (20) and (22) into ?S2Nx-$
SIW,~~ -j$(D22-+3
operator
Q(w,~@ + w) + RNx,aa = R=P*
11
+ N,a*
(23)
where p* = -Gx COE 1;4-c
+
qY,o
- aw*
(24)
V. BIRMAN
230
and M. G. MAGID
The expression for QNX can be substituted from the second equation (23) into the first equation (23) that has been previously subject to the differential operator 0. This yields the single partial differential equation D22QWw,flj3 + WI + D22QQw,LYa + ~~l~Q~,amxcY + 24*~wv,&3
+ N,aa
+ A~lR2(w,~p+ w),~~~~= - R4i2P*.
WI
Excluding the operator ( ),BB + 1, one obtains D22aQw
+ Q2Qyc43 z
-R4p*
+
41Rw,amm
+
=3,2RWm
+ &R2yaaaa
(261
*Iv*
If the shell is isotropic, D22 = D, D12= BII = B12= 0, AlI = Eh, and (26) coincides with the corresponding equation presented by Vol’mir (1967). One can easily derive the counterparts of eqns (8), (11) and (17) if (24,w) = (U, W) sin n/3 (27)
u = Vcosnj3 are used to specify the right side of (26) in terms of W. The differential equation for the buckling load is
(AIIR - BIIn2)RW’aaaa+ n2[(l - n2)(n2D12- 2B12R)+ (n2 + l)ZVIR2]FVaa + n4(n2- l)zD22 W = 0. In particular,
WI
if the shell is simply supported, W=fsini%,
M=- rnnR
L
and
NICr= {(A,IR - BI,n2)RM4 - n2[(l - n2)(n2D12- 2B12R)]M2 + n4(l - n2)2D22J[n2(n2 + l)R2M2]-‘.
(30)
In the vibration problem the differential equation of motion can be developed if the derivatives of in-surface displacements in
P* = pC02(-I.4a + v,fl - W,BO)
(31)
are expressed in terms of W. This can be easily done using the conditions that the middle surface is inextensible in the circumferential direction and its shearing deformations are negligible. Then the differential equation obtained after some transformations becomes
(A,,R - BII n2)RFVaaaa + n2 (1 - n2)(n2D12- 2B12R)- p$R4 + n2[(l - n2)2n2D22 + pa2R4(n2+ l)]W = 0. The natural frequencies of a simply-supported
I
FVaa (32)
shell are
a2 = {(AIIR - BIIn2)RM4 - n2[(l - n2)(n2D,z- 2B12R)]M2+ n4(l - n2)2D22J x {pR4[n2(n2+ 1) + M2]jm’.
(33)
Notably, the second method presented here results in much simpler equations. It is also emphasized that an effect of in-surface inertias on the frequencies of radial vibrations is accounted for in (33). NUMERICAL
ANALYSIS
The accuracy and limits of applicability of Vlasov’s semi-membrane shell theory extended to composite shells were analyzed by comparing the natural frequencies obtained
Closed-form solutions of cylindrical shells
231
using(33)with the frequenciescalculatedusingLove’s shelltheory. Note that the frequencies obtainedusing (33) practically coincidedwith thosecalculatedusing (19)in several representativeexamplesconsidered.The results were obtained for simply-supported shells,where Nx = v = w = w,~~= 0 at x = 0, L. (34) This is the only type of boundary condition that allows a closed-form solution using Love’s shell theory if each displacementcomponent(u, v and w) is representedby a productof two trigonometricfunctions(onefunction of x andanotherfunction of y). The correspondingsolution was obtainedby Bert and Reddy(1982). The calculationswereperformed for multilayered symmetrically-laminatedboronepoxyshells(& = 206.850MPa, ET= 20.685MPa, vLT = 0.3, Grr = 6.895MPa). The radius of the shell was R = 0.508m and its thicknesswas equal to 0.00127m. The results of the calculationsare presentedin Figs l-5. As follows from these figures,thedifferencebetweentheresultsobtainedby Love’sandVlasov’ssemi-membrane theoriesdecreases for longershells.The worstconvergenceof resultsobtainedby two shell theoriescorrespondsto n = 2 (two halfwavesof deformationin the circumferentialdirection). The bestconvergence correspondsto vibrationswith onehalfwaveof deformationin
Love - Vlasov Ratio m=l
Fig. 1. Ratio of squared natural frequencies calculated using Love’s shell theory to those obtained by Vlasov’s semi-membrane theory; m = 1,
Love - Vlasov Ratio m=2 .- --* -. -I n=2 + n=3 -a+ n=5 -f3n=10
Fig. 2. Ratio of squared natural frquencies calculated using Love’s shell theory to those obtained by Vlasov’s semi-membrane theory; m = 2.
V. BIICMANand M. G. MAOID
232
Love - Vlasov Ratio m=3 -an=2 -+ n=3 -*.. n=5 -t3-n=10
Fig. 3. Ratio of squared natural frequencies calculated using Love’s shell theory to those obtained by Vlasov’s semi-membrane theory; m = 3.
Love - Vlasov Ratio m=5
n=lO ---
0
5
10
15
20
26
30
of squared natural frequencies calculated using Love’s shell theory to those by Vlasov’s semi-membrane theory; m = 5.
Love - ;lyv
Ratio --.. .-m-
n=2 + n=3 -4. n=6 -Hn=lC - --
0
6
Fig. 5. Ratio of squared natural frequencies calculated using Love’s shell theory to those obtained by Vlasov’s semi-membrane theory; m = 10.
Closed-form solutions of cylindrical shells
233
the axial direction (m = 1). Vlasov’s semi-membrane shell theory provides practically accurate results for shells with a length-to-radius ratio larger than 18-20, vibrating with one halfwave in the axial direction. It is emphasized that the length-to-radius ratios mentioned here refer to the shells considered in this example. Different materials, laminations and geometry could change the values of minimum length that limit an acceptability range of the semi-membrane theory. The usefulness of Vlasov’s semi-membrane theory can be established only based on an assessment of the mode shapes corresponding to the lowest frequencies of vibration. At all ratios L/R the lowest frequencies corresponded to m = 1 and small numbers n. Therefore, Vlasov’s semi-membrane theory can accurately predict the lowest frequencies of vibration of long, composite cylindrical shells. CONCLUSIONS
A relatively-simple theory of composite cylindrical shells with a large length-to-radius ratio is proposed. The theory represents an extension of Vlasov’s semi-membrane theory to composite material shells. The advantage of the new theory compared to existing theories of shells is that it can be used to generate closed-form solutions for arbitrary boundary conditions. Two versions of the theory are proposed. One of them employs equations of motion (equilibrium) in term of displacements, while the other includes the compatibility condition. The natural frequencies of simply-supported shells obtained using the second version of the theory were compared to those calculated using Love’s shell theory. It was shown that the semi-membrane theory can provide accurate results for the lowest frequencies of vibration of the shells with large length-to-radius ratios. REFERENCES Ambarsumian, S. A. (1974). Generut ?%eory of Anisotropic Shells. Nauka, Moscow. (In Russian.) Bert, C. W. and Reddy, V. S. (1982). Composite shells of bimodulus composite materials. AXE J. Engng Mech. 108, EM5, 675-688. Vlasov, V. Z. (1944). Governing differential equations of a general theory of elastic shells. AppL M&h. Mech. 8(2). (In Russian.) Vol’mir, A. S. (1967). Stability of Deformable Systems, pp. 51 I-521. Nauka, Moscow. (In Russian.)
0961-9$26/91 S3.W+ .OO @ I991 Pergamon Press plc
CLOSED-FORM SOLUTIONS OF STATIC AND DYNAMIC PROBLEMS OF LONG, COMPOSITE CYLINDRICAL SHELLS University of Missouri-Rolla,
VICTOR BIRMAN Engineering Education Center, 8001 Natural Bridge Road, St Louis, MO 63121, U.S.A.
and MARK G. MAGID MICO, St Louis, MO 63130, U.S.A. (Received
21 Mu&
1991)
Abstract-Vlasov’s semi-membrane isotropic shells theory is extended to composite cylindrical shells. The resulting theory can be used to obtain closed-form solutions for the statics and dynamics of composite cylindrical shells. Two versions of the theory are proposed. The range of applicability and the accuracy of the new theory are established via a comparison of natural frequencies calculated by the theory with those obtained by Love’s first approximation theory. It is shown that the semi-membrane theory accurately predicts the lowest frequencies of long, composite cychndrical shells. NOTATION extensional, coupling and bending stiffnesses width of cylindrical panel shell length stress couples integers stress resultants load intensities shell middle surface radius axial, circumferential and radial displacements axial and circumferential coordinates mass per unit area natural frequency.
INTRODUCTION
Long cylindrical shells are used in pipelines, missiles, pressure vessels and other applications. It is often advantageous to manufacture these shells from advanced composite materials. Therefore, knowledge of the static and dynamic behavior of composite cylindrical shells of large length-to-radius ratio is important for their application in engineering. A large number of shell theories have been developed, e.g. the Donnell, Morley, Love’s first approximation, Sanders, Fhigge and Novozhilov theories. Although all these theories were originally developed for isotropic shells, most of them have been extended to composite material structures. A comparison of results obtained based on the first four theories cited above was performed for composite cylindrical shells by Bert and Reddy (1982), who showed that a relatively-simple Donnell’s theory yields an acceptable accuracy if the radius-to-thickness ratio of the shell exceeds 20. This is an important conclusion since an eighth-order differential equation for a potential function was derived based on Donnell’s theory in the monograph of Ambartsumian (1974). This equation can be used to obtain closed-form solutions of static and dynamic problems for arbitrary boundary conditions. 225
226
V. BIRMAN
and M. G. MAGID
In spite of this important development, the creation of closed-form solutions for composite shells based on the potential function is so laborious that the method developed by Ambartsumian has mostly theoretical importance, i.e. no analytical solutions utilizing this approach have appeared in the literature. It should also be mentioned that Donnell’s shell theory is not accurate for long shells. A simplified theory is necessary to develop closed-form analytical solutions for long composite shells and to justify their advantage compared to results available using computer codes. In the present paper such a simplified theory is proposed, based on an extension of Vlasov’s semi-membrane theory (Vlasov, 1944). According to the proposed theory, stress couples MX, MW and, therefore, the transverse shear stress resultant QX are negligible due to a large length of the shell. In addition, the circumference of the shell is inextensible. An additional Vlasov’s assumption that in-surface shearing deformations are negligible is adopted only for the solution obtained using the compatibility equation (Method 2). The governing equations are developed based on Love’s shell theory. The deflection pattern being assumed harmonic in the circumferential direction, these equations are reduced to a single fourth-order ordinary differential equation for radial deflection. The solution of this equation includes four constants of integration which are related via boundary conditions. In this paper, the stability and vibration of shells with arbitrary boundary conditions are considered and closed-form analytical solutions are obtained. The problem of the range of applicability of the new theory is addressed by comparing solutions derived using this theory with results obtained using Love’s first approximation theory of shells for the only type of boundary conditions where a closed-form solution based on Love’s theory is available. The limitations of the new theory being established, it can be used to obtain closed-form solutions for various boundary conditions. Customary notation of the the theories of shells and composite material structures is used throughout the paper. SEMI-MEMBRANE
Method
SHELL
THEORY
FOR
COMPOSITE
CYLINDRICAL
SHELLS
1: Equations in terms of displacements
The semi-membrane theory is developed based on Love’s first approximation theory. The equations of equilibrium of an infinitesimal shell element are
shell
N&X + Ny,y = -a
The positive direction of the radial coordinate is taken toward the center of curvature. The assumption MX = MW = 0 results in a simplified system of equations: Nx,x + Nw,y = -qx M N .vJ + 4.Y - - Y*Y -- -qY R M y,yy + 2
= -q.
The shell is laminated in such a manner that the stiffnesses Al6 = AZ6 = BIG = Bz6 = Dr6 = Dz6 = 0. This situation occurs if the shell is formed ofi -multiple isotropic layers; -antisymmetric cross-ply layers (B,* = Be6 = 0, Bz2 = -Bll); -a large number of symmetric angle-ply layers (Bij = 0); -symmetric cross-ply or specially-orthotropic layers (Bu = 0).
221
Closed-formsolutionsof cylindricalshells
The stress resultants and the stress couple MY are the following displacements:
functions
of the
(3)
My = B12u.x - 42w.m The substitution
- D22(w,w + 3.
of (3) into (2) yields the system of equilibrium
equations:
A closed-form solution of (4) is possible if the external loads and the displacement field can be represented as harmonic functions of the circumferential coordinate:
b4w,qx,d
= iU,W,&,qJsinb
(9
tu9 $4 = iv9 c&Jax AY where A = n/R.
where
k4 =
(B,2 + B66)A2 - 2 L
ks = [(&
k5 I
-$A]-’
k, = A66 - $f
V. BIUAN
228
and M. G. MACXO
The substitution of the first and the third equations (6) into the second equation yields a fourth-order differential equation for W: n Y@Lx + n~bVxx + n:W = Q
@I
where
Q = -k,k,hzqxx
- (kzk, + kcjhg
- gy - k5k,h
no1 = k,k,klo
+ k 3 k7
n: = klk,kll
+ (k2k, + k6)k,o + k4k, + k8
(9)
nt = (k2 k, + k6)kll + kg. Suppose that axial compression of cylindrical shells is considered. Then consideration of an equilibrium of an infinitesimal shell element yields (Vol’mir, 1967)
where N, is an applied load. The substitution
of (10) into (8) and (9) yields
n I Yxxxx + n2yxx + n3W = 0 where n, = ny - N,k,k,z n2 = ni
+
k,k,o
+ WV,
+ V&
- Wzb + W,z + k, k,, + bk,o + kW, + W,&
n3 = n$’ - k2kllNl. The integral of (11) must satisfy the boundary conditions, If the shell is simply supported, one can use the expression for iWX:
Equation (13) incorporates the assumption that the shell is inextensible in the circumferential direction. The requirement 1?4~= 0 in conjunction with the last equations (6) and (10) yields the boundary condition yXX = 0. Note that this also implies that u,~ = NX = 0, i.e. the ends of the shell are not restrained against axial movements. Other boundary conditions such as clamping, elastic support or elastic clamping can be included in the analysis. Four constants of integration in the integral of (11) are related via boundary conditions that yield a system of homogeneous algebraic equations. The nonzero requirement for constants of integration provides the buckling equation. If the shell is simply supported, W = f sin vx,
(14)
Closed-form solutions of cylindrical shells
229
satisfies the boundary conditions. The substitution of (14) into (11) yields an equation for the buckling load that is omitted here for brevity. The present method can be also used if vibrations of the shell are considered. In vibration problems of long shells that satisfy Vlasov’s theory assumptions regarding inextensibility of the middle surface, circumferential inertia cannot be disregarded as compared to radial inertia. The requirement &y = v,y - !I! = 0 R
WI
indicated that if the number of deformation halfwaves in the circumferential small, I’ is of the same order as W. The substitution of q = pw2W, q,, = -pw2 W/AR into the expression for Q given by (9) yields
1
Q = -pw2 Therefore, the differential of the shell is
direction is & = 0 and
k,k,k12FVxx - (k2k, + k6)k,2W - g . Ufd L equation that can be used to evaluate the natural frequencies rl; yxxxx + n;bvxx + n;w = 0
(17)
where ?I; = rly ni = r$ + pwzkl k, klZ
w
n; = ni + pw2 k12(k2kT + k6) + & . L I If the shell is simply supported, the natural frequencies are given by -1
pw2 = (nyv4
k12(kl k,v2 - k2k, - k6) - &
. I
Method 2; Utilization
of compatibility
equation
This method follows an approach illustrated by Vol’mir Equations (2) are transformed to
(1967) for isotropic shells.
WV
RMy.yyyy + ;My ,yy + Nx,xx = FJ where
(21)
IJ = -qx,x + qy,y - Rqyy.
The assumptions that the shell circumference is inextensible and in-surface shearing deformations are negligible yield v,~ = w/R and u,~ = -v,~. These relations, substituted into expressions for the axial strain .sX= u,~ and for the change of curvature in the crosssections perpendicular to the shell axis JC~= -(w,~,, + w/R), yield the compatibility conditions: &,yyyy
+ ; %yy =
KYJX
Gw
*
Introducing nondimensional coordinates (CY,/?) = (x, y)/R and the differential Q = ( ~m3L3 + ( ).Lw~ one can transform (20) and (22) into ?S2Nx-$
SIW,~~ -j$(D22-+3
operator
Q(w,~@ + w) + RNx,aa = R=P*
11
+ N,a*
(23)
where p* = -Gx COE 1;4-c
+
qY,o
- aw*
(24)
V. BIRMAN
230
and M. G. MAGID
The expression for QNX can be substituted from the second equation (23) into the first equation (23) that has been previously subject to the differential operator 0. This yields the single partial differential equation D22QWw,flj3 + WI + D22QQw,LYa + ~~l~Q~,amxcY + 24*~wv,&3
+ N,aa
+ A~lR2(w,~p+ w),~~~~= - R4i2P*.
WI
Excluding the operator ( ),BB + 1, one obtains D22aQw
+ Q2Qyc43 z
-R4p*
+
41Rw,amm
+
=3,2RWm
+ &R2yaaaa
(261
*Iv*
If the shell is isotropic, D22 = D, D12= BII = B12= 0, AlI = Eh, and (26) coincides with the corresponding equation presented by Vol’mir (1967). One can easily derive the counterparts of eqns (8), (11) and (17) if (24,w) = (U, W) sin n/3 (27)
u = Vcosnj3 are used to specify the right side of (26) in terms of W. The differential equation for the buckling load is
(AIIR - BIIn2)RW’aaaa+ n2[(l - n2)(n2D12- 2B12R)+ (n2 + l)ZVIR2]FVaa + n4(n2- l)zD22 W = 0. In particular,
WI
if the shell is simply supported, W=fsini%,
M=- rnnR
L
and
NICr= {(A,IR - BI,n2)RM4 - n2[(l - n2)(n2D12- 2B12R)]M2 + n4(l - n2)2D22J[n2(n2 + l)R2M2]-‘.
(30)
In the vibration problem the differential equation of motion can be developed if the derivatives of in-surface displacements in
P* = pC02(-I.4a + v,fl - W,BO)
(31)
are expressed in terms of W. This can be easily done using the conditions that the middle surface is inextensible in the circumferential direction and its shearing deformations are negligible. Then the differential equation obtained after some transformations becomes
(A,,R - BII n2)RFVaaaa + n2 (1 - n2)(n2D12- 2B12R)- p$R4 + n2[(l - n2)2n2D22 + pa2R4(n2+ l)]W = 0. The natural frequencies of a simply-supported
I
FVaa (32)
shell are
a2 = {(AIIR - BIIn2)RM4 - n2[(l - n2)(n2D,z- 2B12R)]M2+ n4(l - n2)2D22J x {pR4[n2(n2+ 1) + M2]jm’.
(33)
Notably, the second method presented here results in much simpler equations. It is also emphasized that an effect of in-surface inertias on the frequencies of radial vibrations is accounted for in (33). NUMERICAL
ANALYSIS
The accuracy and limits of applicability of Vlasov’s semi-membrane shell theory extended to composite shells were analyzed by comparing the natural frequencies obtained
Closed-form solutions of cylindrical shells
231
using(33)with the frequenciescalculatedusingLove’s shelltheory. Note that the frequencies obtainedusing (33) practically coincidedwith thosecalculatedusing (19)in several representativeexamplesconsidered.The results were obtained for simply-supported shells,where Nx = v = w = w,~~= 0 at x = 0, L. (34) This is the only type of boundary condition that allows a closed-form solution using Love’s shell theory if each displacementcomponent(u, v and w) is representedby a productof two trigonometricfunctions(onefunction of x andanotherfunction of y). The correspondingsolution was obtainedby Bert and Reddy(1982). The calculationswereperformed for multilayered symmetrically-laminatedboronepoxyshells(& = 206.850MPa, ET= 20.685MPa, vLT = 0.3, Grr = 6.895MPa). The radius of the shell was R = 0.508m and its thicknesswas equal to 0.00127m. The results of the calculationsare presentedin Figs l-5. As follows from these figures,thedifferencebetweentheresultsobtainedby Love’sandVlasov’ssemi-membrane theoriesdecreases for longershells.The worstconvergenceof resultsobtainedby two shell theoriescorrespondsto n = 2 (two halfwavesof deformationin the circumferentialdirection). The bestconvergence correspondsto vibrationswith onehalfwaveof deformationin
Love - Vlasov Ratio m=l
Fig. 1. Ratio of squared natural frequencies calculated using Love’s shell theory to those obtained by Vlasov’s semi-membrane theory; m = 1,
Love - Vlasov Ratio m=2 .- --* -. -I n=2 + n=3 -a+ n=5 -f3n=10
Fig. 2. Ratio of squared natural frquencies calculated using Love’s shell theory to those obtained by Vlasov’s semi-membrane theory; m = 2.
V. BIICMANand M. G. MAOID
232
Love - Vlasov Ratio m=3 -an=2 -+ n=3 -*.. n=5 -t3-n=10
Fig. 3. Ratio of squared natural frequencies calculated using Love’s shell theory to those obtained by Vlasov’s semi-membrane theory; m = 3.
Love - Vlasov Ratio m=5
n=lO ---
0
5
10
15
20
26
30
of squared natural frequencies calculated using Love’s shell theory to those by Vlasov’s semi-membrane theory; m = 5.
Love - ;lyv
Ratio --.. .-m-
n=2 + n=3 -4. n=6 -Hn=lC - --
0
6
Fig. 5. Ratio of squared natural frequencies calculated using Love’s shell theory to those obtained by Vlasov’s semi-membrane theory; m = 10.
Closed-form solutions of cylindrical shells
233
the axial direction (m = 1). Vlasov’s semi-membrane shell theory provides practically accurate results for shells with a length-to-radius ratio larger than 18-20, vibrating with one halfwave in the axial direction. It is emphasized that the length-to-radius ratios mentioned here refer to the shells considered in this example. Different materials, laminations and geometry could change the values of minimum length that limit an acceptability range of the semi-membrane theory. The usefulness of Vlasov’s semi-membrane theory can be established only based on an assessment of the mode shapes corresponding to the lowest frequencies of vibration. At all ratios L/R the lowest frequencies corresponded to m = 1 and small numbers n. Therefore, Vlasov’s semi-membrane theory can accurately predict the lowest frequencies of vibration of long, composite cylindrical shells. CONCLUSIONS
A relatively-simple theory of composite cylindrical shells with a large length-to-radius ratio is proposed. The theory represents an extension of Vlasov’s semi-membrane theory to composite material shells. The advantage of the new theory compared to existing theories of shells is that it can be used to generate closed-form solutions for arbitrary boundary conditions. Two versions of the theory are proposed. One of them employs equations of motion (equilibrium) in term of displacements, while the other includes the compatibility condition. The natural frequencies of simply-supported shells obtained using the second version of the theory were compared to those calculated using Love’s shell theory. It was shown that the semi-membrane theory can provide accurate results for the lowest frequencies of vibration of the shells with large length-to-radius ratios. REFERENCES Ambarsumian, S. A. (1974). Generut ?%eory of Anisotropic Shells. Nauka, Moscow. (In Russian.) Bert, C. W. and Reddy, V. S. (1982). Composite shells of bimodulus composite materials. AXE J. Engng Mech. 108, EM5, 675-688. Vlasov, V. Z. (1944). Governing differential equations of a general theory of elastic shells. AppL M&h. Mech. 8(2). (In Russian.) Vol’mir, A. S. (1967). Stability of Deformable Systems, pp. 51 I-521. Nauka, Moscow. (In Russian.)