- Email: [email protected]
Minimum weight design of circular plates with limited thickness
tni
J Non-L.inrar
.Merhamcr.
MINIMUM
Vol. 7. pp. 353-W
WEIGHT
Tartu
Pergam~n
Press 1972.
Pmted
tn Great
Bnlain
DESIGN OF CIRCULAR LIMITED THICKNESS
State University,
Estonian
S.S.R.,
PLATES
WITH
U.S.S.R
Abstract-The problem of minimum weight design of circular plates whose thickness is limited from below and above is studied. The material of the plate is rigid-plastic and has different yield stresses in tension and compression. The problem is solved with the aid of Pontryagin’s maximum principle. Three types of solutions arc found.
1. INTRODUCTION weight design for a uniformly loaded circular plate in the case of Tresca’s yield condition was solved by Hopkins and Prager [ 11. In the present paper some extensions of this problem are considered. Firstly, the analysis is carried out for a material which has different yield stresses in tension and compression. The condition proposed by Prager [Z] is taken for the yield condition of such material (Fig. 1) Secondly, the problem THE PROBLEM of minimum
c 0
t ..
Y-
4 f
Y
0’
I FIG. 1. Prager’s
yield condition.
of optimal design is solved with the complementary condition that the thickness of the plate is limited from below and above. Thus, the mathematical problem can be formulated as that of designing a pIate of minimum weight (or minimum volume) for a given load q so that its thickness h does not exceed a given value h,,, and is not less from another given value hmi,. This paper has also a methodical aim. It endeavours to demonstrate that in solving problems of optimal design Pontryagin’s maximum principle can be of great value. 353
354
iiL0
2. BASK
LEPIK
EQUATIONS
OF TtltC PROBLEM
Let us consider a simply supported plate of radius u under a uniform lateral load of magnitude 4. The material of the plate is regarded as rigid-plastic (without strain hardening). The yield stresses in tension and compression will be denoted by the symbols CJ~, a;, respectively. In the plate there is a surface z = zO at which the stresses c,., c6 change their signs (the neutral surface). Let us denote the distances from this surface to the bounding surfaces ofthe plate by the symbols h, and h,. Thus, the thickness of the plate is h = h, + h,. Since we have confined ourselves to an axi-symmetric problem, the thickness h will depend on the current radius r only. It is convenient to use the following dimensionless quantities: _ p =; A+ yli =+ (i = 1,2), max max
?=II’
‘r’
(2.1) The yield condition of Prager has the form in the plane (K,, K2) as shown in Fig. 1. Instead of stresses T,, T2 and bending moments M,, M, we shall use their non-dimensional values
(2.2) i-1,2. 1 Now, the equilibrium
equations rz -
dt,_ supported
t1
P'
dp
For a simply states can exist:
of the plate take the form
plate
dm1
m2
-ml
dp-
at a given
1
-
P
TPP.
(2.3)
p = const
cross-section
the following
stress
(4 x2 = - Y,
(b)
K2
=
1,
K,
=
K2
=
-
y,
v E ho
-
II E ho3
ylo
‘II,
%J
+
r2).
(2.5) K ‘=K2=
1,
For brevity let us call these state regimes DE-A? and D-A according to Fig. 1. By virtue of equation (2.2) we obtain the following formulae for these regimes t2=Y12--lV1; The plastic flow law applied
&r =
m2 =
yloh2
for regime DE-AB 1 du 10, adp
-
Y’Il)
+
tcv;
+
VI:).
(2.6)
gives:
K, = _C!!?=(). a2 dp2 -
(2.7)
Minimum
Here u and w are displacement
rates Ed, K,-radial
unit extension
for the layer q = 0. It follows from these formulae that u = const, coordinate of the neutral surface q0 can be found from the formula
”
355
weight design of circular plates with limited thickness
’ ”
= - hmaxT
a A--
and curvature dw/dp
= con&.
= h,,,dw/dp
= const.
rates The
(2.8)
Thus we see that the coordinate q. has a constant value in the case of regime DE-M. Now let us consider the problem to be solved in terms of optimal control. We take t, and m, for state constraints; ql, y12are control constraints, and q. is a parameter which belongs to optimization. The control constraints ql, q2 must satisfy the inequalities: 6 d rl + r2 d 1,
V&a 0;
y11,3 0,
(2.9)
consequently, admissible values of ql, y/z must lie in the shaded region in Fig. 2. Since we have to design a plate of minimum volume the performance index is expressed by (2.10)
J = i (q, + q2) pdp = min. 0
FIG. 2. Admissible
3. SOLUTION
OF OPTIMAL
values for control
CONTROL
constraints
PROBLEM
ql, q2.
FOR REGIME
D/GAB
Let us suppose that in the plate there exists a region CI< p < B where the plastic regime DE-AB is realized. As p is the independent variable in the optimal control problem, the system (2.3) is non-autonomous and the Pontryagin Hamiltonian is [3,4] :
The adjoint
system (dots denote
differentiation
with respect to p) is: (3.2)
In these formulae quantities belong to the system (3.2):
t,, m, should
by (2.6). The boundary
conditions (3.3)
$‘&a) = ~~(~~) = 0.
~~(~~ = 0, The two first equations
be replaced
of (3.2) can easily be integrated: *1 = C,l%
(3.4)
*2 = C,P.
According to the maximum principle admissible control constraints qi, q2 should be found which give the Hamiltonian (3.1) a maximum value for Vp E [a, p]. Let us show that this maximum value cannot be achieved for the interior points of the shaded region in Fig. 2. At the interior maximum point the foilowing conditions must be fulfilled: i/?.Z ---=-= h, These conditions
Adding
a.9 %,
0,
d2.Gf c?2_#
(‘2.P ^2 < 0,
z2fl
c?q; ay;
0'11
>
0.
df?,%,
with respect to (3.4) give
the last two inequalities,
one gets
It is not possible to satisfy this inequality since it follows from the theory of optimal control that rl/, = const. < 0. Consequently, the maximum of .X can occur only for the boundary points of the region in Fig. 2. One can easily see that the values qt = 0, q2 E [6, I] or q2 = 0, yr E [6, 1J cannot be taken into account either (for these cases the boundary conditions for li/, cannot be satisfied). Thus we have proved that the maximum of the Hamiltonian is realizable only in the cases ye = v1 + v2 = 6*, where 6* has the values (3.1) depends only upon the variable vi. It 6 or 1. Since q2 = 6* - y~i the Hamiltonian follows from the extreme point condition 8X/‘li?y, = 0, that 6% 41 =
vo +
1 +
c
?/ +d C,'
Since ye* = const. for regime DE-A& q1 = const > 0 and q2 = const > 0. Now it follows from the last equation of (3.2) and from the boundary conditions (3.3) that r12 = yyr, and consequently the optimal control constraints are (3.5) Making
use of equations
(2.6) and (3.9, we can integrate
the equilibrium
r,_l/p,r,l,=_?ii”Z_Lpp2+~ 2(1+y)
6
P’
equations
to get (3.6)
where ,f and 1 are constants of integration. Let us sum up the results achieved in this paragraph: When the plastic regime DE-AB is realized for p E [a, p], the thickness of the plate is piecewise constant for an optimal solution and has either the maximal or minimal allowable value (y = 1 or q = 6, respectively).
Minimum weight design of circular plates with limited thickness 4. SOLUTION
351
FOR REGIME D-d
For regime D-A we have t, = t,, m, = m2. It follows from (2.4) that now we must solve a problem with state-dependent control constraints. Besides, quantity q0 is no more constant. All this impels us to reformulate the problem of optimal control. Substituting the expressions t, = t,, ml = m2 in the equilibrium equations and making use of (2.6) one obtains .
q2
-
yql
=
a =
‘wo+ 4PP * q1 = - y[u + (1 + Y)tilJ’
const,
(4.1)
Let us take now q1 for the state constraint and Go for the control constraint. The Hamiltonian is (4.2) The control constraint G-pmust be chosen so that the Hamiltonian would have a maxrmum value for Vp. Since no is unrestricted, it is necessary for the maximum of ,%? that CI= 0. Now equations (2.6) and (4.1) give Y2
=
YYlT
t,
=
t,
=
0,
m,
=
m2 =
+y(l + y) $(p) = C - &7”.
(4.3)
Consequently, in the case of regime D-A the plate has a variable thickness ‘1 = r1 + q.2 = &2/Y) (1 +
r,l . Jcc - hP2f.
(4.4)
It is interesting to note that the optimal solutions which were obtained in Sections 3 and 4 do not depend upon the choice of the coordinate of that neutral surface q0 and we can take q* = 0. 5. SYNTHESIS
OF THE SOLUTIONS
OBTAINED
The problem of minimum volume design of the plate formulated in the Introduction can be solved with the aid of the optimal solutions obtained in Sections 3 and 4. A more detailed analysis shows that there are three types of solutions for which the plate has a minimum weight (Fig. 3). Before passing on to the analysis of these cases, let us find the bounds between which the given load p must lie. We obtain these bounds if we examine the plate of constant thickness. When the non-dimensional thickness of the plate is q = 6, we have pmin= 3yd2(1 + y)- ’ ; by analogy we find the value pma, = 3y(l + y)- ’ for q = 1. All our results which will be given below are valid for p E [pmin,p,,]. (a) First type (~s5l~ti5?1 Here the plate of minimum volume has a variable thickness in the central part (0, p2), but a constant thickness q = 6 in the outer part (pz, 1) (Fig. 3a). We have equations (4.3) and (4.4) for the central part (0, p&. The constant C can be found from the continuity condition q(p2) = 6. Now, equation (4.4) takes the form
4(P)=
1+ y
J(
~l@4
1
- P2)+ is2
(5.1)
The moment m, can be calculated in the region (pz, 1) from equation (3.6). Satisfying the boundary condition m,(l) = 0 and the continuity condition for m, at p = pz, one obtains
3.58
The coordinate p2 can be calculated from this formula for the given y, 6 and p. Then the optimal shape of the plate can be found from (5.1). For the special case y = t,6 = 0 we have pz = 1, and our solution reduces to the solution given by kopkins and Prager [I].
(a) Solution
FIG. 3. Optimal shapes of the plate for y = 1.5. for P,,,~,,< p SCp*; (b) solution for p* -c p -c p*+; (c) solution
for p_
< p < pm,,.
The solutions (5.1) and (5.2) are applicable when g(O) < 1. Let us denote the values ofp and p2, for which ~(0) = 1, by p* and p*. For py, we get the cubic equation 2(1 - S2)(1 - pi) = 362p$
(5.3)
The quantity p* can be found from (5.2) if we substitute p* for p2. Thus we see that the first type of solution is valid for non-dimensional loads p E [pmin, p,]. (b) Second type of solution When p > p*, near the centre p = 0 there appears a region (0, pl) where the plate has a maximum allowable thickness g = 1. In the region (pi, p2) the thickness is variable and is determined by equation (4.4). For p E (p2, 1) the thickness of the plate has the minimum value (Fig. 3b). Now we have to satisfy the continuity conditions at pl, p1 for m,, the boundary equation m,(l) = 0 and the continuity condition q(pz) = 6. Using equations (3.6), (4.3) and (4.4) and fulfilling all these conditions, we see that equation (5.1) remains valid for p E (pl, pz] ; the values pi and p2 can be calculated from the equations 2y(l
P(P: - iP_p:, = -
-
S2)
1 + ‘r‘
(5.4)
p(1 -
pi)
= .i”::;.
Minimum
weight design of&c&r
359
plates with limited thickness
Since in the section p = pi the regime D-A must be realized, the continuity condition for q at p = pi cannot be fulfilled, in fact the thickness 11has at p = p1 a jump of magnitude 1 - ~(~~~~~). The analysis of equations (5.4) shows that when increasing p the values of pi and p2 approach one another and there exists a load p = p** for which p1 = p2 = p**. The coordinate p** may be calculated from the equation (1 - pZ*)(l - 82) = Pp;,,
(5.5)
and the load p*.+ from one of the equations (5.4). (c) Third type of solution When P E (P,, , pma,)Tthe optimal shape of the plate has the form shown in Fig. 3c. Now
the moment m, is to be calculated from equations (3.6). Satisfying the continuity condition at p = p2 and the boundary condition ml(l) = 0, we get the formula for p2: (1 f YIP - 3Yd2 3y(l - 82) .
P2 =
(5.6)
Thus we have found all optimal shapes for the loads from pmin to p,,, . In Table 1 the values of p*, p*, p._, p** which were calculated for some values of 6 are presented. This table allows us to find easily which type of solution exists for given parameters. For example, ify = 1,6 = 0.6 and p = 1.2, we can find from the table that p* = 1.041 and p** = 1.350; consequently the given value p = 1.2 lies in the interval (p,, p*J and we have the second type of solution. From equation (5.4) the values of pi and pZ are found to be p1 = 0.643, p2 = 0819. TABLE1
0.1
09949
0.2 0.3 0.4 0.5 @6 0.7 0.8 0‘9
0.9795 0.953 @913 0.858 0,784 0686 0.557 0.384
@9966 0.9863 o-968 0940 0.900 0.843 0.762 0.643 0.460
1wO 1xlol I+)02
1.495 1.480 1.456 l-425 1.388 1.350 1,318 l-307 1,346
I.007 1.019 1,041
I.084 1.160 1,289
REFERENCES [l]
H. G. HOPKINS and W. PRAGER, Limits of economy of material in plates. J. uppl. Me&. 22, 3, 372
[2]
(1955). R. IIparep.
[3]
cnnoru~oB cpe&b~e. MocKsa AH CCCP JI. C. IIOHT~RI‘HH, R. IT. ~o~T~H~~~~,
0
nnacTMsecIEoh1
mami3e
CJI~I~~T~IX
K~HCTP-+IJM~~.
R IEII.rrIp06ncME4 MeXaHLllEa
(1961).
P. B. ~a~~p~.~~~~~e, E. @.
~~~~~~H~O.
URCIECLC meopta ~nnz~ua~.~u~~npoyeccos, E-11 G. LEITMAN, An Introduction
to Optimur
Mowua (1969). Control. McGraw-Hill,
(Received
1 July 1971)
New York (1966).
~U~//e~~~ZU-
360
ul.0
LEPIK
R&urn6 On etudie le probleme du dessin de poids minrmal de plaques tit-culaires dont I’epaisseur est limitee inftrieurement et superieurement. Le materiau de la plaque est rigide-plastique et a des resistances a la traction et a la compression dtfferentes. On r&out le probleme a I’aidc du principe du maximum de Pontryagin. On trouve trois types de solutions.
Zusammenfassung -Ein Problem iiber das Mmimalgcwicht von Kreisplattcn, derer Drcke von oben und unter begrenzt ist, wird gelost. Das Material der Platte ist Starr-plastisch und hat verschiedene Fliessgrenzen bei Zug and Druck. Fur die Losung des Problemes wird das Maximum-printsip von Pontryagin angewandt. Drei verschiedene Typen von Llisungen sind gefunden.
J Non-L.inrar
.Merhamcr.
MINIMUM
Vol. 7. pp. 353-W
WEIGHT
Tartu
Pergam~n
Press 1972.
Pmted
tn Great
Bnlain
DESIGN OF CIRCULAR LIMITED THICKNESS
State University,
Estonian
S.S.R.,
PLATES
WITH
U.S.S.R
Abstract-The problem of minimum weight design of circular plates whose thickness is limited from below and above is studied. The material of the plate is rigid-plastic and has different yield stresses in tension and compression. The problem is solved with the aid of Pontryagin’s maximum principle. Three types of solutions arc found.
1. INTRODUCTION weight design for a uniformly loaded circular plate in the case of Tresca’s yield condition was solved by Hopkins and Prager [ 11. In the present paper some extensions of this problem are considered. Firstly, the analysis is carried out for a material which has different yield stresses in tension and compression. The condition proposed by Prager [Z] is taken for the yield condition of such material (Fig. 1) Secondly, the problem THE PROBLEM of minimum
c 0
t ..
Y-
4 f
Y
0’
I FIG. 1. Prager’s
yield condition.
of optimal design is solved with the complementary condition that the thickness of the plate is limited from below and above. Thus, the mathematical problem can be formulated as that of designing a pIate of minimum weight (or minimum volume) for a given load q so that its thickness h does not exceed a given value h,,, and is not less from another given value hmi,. This paper has also a methodical aim. It endeavours to demonstrate that in solving problems of optimal design Pontryagin’s maximum principle can be of great value. 353
354
iiL0
2. BASK
LEPIK
EQUATIONS
OF TtltC PROBLEM
Let us consider a simply supported plate of radius u under a uniform lateral load of magnitude 4. The material of the plate is regarded as rigid-plastic (without strain hardening). The yield stresses in tension and compression will be denoted by the symbols CJ~, a;, respectively. In the plate there is a surface z = zO at which the stresses c,., c6 change their signs (the neutral surface). Let us denote the distances from this surface to the bounding surfaces ofthe plate by the symbols h, and h,. Thus, the thickness of the plate is h = h, + h,. Since we have confined ourselves to an axi-symmetric problem, the thickness h will depend on the current radius r only. It is convenient to use the following dimensionless quantities: _ p =; A+ yli =+ (i = 1,2), max max
?=II’
‘r’
(2.1) The yield condition of Prager has the form in the plane (K,, K2) as shown in Fig. 1. Instead of stresses T,, T2 and bending moments M,, M, we shall use their non-dimensional values
(2.2) i-1,2. 1 Now, the equilibrium
equations rz -
dt,_ supported
t1
P'
dp
For a simply states can exist:
of the plate take the form
plate
dm1
m2
-ml
dp-
at a given
1
-
P
TPP.
(2.3)
p = const
cross-section
the following
stress
(4 x2 = - Y,
(b)
K2
=
1,
K,
=
K2
=
-
y,
v E ho
-
II E ho3
ylo
‘II,
%J
+
r2).
(2.5) K ‘=K2=
1,
For brevity let us call these state regimes DE-A? and D-A according to Fig. 1. By virtue of equation (2.2) we obtain the following formulae for these regimes t2=Y12--lV1; The plastic flow law applied
&r =
m2 =
yloh2
for regime DE-AB 1 du 10, adp
-
Y’Il)
+
tcv;
+
VI:).
(2.6)
gives:
K, = _C!!?=(). a2 dp2 -
(2.7)
Minimum
Here u and w are displacement
rates Ed, K,-radial
unit extension
for the layer q = 0. It follows from these formulae that u = const, coordinate of the neutral surface q0 can be found from the formula
”
355
weight design of circular plates with limited thickness
’ ”
= - hmaxT
a A--
and curvature dw/dp
= con&.
= h,,,dw/dp
= const.
rates The
(2.8)
Thus we see that the coordinate q. has a constant value in the case of regime DE-M. Now let us consider the problem to be solved in terms of optimal control. We take t, and m, for state constraints; ql, y12are control constraints, and q. is a parameter which belongs to optimization. The control constraints ql, q2 must satisfy the inequalities: 6 d rl + r2 d 1,
V&a 0;
y11,3 0,
(2.9)
consequently, admissible values of ql, y/z must lie in the shaded region in Fig. 2. Since we have to design a plate of minimum volume the performance index is expressed by (2.10)
J = i (q, + q2) pdp = min. 0
FIG. 2. Admissible
3. SOLUTION
OF OPTIMAL
values for control
CONTROL
constraints
PROBLEM
ql, q2.
FOR REGIME
D/GAB
Let us suppose that in the plate there exists a region CI< p < B where the plastic regime DE-AB is realized. As p is the independent variable in the optimal control problem, the system (2.3) is non-autonomous and the Pontryagin Hamiltonian is [3,4] :
The adjoint
system (dots denote
differentiation
with respect to p) is: (3.2)
In these formulae quantities belong to the system (3.2):
t,, m, should
by (2.6). The boundary
conditions (3.3)
$‘&a) = ~~(~~) = 0.
~~(~~ = 0, The two first equations
be replaced
of (3.2) can easily be integrated: *1 = C,l%
(3.4)
*2 = C,P.
According to the maximum principle admissible control constraints qi, q2 should be found which give the Hamiltonian (3.1) a maximum value for Vp E [a, p]. Let us show that this maximum value cannot be achieved for the interior points of the shaded region in Fig. 2. At the interior maximum point the foilowing conditions must be fulfilled: i/?.Z ---=-= h, These conditions
Adding
a.9 %,
0,
d2.Gf c?2_#
(‘2.P ^2 < 0,
z2fl
c?q; ay;
0'11
>
0.
df?,%,
with respect to (3.4) give
the last two inequalities,
one gets
It is not possible to satisfy this inequality since it follows from the theory of optimal control that rl/, = const. < 0. Consequently, the maximum of .X can occur only for the boundary points of the region in Fig. 2. One can easily see that the values qt = 0, q2 E [6, I] or q2 = 0, yr E [6, 1J cannot be taken into account either (for these cases the boundary conditions for li/, cannot be satisfied). Thus we have proved that the maximum of the Hamiltonian is realizable only in the cases ye = v1 + v2 = 6*, where 6* has the values (3.1) depends only upon the variable vi. It 6 or 1. Since q2 = 6* - y~i the Hamiltonian follows from the extreme point condition 8X/‘li?y, = 0, that 6% 41 =
vo +
1 +
c
?/ +d C,'
Since ye* = const. for regime DE-A& q1 = const > 0 and q2 = const > 0. Now it follows from the last equation of (3.2) and from the boundary conditions (3.3) that r12 = yyr, and consequently the optimal control constraints are (3.5) Making
use of equations
(2.6) and (3.9, we can integrate
the equilibrium
r,_l/p,r,l,=_?ii”Z_Lpp2+~ 2(1+y)
6
P’
equations
to get (3.6)
where ,f and 1 are constants of integration. Let us sum up the results achieved in this paragraph: When the plastic regime DE-AB is realized for p E [a, p], the thickness of the plate is piecewise constant for an optimal solution and has either the maximal or minimal allowable value (y = 1 or q = 6, respectively).
Minimum weight design of circular plates with limited thickness 4. SOLUTION
351
FOR REGIME D-d
For regime D-A we have t, = t,, m, = m2. It follows from (2.4) that now we must solve a problem with state-dependent control constraints. Besides, quantity q0 is no more constant. All this impels us to reformulate the problem of optimal control. Substituting the expressions t, = t,, ml = m2 in the equilibrium equations and making use of (2.6) one obtains .
q2
-
yql
=
a =
‘wo+ 4PP * q1 = - y[u + (1 + Y)tilJ’
const,
(4.1)
Let us take now q1 for the state constraint and Go for the control constraint. The Hamiltonian is (4.2) The control constraint G-pmust be chosen so that the Hamiltonian would have a maxrmum value for Vp. Since no is unrestricted, it is necessary for the maximum of ,%? that CI= 0. Now equations (2.6) and (4.1) give Y2
=
YYlT
t,
=
t,
=
0,
m,
=
m2 =
+y(l + y) $(p) = C - &7”.
(4.3)
Consequently, in the case of regime D-A the plate has a variable thickness ‘1 = r1 + q.2 = &2/Y) (1 +
r,l . Jcc - hP2f.
(4.4)
It is interesting to note that the optimal solutions which were obtained in Sections 3 and 4 do not depend upon the choice of the coordinate of that neutral surface q0 and we can take q* = 0. 5. SYNTHESIS
OF THE SOLUTIONS
OBTAINED
The problem of minimum volume design of the plate formulated in the Introduction can be solved with the aid of the optimal solutions obtained in Sections 3 and 4. A more detailed analysis shows that there are three types of solutions for which the plate has a minimum weight (Fig. 3). Before passing on to the analysis of these cases, let us find the bounds between which the given load p must lie. We obtain these bounds if we examine the plate of constant thickness. When the non-dimensional thickness of the plate is q = 6, we have pmin= 3yd2(1 + y)- ’ ; by analogy we find the value pma, = 3y(l + y)- ’ for q = 1. All our results which will be given below are valid for p E [pmin,p,,]. (a) First type (~s5l~ti5?1 Here the plate of minimum volume has a variable thickness in the central part (0, p2), but a constant thickness q = 6 in the outer part (pz, 1) (Fig. 3a). We have equations (4.3) and (4.4) for the central part (0, p&. The constant C can be found from the continuity condition q(p2) = 6. Now, equation (4.4) takes the form
4(P)=
1+ y
J(
~l@4
1
- P2)+ is2
(5.1)
The moment m, can be calculated in the region (pz, 1) from equation (3.6). Satisfying the boundary condition m,(l) = 0 and the continuity condition for m, at p = pz, one obtains
3.58
The coordinate p2 can be calculated from this formula for the given y, 6 and p. Then the optimal shape of the plate can be found from (5.1). For the special case y = t,6 = 0 we have pz = 1, and our solution reduces to the solution given by kopkins and Prager [I].
(a) Solution
FIG. 3. Optimal shapes of the plate for y = 1.5. for P,,,~,,< p SCp*; (b) solution for p* -c p -c p*+; (c) solution
for p_
< p < pm,,.
The solutions (5.1) and (5.2) are applicable when g(O) < 1. Let us denote the values ofp and p2, for which ~(0) = 1, by p* and p*. For py, we get the cubic equation 2(1 - S2)(1 - pi) = 362p$
(5.3)
The quantity p* can be found from (5.2) if we substitute p* for p2. Thus we see that the first type of solution is valid for non-dimensional loads p E [pmin, p,]. (b) Second type of solution When p > p*, near the centre p = 0 there appears a region (0, pl) where the plate has a maximum allowable thickness g = 1. In the region (pi, p2) the thickness is variable and is determined by equation (4.4). For p E (p2, 1) the thickness of the plate has the minimum value (Fig. 3b). Now we have to satisfy the continuity conditions at pl, p1 for m,, the boundary equation m,(l) = 0 and the continuity condition q(pz) = 6. Using equations (3.6), (4.3) and (4.4) and fulfilling all these conditions, we see that equation (5.1) remains valid for p E (pl, pz] ; the values pi and p2 can be calculated from the equations 2y(l
P(P: - iP_p:, = -
-
S2)
1 + ‘r‘
(5.4)
p(1 -
pi)
= .i”::;.
Minimum
weight design of&c&r
359
plates with limited thickness
Since in the section p = pi the regime D-A must be realized, the continuity condition for q at p = pi cannot be fulfilled, in fact the thickness 11has at p = p1 a jump of magnitude 1 - ~(~~~~~). The analysis of equations (5.4) shows that when increasing p the values of pi and p2 approach one another and there exists a load p = p** for which p1 = p2 = p**. The coordinate p** may be calculated from the equation (1 - pZ*)(l - 82) = Pp;,,
(5.5)
and the load p*.+ from one of the equations (5.4). (c) Third type of solution When P E (P,, , pma,)Tthe optimal shape of the plate has the form shown in Fig. 3c. Now
the moment m, is to be calculated from equations (3.6). Satisfying the continuity condition at p = p2 and the boundary condition ml(l) = 0, we get the formula for p2: (1 f YIP - 3Yd2 3y(l - 82) .
P2 =
(5.6)
Thus we have found all optimal shapes for the loads from pmin to p,,, . In Table 1 the values of p*, p*, p._, p** which were calculated for some values of 6 are presented. This table allows us to find easily which type of solution exists for given parameters. For example, ify = 1,6 = 0.6 and p = 1.2, we can find from the table that p* = 1.041 and p** = 1.350; consequently the given value p = 1.2 lies in the interval (p,, p*J and we have the second type of solution. From equation (5.4) the values of pi and pZ are found to be p1 = 0.643, p2 = 0819. TABLE1
0.1
09949
0.2 0.3 0.4 0.5 @6 0.7 0.8 0‘9
0.9795 0.953 @913 0.858 0,784 0686 0.557 0.384
@9966 0.9863 o-968 0940 0.900 0.843 0.762 0.643 0.460
1wO 1xlol I+)02
1.495 1.480 1.456 l-425 1.388 1.350 1,318 l-307 1,346
I.007 1.019 1,041
I.084 1.160 1,289
REFERENCES [l]
H. G. HOPKINS and W. PRAGER, Limits of economy of material in plates. J. uppl. Me&. 22, 3, 372
[2]
(1955). R. IIparep.
[3]
cnnoru~oB cpe&b~e. MocKsa AH CCCP JI. C. IIOHT~RI‘HH, R. IT. ~o~T~H~~~~,
0
nnacTMsecIEoh1
mami3e
CJI~I~~T~IX
K~HCTP-+IJM~~.
R IEII.rrIp06ncME4 MeXaHLllEa
(1961).
P. B. ~a~~p~.~~~~~e, E. @.
~~~~~~H~O.
URCIECLC meopta ~nnz~ua~.~u~~npoyeccos, E-11 G. LEITMAN, An Introduction
to Optimur
Mowua (1969). Control. McGraw-Hill,
(Received
1 July 1971)
New York (1966).
~U~//e~~~ZU-
360
ul.0
LEPIK
R&urn6 On etudie le probleme du dessin de poids minrmal de plaques tit-culaires dont I’epaisseur est limitee inftrieurement et superieurement. Le materiau de la plaque est rigide-plastique et a des resistances a la traction et a la compression dtfferentes. On r&out le probleme a I’aidc du principe du maximum de Pontryagin. On trouve trois types de solutions.
Zusammenfassung -Ein Problem iiber das Mmimalgcwicht von Kreisplattcn, derer Drcke von oben und unter begrenzt ist, wird gelost. Das Material der Platte ist Starr-plastisch und hat verschiedene Fliessgrenzen bei Zug and Druck. Fur die Losung des Problemes wird das Maximum-printsip von Pontryagin angewandt. Drei verschiedene Typen von Llisungen sind gefunden.