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A note on the creep of variable thickness circular plates
A Note on the Creep o f Variable Thickness Circular Plates* b y B. W N K A T r ~ M A N and COLEMAN tlAPtt.4.EL'~ Polytechnic Institute of Brooklyn, Brooklyn, New York Introduction The creep bending of laterally loaded circular and a n m d a r plates of uniform thickness were analyzed in (1, 2) respectively. These analyses are based on a nonlinear elastic law analogous to a creep law formulated in terms of the Tresea criterion and the associated flow rule of plasticity theory. The present note considers the inclusion of the effects of radial variation in the thickness of the plates. Since this investigation is essentially an extension of the preceding analyses, the discussion here is k e p t brief and the pertinent governing equations are reproduced from the references. Basic Equations
Problems of circular plates under a uniform lateral pressure are governed b y the following equilibrium equation and curvature-displacement relation, (2)
~
(rm~)' - m0 + 3r 2 = 0
(la)
= -w",
(lb)
~o = - w ' / r
where primes indicate differentiation with respect to r and the center of the plate is taken as the origin of coordinates. The dimensionless variables in these equations are defined b y
r = R/R~, m~ = 6M~/pRo ~, mo= 6Me/pRo2, w = (6k/pRo~)2'n+l(Ro/H/"+~(W/Ro2).
(2)
Here, R0 is the outer radius of the plate, M~ and M0 are, respectively, the radial and circumferential moments, m is a material constant, and k depends upon the material and the plate thickness H . The pressure p and the displacement W are considered positive vertically downwards and the bending moments and curvatures are positive ff they correspond to tension on the lower surface of the plate. The moment-curvature relations for the problem under consideration is conveniently formulated with reference to a stress space whose coordinates are m~ aml m0 (Fig. 1). These relations are derived on the basis of the Tresea criterion * This research was sponsored by the Air Force Office of Scientific Research under Contract No. AF49(638)1360. The paper is based upon a dissertation submitted by the second author to the Polytechnic Institute of Brooklyn in partial fulfillment of the requh'ements for a Ph.D. degree in applied mechanics. t Now with Space Systems Division, Fairchild Hiller Corporation.
391
Brief Communications ko--.
I 0.5~1
I e--,
me
I
he -i.o--
-2.O
~
--
i
~
-]
// STRESS PROFILE
- 2~
o'.t 012 &3 0.4 0.5 o.e o.7 o.8 0.9 Lo
s ~VS. r,
Fig. ]
5 Fig. 2
Fsee (2), Table 1]. When a particular problem is considered, the appropriate row from the table and Eqs. 1 provide the three equations for the determination of m~, rne and w. For the solution to be acceptable, it m u s t satisfy the corresponding inequalities in the table. Solution The specific problem under consideration is a plate on simple supports m~der a uniform lateral pressure. I t is assumed t h a t the dimensionless thickness h = H/Ro of the plate varies as a power function of r and can be expressed in the form (h/ho) = 1 -- (S/ho)r o.
(3)
Here, h0 corresponds to the center of the plate (r = 0) and S and c are some numbers to be assigned various values. While positive values of S/ho correspond to a convex plate, negative values correspond to plates that are concave. Further, any assumed values of c determine the nature of thickness variation. Since the problem has radial s y m m e t r y and the plate is on simple supports, m~(0) = m0(0) and m~(1) = 0. Therefore, it is first assumed t h a t the stress profile for the problem can be represented in the stress space b y the line P R ' (Fig. 1). I~owever, this assumption leads to an uz/acceptable solution, ttence, as a modification, it is assumed that the stress profile for the plate corresponds to the lines PQ and QR. Here, a central circular region (0 to r~) corresponds to PQ and an outer annular region (r~ to 1) corresponds to QR. This stress profile
392
Journal of The FranklinIastitut~
Brief Commumcations
together with Eq. 1 and Table I of Ref. (2) are used to formulate the following mathematical problem: r r a / + mr -- mo + 3r ~ = 0
O < r <_1: 0 <_ r < rl:
rl ~ r < 1:
K, + Ko = -- ( r w ' ) ' / r = m,2m+l/h ~ - ~ ,
~, = - - w " = O,
(4) m~ = me
~o = - - w ' / r = mo~'~+l/h 4m+3.
(5) (6)
The boundary conditions are m,(1) = 0, w(1) = 0, and w'(0) = 0 and it is further required that the continuity of m,, me, w and w' be satisfied at r = rl. Theoretically, the integrations of Eqs. 4 through 6 would contain six eoustunts and these together with the radius rx constitute the seven unkuowns of the problem. T h e three boundary conditions and the four continuity conditions provide the necessary equations for the evaluation of the unknowns. Such a procedure was used in (1) to obtain closed feral solutions for moments and deflections for a plate of uniform thickness. The same procedure in the present consideration of variable thickness leads to the following equation for the determination of r~:
--Dr1
~_
__ F'h(~,,1)"](4~-~)](2m+1)\rl]
j
where
]?or assumed values of the parameters m, 8/ho, and c, Eq. 7 can be solved numerically for rl. W i t h ~he determination of ra, the constants of integTatiou of the problem are easily evaluated and thus lead to the following moment and deflection solutions: O<~ r < _ r l :
(3r~/2)
m~ = me ~ A -
~
r
(8a) 3r ~
qJ)= Llr~l [ L (h~m+,~)(~_ __ A) .... drldr ~- O(l - rl)
(Sb)
rl~r~l: (9a) w = D ( 1 - r)
Vol. 285, ~o. 5, ~ y
1968
(9b)
393
Brief Communications
L6--
o -O.5 -I
1.4
~'"~° ,.o
"~
s/ho
_ _ ................. . . . .
O.e
-I
.
.
.
. . . .
~".,~, z:: ....
~l~
~ \ \ ~ ~-~':':':..... ~ x......... j : \ \ ...... \ \ ~
0.8
= + 0.5 . -o.5
__~j
................
tno
;y ",\\
0.4
O.2
nr
o~ mr
0.2
AND
tn o
0.4 o.o i~_ R / R o VS. r FOR
0.8
~..~
i.o
rn ~ 3 ~
C ~
0.2 I
mr
AND
m~
0.4
0,6 r = R/R o
VS. r
Fig. 3
FOR
-m r
0,8
1.0
rn = 3 ,
C =2
Fig. 4
where A = {EDh4~+~(rl)/r1~} ~/(2~+1) % (3r1:/2).
(10)
Plots of rl vs. S/ho for values of m and c given in Fig. 2 show that the behavior of the plate is not significantly affected by the thickness exponent c. The moment solutions are graphically illustrated in Figs. 3 and 4 for m = 3, c = 1, 2, and S / h e = 0.5, 0, - 0 . 5 , and 1. The corresponding plate deflections given in Table I are correct up to 0.01 : TABLE I. Plate Deflection who4~+~(m = 3) S/ho
0.5
0
--0. 5
--1.0
r/c
1
2
1
2
1
2
1
2
O 0.2 0.4 0.6 0.8 1.0
19.40 19.02 17.08 12.64 6.63 0
2.93 2.78 2.35 1.06 0.86 0
0.31 0.28 0.21 0.13 0.08 0
0.31 0.28 0.21 0.13 0.08 0
0.01 0.01 0.01 0 0 0
0.04 0.03 0.02 0.02 0.01 0
0 0 0 0 0 0
0.01 0 0 0 0 0
~ef~r~nc~$ (1) B. Venhatramaa and P. G. Ho¢lge, Jr., "Creep Behavior of Circular Plates," J. Mech. Phys. Solids, VoI. 6, p. 163, 1958. (2) B. Venkatraman and Sharad A. Patel, "Creep Analysis of Annular Plates," J. Franklin lnst., Vol. 275, No. 1, p. 13, 1963.
394
Journal of The Franklin Institute
Problems of circular plates under a uniform lateral pressure are governed b y the following equilibrium equation and curvature-displacement relation, (2)
~
(rm~)' - m0 + 3r 2 = 0
(la)
= -w",
(lb)
~o = - w ' / r
where primes indicate differentiation with respect to r and the center of the plate is taken as the origin of coordinates. The dimensionless variables in these equations are defined b y
r = R/R~, m~ = 6M~/pRo ~, mo= 6Me/pRo2, w = (6k/pRo~)2'n+l(Ro/H/"+~(W/Ro2).
(2)
Here, R0 is the outer radius of the plate, M~ and M0 are, respectively, the radial and circumferential moments, m is a material constant, and k depends upon the material and the plate thickness H . The pressure p and the displacement W are considered positive vertically downwards and the bending moments and curvatures are positive ff they correspond to tension on the lower surface of the plate. The moment-curvature relations for the problem under consideration is conveniently formulated with reference to a stress space whose coordinates are m~ aml m0 (Fig. 1). These relations are derived on the basis of the Tresea criterion * This research was sponsored by the Air Force Office of Scientific Research under Contract No. AF49(638)1360. The paper is based upon a dissertation submitted by the second author to the Polytechnic Institute of Brooklyn in partial fulfillment of the requh'ements for a Ph.D. degree in applied mechanics. t Now with Space Systems Division, Fairchild Hiller Corporation.
391
Brief Communications ko--.
I 0.5~1
I e--,
me
I
he -i.o--
-2.O
~
--
i
~
-]
// STRESS PROFILE
- 2~
o'.t 012 &3 0.4 0.5 o.e o.7 o.8 0.9 Lo
s ~VS. r,
Fig. ]
5 Fig. 2
Fsee (2), Table 1]. When a particular problem is considered, the appropriate row from the table and Eqs. 1 provide the three equations for the determination of m~, rne and w. For the solution to be acceptable, it m u s t satisfy the corresponding inequalities in the table. Solution The specific problem under consideration is a plate on simple supports m~der a uniform lateral pressure. I t is assumed t h a t the dimensionless thickness h = H/Ro of the plate varies as a power function of r and can be expressed in the form (h/ho) = 1 -- (S/ho)r o.
(3)
Here, h0 corresponds to the center of the plate (r = 0) and S and c are some numbers to be assigned various values. While positive values of S/ho correspond to a convex plate, negative values correspond to plates that are concave. Further, any assumed values of c determine the nature of thickness variation. Since the problem has radial s y m m e t r y and the plate is on simple supports, m~(0) = m0(0) and m~(1) = 0. Therefore, it is first assumed t h a t the stress profile for the problem can be represented in the stress space b y the line P R ' (Fig. 1). I~owever, this assumption leads to an uz/acceptable solution, ttence, as a modification, it is assumed that the stress profile for the plate corresponds to the lines PQ and QR. Here, a central circular region (0 to r~) corresponds to PQ and an outer annular region (r~ to 1) corresponds to QR. This stress profile
392
Journal of The FranklinIastitut~
Brief Commumcations
together with Eq. 1 and Table I of Ref. (2) are used to formulate the following mathematical problem: r r a / + mr -- mo + 3r ~ = 0
O < r <_1: 0 <_ r < rl:
rl ~ r < 1:
K, + Ko = -- ( r w ' ) ' / r = m,2m+l/h ~ - ~ ,
~, = - - w " = O,
(4) m~ = me
~o = - - w ' / r = mo~'~+l/h 4m+3.
(5) (6)
The boundary conditions are m,(1) = 0, w(1) = 0, and w'(0) = 0 and it is further required that the continuity of m,, me, w and w' be satisfied at r = rl. Theoretically, the integrations of Eqs. 4 through 6 would contain six eoustunts and these together with the radius rx constitute the seven unkuowns of the problem. T h e three boundary conditions and the four continuity conditions provide the necessary equations for the evaluation of the unknowns. Such a procedure was used in (1) to obtain closed feral solutions for moments and deflections for a plate of uniform thickness. The same procedure in the present consideration of variable thickness leads to the following equation for the determination of r~:
--Dr1
~_
__ F'h(~,,1)"](4~-~)](2m+1)\rl]
j
where
]?or assumed values of the parameters m, 8/ho, and c, Eq. 7 can be solved numerically for rl. W i t h ~he determination of ra, the constants of integTatiou of the problem are easily evaluated and thus lead to the following moment and deflection solutions: O<~ r < _ r l :
(3r~/2)
m~ = me ~ A -
~
r
(8a) 3r ~
qJ)= Llr~l [ L (h~m+,~)(~_ __ A) .... drldr ~- O(l - rl)
(Sb)
rl~r~l: (9a) w = D ( 1 - r)
Vol. 285, ~o. 5, ~ y
1968
(9b)
393
Brief Communications
L6--
o -O.5 -I
1.4
~'"~° ,.o
"~
s/ho
_ _ ................. . . . .
O.e
-I
.
.
.
. . . .
~".,~, z:: ....
~l~
~ \ \ ~ ~-~':':':..... ~ x......... j : \ \ ...... \ \ ~
0.8
= + 0.5 . -o.5
__~j
................
tno
;y ",\\
0.4
O.2
nr
o~ mr
0.2
AND
tn o
0.4 o.o i~_ R / R o VS. r FOR
0.8
~..~
i.o
rn ~ 3 ~
C ~
0.2 I
mr
AND
m~
0.4
0,6 r = R/R o
VS. r
Fig. 3
FOR
-m r
0,8
1.0
rn = 3 ,
C =2
Fig. 4
where A = {EDh4~+~(rl)/r1~} ~/(2~+1) % (3r1:/2).
(10)
Plots of rl vs. S/ho for values of m and c given in Fig. 2 show that the behavior of the plate is not significantly affected by the thickness exponent c. The moment solutions are graphically illustrated in Figs. 3 and 4 for m = 3, c = 1, 2, and S / h e = 0.5, 0, - 0 . 5 , and 1. The corresponding plate deflections given in Table I are correct up to 0.01 : TABLE I. Plate Deflection who4~+~(m = 3) S/ho
0.5
0
--0. 5
--1.0
r/c
1
2
1
2
1
2
1
2
O 0.2 0.4 0.6 0.8 1.0
19.40 19.02 17.08 12.64 6.63 0
2.93 2.78 2.35 1.06 0.86 0
0.31 0.28 0.21 0.13 0.08 0
0.31 0.28 0.21 0.13 0.08 0
0.01 0.01 0.01 0 0 0
0.04 0.03 0.02 0.02 0.01 0
0 0 0 0 0 0
0.01 0 0 0 0 0
~ef~r~nc~$ (1) B. Venhatramaa and P. G. Ho¢lge, Jr., "Creep Behavior of Circular Plates," J. Mech. Phys. Solids, VoI. 6, p. 163, 1958. (2) B. Venkatraman and Sharad A. Patel, "Creep Analysis of Annular Plates," J. Franklin lnst., Vol. 275, No. 1, p. 13, 1963.
394
Journal of The Franklin Institute