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Correction to the solution of orthotropic circular plates
Build. Sci. Vol. 7, pp. 33-36. Pergamon Press 1972. Printed in Great Britain
I
I
I Ry I(K) i[
Correction to the Solution of Orthotropic Circular Plates* WALID H. R I M A W I f BURAK ERMAN,
The small-deflection theory of uniformly loaded circular plates with cylindrical orthotropy leads to irregular behaviour at the center of the plate. Both the moments and the deflection will be either zero or infinite depending on the values of the material constants. In this paper, the solution based on the above theory is examined and a correction utilizing a small isotropic core is presented. The corrected and uncorrected solutions for a material with a specific orthotropy are compared. Also, moments profiles for materials with different degrees of orthotropy are provided. The method can be extended to other cases of symmetric loadings and boundary conditions.
NOMENCLATURE al, a2, a3, a4 all, a22, a12 AI, A2, B~ C , 1,'
D, Di es, e,
E,~ H
M,,M~ mr, mo
p r,O, z R W, W~
INTRODUCTION
undetermined coefficients material constants (see equation 1) defined by equation 8 ail/a22 and -a12/a22 respectively orthotropic and isotropic flexural rigidities strain components Young modulus and Poisson ratio thickness of the plate moment in the r and 0 directions non-dimensional moments
A C I R C U L A R plate with material constants in the radial direction having different values from those in the tangential direction is considered to possess cylindrical orthotropy. This kind of orthotropy occurs in wood, in drawn wire and in plates with different radial and circumferential reinforcing. This property should be taken into account in the stress analysis of such structural elements. It is known[l, 3] that the small-deflection theory of the orthotropic circular plate with uniform loading, leads either to infinite moments or zero moments at the center of the plate depending on the values of the material constants. In this paper the behavior at the center is corrected by realizing that the center of the plate must be isotropic due to symmetry. Hence, by assuming an isotropic core of a small radius new corrected results are obtained which are more realistic in representing the behavior of the plate.
unit lateral load on plate cylindrical coordinate system at origin of plate radius of plate orthotropic and isotropic displacements in the z-direction
~/O/c) ao, a, Q
UNCORRECTED S O L U ~ O N
stress components shearing force per unit length along the circumference of a circle of radius r.
Consider a simply supported cylindrically orthotropic circular plate with radius R and thickness H, and loaded uniformly with load of intensity p. Attaching a cylindrical co-ordinate system r, 0, z to the center of the plate the stress-strain relations in this case may be written in the following form[2]:
* This research was partly supported by the Research Center of Robert College. Prof. of Civil Eng., Robert College, lstanbul, Turkey. Instructor of Civil Eng., Robert College, Istanbul, Turkey.
eO ~
allO.O-~-al2O-r
er = a12as+a22a, 33
(I)
Walid H. Rimawi and Burak Erman
34
where e 0 and e, are the strain components, a 0 and a, are the stress components and air, at2 and a22 are the material's constants. Making the same assumptions used in the small bending theory of isotropic plates, it is clear that the strain displacement relationships and the equations of equilibrium in terms o f stress resultants remain the same as in the case of isotropy[3]. However, in view o f equation 1 the expression for the stress resultants Mr and Mo and the shearing force Q assume the f o r m : dW\
Mr = - D t c'-~5-rz + r
/
dzW
(law %
|
C~09
~-o.o2
0.08
(2)
0.07 ~ L . J ' ~ ~ =0.05 O.06
cd2W
d3W
W .o.oo,
0.10
Mo = - D r "dr-r+ v dr 2 ]
/
rI
0.12 0.11
v
/
where mr and m o thus defined are non-dimensional moments. For isotropic plates, since c = ~ = 1 and v = Poisson ratio p, equation 5 reduces to the wellknown solution[3]. For values of c other than unity it is obvious that Mr and Mo have the values of either zero or infinite at r = 0. Values of mr and mo, given by equation 5 are plotted in figure I for v = 0.3 and c = 1-3.
dW\ r 2 ~rr l
r --~ =0. I0 n
. . . . Represents uncorrected solution
0.05
)
0O4
Furthermore, the field equation becomes:
0.02
Q = Dtc---~r34 -r dr z
~.~/
~
~1~r
~
~
0.03 0,01
d4W c~+
1 dzw 1 dW P r 2 ~dr q r 3 d ~ - = - - D
2cd3W r dr 3
[
0.1
(3)
all =
a22
,
L
0.4
[
0.5
o.g
O.7
0.e
0.9
Lo
Fig. 1. Moments profiles for C = 1-3.
H3a22
a12,
V-
1
0.3
r/R
In equations 2 and 3, C
1
o.z
a22
CORRECTED SOLUTION
D = 12(alia22_a22)
and W is the deflection in the positive z-direction. The solution of equation 3 is given by:
pr 4 W = atrt+~+a2rl-=+a3r2+a4q (72c-8----~ (4) Where a~ to a , are undetermined coefficients and
o~ = Vq/c. Since dW/dr = 0 at r = 0, the coefficient a 2 must vanish. The equation of equilibrium
O r t h o t r o p y cannot be realized at the center of the plate due to the point symmetry. Therefore, the solution given by equation 5 should not be applied in the n e i g h b o u r h o o d of the center. A more realistic solution is obtained by assuming a small isotropic core of radius r 1 in the orthotropic plate. Thus, the plate is divided into two regions. Modifying the material constants for the isotropic core yields the following two equations for the isotropic and orthotropic regions respectively:
dM,
r-~r + M r - M o = ~o-prdr
W i -= b 2 + b l r 2 +
(6)
Pr4
64Di yields M ~ - M o = 0 at r = 0. On the other hand, substituting equation 4 into equation 2 leads to M , - Mo = 2 a 3 ( c - l) at r = 0 also. Since ( c - 1)# 0 for the orthotropic plate a 3 must vanish. The parameter a~ is obtained f r o m the boundary condition Mr = 0 at r = R. Finally, for the values of the parameters a~, a2, and a 3, thus obtained, equation 2 transforms into
M~
(v+3c' r ( r ) ~ - I
m, = 4 - ~
= (7~c-8)
m° = 4 - ~
= ( 7 2 c - 8) L
(R)Z 1
(v + c~c)
(s)
'
-(1 + 3V,(R)Zl
pr 4
W = alrl+~+a2r I ~+a3rZ+a4q
(72C - 8)D
(7)
where W~ is the deflection of the isotropic core, and Di =
EH3/12(1-/,/2).
Equating the shear forces given by last of equation 2 at the interface, r = r 1, yields a 3 = 0. The parameters b~, a~ and az are determined f r o m the boundary conditions Mr = 0 at r = R and f r o m the remaining two continuity conditions at the ortho-isotropic interface, r = r 1, (Mr) = ( M J i and dW/dr = dWi/dr. These three conditions yield the following three simultaneous equations:
(ca + v)A ~+ (v - c~)A2 =
3c+ v 72c- 8
Correction to the Solution of Orthotropic Circular Plates Ir \ . - a
[rl\-~-3
35
o.o;,i.._._, x /c= 1.5
3c+v
3+/z 64
72c-8
0 . 0 6 ~ . 0 o
(8) r 1~-
3
-
-
-
0.03 ~
(D/D,)
1
64
72c-8
002
\
0.OI
where R ' - 3(I +~)D A1 =
R-=-3(1 - ~ ) D al,
4,0
A2 =
4e
I
[
I
[
[
1
I
I
0.1
0"2
0"3
0'4
0'5
0"6
0"7
0'8
02,
INNN 0"9
I'0
I
l
r/R Fig. 2. m. profiles for various values o f e.
(1 + # ) D ~ , ~ 01
B1 =
0.07 ~
and
c = 1.5
I
0.06 ~ . . . . ~
D
l
c = I, I
(1 --#z~
E = Ea2z Itc--v2 ]
0'0"~~
The moment equations in terms of the new nondimensional constants A1, A2 and B 1 take the form:
005
mr = B , ( r " ~ 2 3 + I t ( r ' ~ z
0.02
=
0
.
5
c=0.5
t,~) --A7-~fi)
(9)
0"01
m° = B1
64 ' o
for the isotropic core, and
I
I
!
i
I
I
L
o.~ 02 0.3 0.4 0.5 o.6 0.7 o.B 0.9 J.o r/R
Fig. 3. mo profiles for various values o f c.
E (')-'
(7-' f r V / 3c+v "k-]
(lo) mo
I =
/,r\~_ 1
-
isotropic Young's modulus E, equal to the value of c. The results are plotted in figures 2 and 3. For the value of C = 1.3 the ratio rl/R was varied between 0.1 and 0.001 and the respective values of the corrected m, and mo at r = 0 were equal and fell between 0.6325 and 0.1113.
)
[r'~-~-I DISCUSSION
fr"12f 1 +3v \-] for the remainder of the plate. By the aid of a digital computer, the values of the constants Ax, A 2 and Bt may be first determined and then substituted into equations 9 and 10 to determine the values of the moments m, and m e after selecting values for the parameters involved. The numerical results were obtained for rn, and m0 for c varying between 0.5 and 1.50 with rl/R = 0.1, v = / z = 0.3 and 1/(E.a22), which is the ratio of tangential modulus of elasticity to the
The method proposed for correcting the behavior of the plate at the center may also be used in the case of a plate having an isotropic core. Although the procedure was applied to the uniformly loaded simply supported plate, it can in general be applied to all cases of symmetric loading and constraint conditions by changing the particular solution, the fifth term in equation 4 and considering the proper boundary conditions. The choice of rl/R is arbitrary but any choice of a small value is acceptable. It should be noted that in the case where the moments tend towards infinite at the center, the maximum value of
Walid H. Rimawi and Burak Erman
36
moments may not exceed the plastic isotropic moment. At the interface, rl/R = 0.I, m o is n o t continuous since this is not acontinuity requirement. Physically, this might be due to the abrupt changes in the materials constants. If the determination o f the deflection (W) is desired the parameters b 2 and a 4 in equations 6 and 7 may be found from two additional b o u n d a r y conditions, namely; W = 0 at r = R and W = W i at r = r 1 .
Examining figure 2 which shows the profile o f mr for various values o f the parameter c, it can be seen that for c > 1 the m a x i m u m value of mr occurs at r = 0 and for c < 1 the position of the m a x i m u m value o f m r starts moving towards the edge. For example for c = 0.5 m a x i m u m mr occurs at r/R = 0"4. The behavior o f mo as shown in figure 3 is similar to that of Mr. It is of interest to note that the value o f rno in the n e i g h b o u r h o o d o f r/R = 0.3 does not appreciably vary with the degree of orthotropy.
REFERENCES 1. 2. 3.
G . F . CARgIER, The bending of the cylindrically aeolotropic plate, J. AppL Mech. 2, A129 (1944). S . G . LEKHNITSKII,Theory of Elasticity of an Anisotropic Elastic Body, Holden-Day, San Francisco (1963). S.P. TIMOSHENKOand S. WOINOWSKY-KR1EGER,Theory of Plates and Shells, pp, 365377 and p. 57, McGraw-Hill, New York (1959).
I
I
I Ry I(K) i[
Correction to the Solution of Orthotropic Circular Plates* WALID H. R I M A W I f BURAK ERMAN,
The small-deflection theory of uniformly loaded circular plates with cylindrical orthotropy leads to irregular behaviour at the center of the plate. Both the moments and the deflection will be either zero or infinite depending on the values of the material constants. In this paper, the solution based on the above theory is examined and a correction utilizing a small isotropic core is presented. The corrected and uncorrected solutions for a material with a specific orthotropy are compared. Also, moments profiles for materials with different degrees of orthotropy are provided. The method can be extended to other cases of symmetric loadings and boundary conditions.
NOMENCLATURE al, a2, a3, a4 all, a22, a12 AI, A2, B~ C , 1,'
D, Di es, e,
E,~ H
M,,M~ mr, mo
p r,O, z R W, W~
INTRODUCTION
undetermined coefficients material constants (see equation 1) defined by equation 8 ail/a22 and -a12/a22 respectively orthotropic and isotropic flexural rigidities strain components Young modulus and Poisson ratio thickness of the plate moment in the r and 0 directions non-dimensional moments
A C I R C U L A R plate with material constants in the radial direction having different values from those in the tangential direction is considered to possess cylindrical orthotropy. This kind of orthotropy occurs in wood, in drawn wire and in plates with different radial and circumferential reinforcing. This property should be taken into account in the stress analysis of such structural elements. It is known[l, 3] that the small-deflection theory of the orthotropic circular plate with uniform loading, leads either to infinite moments or zero moments at the center of the plate depending on the values of the material constants. In this paper the behavior at the center is corrected by realizing that the center of the plate must be isotropic due to symmetry. Hence, by assuming an isotropic core of a small radius new corrected results are obtained which are more realistic in representing the behavior of the plate.
unit lateral load on plate cylindrical coordinate system at origin of plate radius of plate orthotropic and isotropic displacements in the z-direction
~/O/c) ao, a, Q
UNCORRECTED S O L U ~ O N
stress components shearing force per unit length along the circumference of a circle of radius r.
Consider a simply supported cylindrically orthotropic circular plate with radius R and thickness H, and loaded uniformly with load of intensity p. Attaching a cylindrical co-ordinate system r, 0, z to the center of the plate the stress-strain relations in this case may be written in the following form[2]:
* This research was partly supported by the Research Center of Robert College. Prof. of Civil Eng., Robert College, lstanbul, Turkey. Instructor of Civil Eng., Robert College, Istanbul, Turkey.
eO ~
allO.O-~-al2O-r
er = a12as+a22a, 33
(I)
Walid H. Rimawi and Burak Erman
34
where e 0 and e, are the strain components, a 0 and a, are the stress components and air, at2 and a22 are the material's constants. Making the same assumptions used in the small bending theory of isotropic plates, it is clear that the strain displacement relationships and the equations of equilibrium in terms o f stress resultants remain the same as in the case of isotropy[3]. However, in view o f equation 1 the expression for the stress resultants Mr and Mo and the shearing force Q assume the f o r m : dW\
Mr = - D t c'-~5-rz + r
/
dzW
(law %
|
C~09
~-o.o2
0.08
(2)
0.07 ~ L . J ' ~ ~ =0.05 O.06
cd2W
d3W
W .o.oo,
0.10
Mo = - D r "dr-r+ v dr 2 ]
/
rI
0.12 0.11
v
/
where mr and m o thus defined are non-dimensional moments. For isotropic plates, since c = ~ = 1 and v = Poisson ratio p, equation 5 reduces to the wellknown solution[3]. For values of c other than unity it is obvious that Mr and Mo have the values of either zero or infinite at r = 0. Values of mr and mo, given by equation 5 are plotted in figure I for v = 0.3 and c = 1-3.
dW\ r 2 ~rr l
r --~ =0. I0 n
. . . . Represents uncorrected solution
0.05
)
0O4
Furthermore, the field equation becomes:
0.02
Q = Dtc---~r34 -r dr z
~.~/
~
~1~r
~
~
0.03 0,01
d4W c~+
1 dzw 1 dW P r 2 ~dr q r 3 d ~ - = - - D
2cd3W r dr 3
[
0.1
(3)
all =
a22
,
L
0.4
[
0.5
o.g
O.7
0.e
0.9
Lo
Fig. 1. Moments profiles for C = 1-3.
H3a22
a12,
V-
1
0.3
r/R
In equations 2 and 3, C
1
o.z
a22
CORRECTED SOLUTION
D = 12(alia22_a22)
and W is the deflection in the positive z-direction. The solution of equation 3 is given by:
pr 4 W = atrt+~+a2rl-=+a3r2+a4q (72c-8----~ (4) Where a~ to a , are undetermined coefficients and
o~ = Vq/c. Since dW/dr = 0 at r = 0, the coefficient a 2 must vanish. The equation of equilibrium
O r t h o t r o p y cannot be realized at the center of the plate due to the point symmetry. Therefore, the solution given by equation 5 should not be applied in the n e i g h b o u r h o o d of the center. A more realistic solution is obtained by assuming a small isotropic core of radius r 1 in the orthotropic plate. Thus, the plate is divided into two regions. Modifying the material constants for the isotropic core yields the following two equations for the isotropic and orthotropic regions respectively:
dM,
r-~r + M r - M o = ~o-prdr
W i -= b 2 + b l r 2 +
(6)
Pr4
64Di yields M ~ - M o = 0 at r = 0. On the other hand, substituting equation 4 into equation 2 leads to M , - Mo = 2 a 3 ( c - l) at r = 0 also. Since ( c - 1)# 0 for the orthotropic plate a 3 must vanish. The parameter a~ is obtained f r o m the boundary condition Mr = 0 at r = R. Finally, for the values of the parameters a~, a2, and a 3, thus obtained, equation 2 transforms into
M~
(v+3c' r ( r ) ~ - I
m, = 4 - ~
= (7~c-8)
m° = 4 - ~
= ( 7 2 c - 8) L
(R)Z 1
(v + c~c)
(s)
'
-(1 + 3V,(R)Zl
pr 4
W = alrl+~+a2r I ~+a3rZ+a4q
(72C - 8)D
(7)
where W~ is the deflection of the isotropic core, and Di =
EH3/12(1-/,/2).
Equating the shear forces given by last of equation 2 at the interface, r = r 1, yields a 3 = 0. The parameters b~, a~ and az are determined f r o m the boundary conditions Mr = 0 at r = R and f r o m the remaining two continuity conditions at the ortho-isotropic interface, r = r 1, (Mr) = ( M J i and dW/dr = dWi/dr. These three conditions yield the following three simultaneous equations:
(ca + v)A ~+ (v - c~)A2 =
3c+ v 72c- 8
Correction to the Solution of Orthotropic Circular Plates Ir \ . - a
[rl\-~-3
35
o.o;,i.._._, x /c= 1.5
3c+v
3+/z 64
72c-8
0 . 0 6 ~ . 0 o
(8) r 1~-
3
-
-
-
0.03 ~
(D/D,)
1
64
72c-8
002
\
0.OI
where R ' - 3(I +~)D A1 =
R-=-3(1 - ~ ) D al,
4,0
A2 =
4e
I
[
I
[
[
1
I
I
0.1
0"2
0"3
0'4
0'5
0"6
0"7
0'8
02,
INNN 0"9
I'0
I
l
r/R Fig. 2. m. profiles for various values o f e.
(1 + # ) D ~ , ~ 01
B1 =
0.07 ~
and
c = 1.5
I
0.06 ~ . . . . ~
D
l
c = I, I
(1 --#z~
E = Ea2z Itc--v2 ]
0'0"~~
The moment equations in terms of the new nondimensional constants A1, A2 and B 1 take the form:
005
mr = B , ( r " ~ 2 3 + I t ( r ' ~ z
0.02
=
0
.
5
c=0.5
t,~) --A7-~fi)
(9)
0"01
m° = B1
64 ' o
for the isotropic core, and
I
I
!
i
I
I
L
o.~ 02 0.3 0.4 0.5 o.6 0.7 o.B 0.9 J.o r/R
Fig. 3. mo profiles for various values o f c.
E (')-'
(7-' f r V / 3c+v "k-]
(lo) mo
I =
/,r\~_ 1
-
isotropic Young's modulus E, equal to the value of c. The results are plotted in figures 2 and 3. For the value of C = 1.3 the ratio rl/R was varied between 0.1 and 0.001 and the respective values of the corrected m, and mo at r = 0 were equal and fell between 0.6325 and 0.1113.
)
[r'~-~-I DISCUSSION
fr"12f 1 +3v \-] for the remainder of the plate. By the aid of a digital computer, the values of the constants Ax, A 2 and Bt may be first determined and then substituted into equations 9 and 10 to determine the values of the moments m, and m e after selecting values for the parameters involved. The numerical results were obtained for rn, and m0 for c varying between 0.5 and 1.50 with rl/R = 0.1, v = / z = 0.3 and 1/(E.a22), which is the ratio of tangential modulus of elasticity to the
The method proposed for correcting the behavior of the plate at the center may also be used in the case of a plate having an isotropic core. Although the procedure was applied to the uniformly loaded simply supported plate, it can in general be applied to all cases of symmetric loading and constraint conditions by changing the particular solution, the fifth term in equation 4 and considering the proper boundary conditions. The choice of rl/R is arbitrary but any choice of a small value is acceptable. It should be noted that in the case where the moments tend towards infinite at the center, the maximum value of
Walid H. Rimawi and Burak Erman
36
moments may not exceed the plastic isotropic moment. At the interface, rl/R = 0.I, m o is n o t continuous since this is not acontinuity requirement. Physically, this might be due to the abrupt changes in the materials constants. If the determination o f the deflection (W) is desired the parameters b 2 and a 4 in equations 6 and 7 may be found from two additional b o u n d a r y conditions, namely; W = 0 at r = R and W = W i at r = r 1 .
Examining figure 2 which shows the profile o f mr for various values o f the parameter c, it can be seen that for c > 1 the m a x i m u m value of mr occurs at r = 0 and for c < 1 the position of the m a x i m u m value o f m r starts moving towards the edge. For example for c = 0.5 m a x i m u m mr occurs at r/R = 0"4. The behavior o f mo as shown in figure 3 is similar to that of Mr. It is of interest to note that the value o f rno in the n e i g h b o u r h o o d o f r/R = 0.3 does not appreciably vary with the degree of orthotropy.
REFERENCES 1. 2. 3.
G . F . CARgIER, The bending of the cylindrically aeolotropic plate, J. AppL Mech. 2, A129 (1944). S . G . LEKHNITSKII,Theory of Elasticity of an Anisotropic Elastic Body, Holden-Day, San Francisco (1963). S.P. TIMOSHENKOand S. WOINOWSKY-KR1EGER,Theory of Plates and Shells, pp, 365377 and p. 57, McGraw-Hill, New York (1959).