- Email: [email protected]
On the formulation of the yield condition under uncertainty
MECH. RES. COMM.
ON THE
Vol.3, 233-236,
FORMULATION
A. B a r a t t a Istituto di
Scienza
OF THE
delle
1976.
YIELD
Pergamon Press.
CONDITION
Costruzioni,
UNDER
Printed in USA.
UNCERTAINTY
University
of N a p l e s ,
Italy
(Received 3 November 1975; accepted as ready for print 26 February 1976)
Introduction In Structural Engineering it is often assumed that materials obey perfectlyplastic constitutive laws, as a tool to get predictions on the ultimate loadcarrying capacity of members and structures. Probabilistic Limit Analysis of perfectly-plastlc structures is then devoted to calculate, or at least to estimate, the probability that these structures undergo plastic collapse under uncertainty of the local strength of materials. [ 1,2,3] The result of such analysis is a real number P , the probability of failure fc ( 0 ~ P f ~ i), which, in the context of probabillstic phylosophy of safety,can be loo~ed at as the conditional p!obability of plastic collapse under given loads. Description of the yield condition Uncertainty in the material strength does not rise any problem from a macroscopical point of view, for uniaxial states of stress. In this case, in fact, experience shows that the yield condition is of the type o
=
~
(i) O
o
being the limit stress of the material in tension, and it is immediate to O
assess that randomness of strength corresponds o
is a random variable ~ O
to assume that the limit value
in every point of the body. For pluriaxial
stress,
O
the yield condition is generally formulated by postulating that a numerical fun ction of stress exists such that a stress state I is on the yield threshold iff ~( Z,~ ) : %
o where ~ summarizes the set of strain-hardening
(2) parameters,
and %
is the limit O
value of ~. For perfectly-plastic written
materials,
% does not depend on ~ , and eq.(2) can be
~( Z ) = ~
(3) 0
Work in Progress and Preliminary Results
233
234
A. BARATTA
If the material
is subject to uncertainty,
Vol.3,
No.3
not only it can be assumed that O
is the realization
of a random variable ~
, but function ~ itself should be alO
lowed to undergo
random variations.
The simplest point of view is to assume that ~ depends on a set ~ of random parameters;
so the yield condition of a material with random strength variations
can be set in the form ~( ~,~ ) = ¢
(4) O
which has the same formal appearence hardening materials, dom parameters
eq.
(2). The only difference
~. In other words,
but cannot vary as the material
the set ~ is unknown
goes through
known at the beginning
for strain-
lies in the fact that the ran-
in ~ do not change during the loading process
dening parameters
generally
of the general yield condition
its history;
of the loading program,
as the strain har-
in the virgin state, on the contrary but evolves
~ is
as plastic
strains grow. It is worthwhile
to note that this analogy is not only formal.
the main sources of uncertainty processes mineral
such as rolling,
extruding,
in the final structural
surface of the material
can be individuated
as it can be verified
Thus,
it is possible
predicted.
exhibit perfectly-plastic
behaviour
that, even if loads are increased
strains under loading are negligible
tic excursions
experienced
An attempt
Properties
of ductile
up to the collapse
in comparison
thre-
of the plas-
(according
the uncertainty
to Drucker's
postulate)
in the
to be inherent
and indifference
to the material,
It follows that stochastic variations
(4), should save convexity
of stress J1" Mutual
express
structural materials.
stress will be assumed
affected by uncertainty. face, eq.
as long
expressions of the r an~omyie•l d function
like stability
to hydrostatic
under loading,
under manufacturing.
is made in this Section to explicitly
yield condition
vicissitudes,
This does not deny the assump-
shold, plastic
Some particular
forming
that the yield
a lot of more or less stochastic
whose final state cannot be uniquely tion that structures
in the technological
etc., which transform the original melt
element.
undergoes
In fact, one of
statistical
and independence
independence
i.e. not
of the yield sur-
on the linear invariant
of the random parameters
involved
Vol.3, No.3
YIELD CONDITION UNDER UNCERTAINTY
in ~ and in L m a y
235
thus be allowed as long as it is consistent with this require-
ment. When the material is also isotropic,
it is well known that the yield surface ve-
rifies some conditions of symmetry with respect to the principal axes of the stress tensor. In particular,
the intersection of the yield surface with the
plane Jl=O should be included, no matter the value of ¢o' between Tresca's and Hill's hexagons.
In view of this result, it may be convenient to exclude uncer-
tainty from the expression of #, and to assume, to within a small spread, that for isotropic behaviour ¢ is a kind of average yield function, expressed for instance by the elastic potential of the stress deviator ¢( g )
(Isotropic uncertainty)
2 42 = Jl - J2 = °o
(5)
A different formulation is obtained assuming that uncertainty involves only the position of the yield surface with respect to the principal axes. Function # depends then on the radius-vector ~ =
( ~i,02,
) of the center C of the yield
surface, assumed to be a random vector. Uncertainty is then
explicitly formulated as follows 2 ¢( z - e I ) = ~
(6)
O
where I is the unit tensor, and o
is, for the present,
assumed to be a deter-
O
ministic constant. This formulation can be named, by analogy with strain-hardening theories, Kinematic uncerta-
FT
inty. It is to be noted that the random parameters ~ cannot be statistically independent,
since the
point @ = 0 must be admissible for any sample volume of the material. m~
%
The inequality ¢( O, @ ) < o --
2
(7)
O
cannot be subjected to uncertainty, i.e. the vector ~ cannot go out of the O-domain drawed in Fig.2 whose boundary is defined by eq.(5) FIG.
I:
Isotropic
Uncertainty
putting ~ = O
o . O
236
A. BARATTA
Vol.3, No.3
A more general formulation can be got by combining isotropic
%
with kinematic uncertainty, thus allowing o
Ly
J
'(~-domain
~."
(8)
0
!
l
I 2
(
J
/"
v
o1
Since,
as b e f o r e ,
¢( O , ~ ) N a
~2
,
0
holds deterministically,
~ and ~ 0
2
t %
uncertainty can then be expressed ¢( ~ ,~ ) = ~2
/
| II
random variations. Iso-kinematic
/
/
in eq.(6) to undergo O
are also mutually
yield surface
the
correlated,
but
O-domain is no more determi-
nistically bounded. More sophisticated formulations of uncertainty can, of course, FIG.
2:
Kinematic
Uncertainty
be derived. In this regard, beca-
use of the analogy pointed out in the previous Section between strain-hardening and uncertainty, any strain-hardening formulation of plasticity can be easily converted into a uncertainty formulation. However, it is to be stressed the prac tical difficulty to get the joint statistics of the random parameters involved both in eqs. (6) and (8), and obviously in more general formulations.
Acknowledgement: Research sponsored by the Italian National Council of Researches (C.N.R.)
References ~] G.Augusti-A.Baratta,Journ. Struct.Mech.l(1):pp.43-62,1972 [2] G.Augusti-A.Baratta,Proc. Int. Symp."Problems of Plasticity",Warsaw,1972 [3] G.Augusti-A.Baratta,Proc.2ndInt.Conf. Struct.Mech.in Reactor Techn.,Berlin, 1973 [4] V.V.Bolotin,Statistical Methods in Structural Mechanics,S.Francisco,1969 [5] A.I.Johnson,Struct.Engrg. Div.,Royal Inst. Technology,Stockholm, 1953. [6] G.A.Alpsten,Int.Conf."Planning and Design of Tall Buildings,Vol. Ib, pp.755-807, Bethlehem, U.S.A.,1972.
ON THE
Vol.3, 233-236,
FORMULATION
A. B a r a t t a Istituto di
Scienza
OF THE
delle
1976.
YIELD
Pergamon Press.
CONDITION
Costruzioni,
UNDER
Printed in USA.
UNCERTAINTY
University
of N a p l e s ,
Italy
(Received 3 November 1975; accepted as ready for print 26 February 1976)
Introduction In Structural Engineering it is often assumed that materials obey perfectlyplastic constitutive laws, as a tool to get predictions on the ultimate loadcarrying capacity of members and structures. Probabilistic Limit Analysis of perfectly-plastlc structures is then devoted to calculate, or at least to estimate, the probability that these structures undergo plastic collapse under uncertainty of the local strength of materials. [ 1,2,3] The result of such analysis is a real number P , the probability of failure fc ( 0 ~ P f ~ i), which, in the context of probabillstic phylosophy of safety,can be loo~ed at as the conditional p!obability of plastic collapse under given loads. Description of the yield condition Uncertainty in the material strength does not rise any problem from a macroscopical point of view, for uniaxial states of stress. In this case, in fact, experience shows that the yield condition is of the type o
=
~
(i) O
o
being the limit stress of the material in tension, and it is immediate to O
assess that randomness of strength corresponds o
is a random variable ~ O
to assume that the limit value
in every point of the body. For pluriaxial
stress,
O
the yield condition is generally formulated by postulating that a numerical fun ction of stress exists such that a stress state I is on the yield threshold iff ~( Z,~ ) : %
o where ~ summarizes the set of strain-hardening
(2) parameters,
and %
is the limit O
value of ~. For perfectly-plastic written
materials,
% does not depend on ~ , and eq.(2) can be
~( Z ) = ~
(3) 0
Work in Progress and Preliminary Results
233
234
A. BARATTA
If the material
is subject to uncertainty,
Vol.3,
No.3
not only it can be assumed that O
is the realization
of a random variable ~
, but function ~ itself should be alO
lowed to undergo
random variations.
The simplest point of view is to assume that ~ depends on a set ~ of random parameters;
so the yield condition of a material with random strength variations
can be set in the form ~( ~,~ ) = ¢
(4) O
which has the same formal appearence hardening materials, dom parameters
eq.
(2). The only difference
~. In other words,
but cannot vary as the material
the set ~ is unknown
goes through
known at the beginning
for strain-
lies in the fact that the ran-
in ~ do not change during the loading process
dening parameters
generally
of the general yield condition
its history;
of the loading program,
as the strain har-
in the virgin state, on the contrary but evolves
~ is
as plastic
strains grow. It is worthwhile
to note that this analogy is not only formal.
the main sources of uncertainty processes mineral
such as rolling,
extruding,
in the final structural
surface of the material
can be individuated
as it can be verified
Thus,
it is possible
predicted.
exhibit perfectly-plastic
behaviour
that, even if loads are increased
strains under loading are negligible
tic excursions
experienced
An attempt
Properties
of ductile
up to the collapse
in comparison
thre-
of the plas-
(according
the uncertainty
to Drucker's
postulate)
in the
to be inherent
and indifference
to the material,
It follows that stochastic variations
(4), should save convexity
of stress J1" Mutual
express
structural materials.
stress will be assumed
affected by uncertainty. face, eq.
as long
expressions of the r an~omyie•l d function
like stability
to hydrostatic
under loading,
under manufacturing.
is made in this Section to explicitly
yield condition
vicissitudes,
This does not deny the assump-
shold, plastic
Some particular
forming
that the yield
a lot of more or less stochastic
whose final state cannot be uniquely tion that structures
in the technological
etc., which transform the original melt
element.
undergoes
In fact, one of
statistical
and independence
independence
i.e. not
of the yield sur-
on the linear invariant
of the random parameters
involved
Vol.3, No.3
YIELD CONDITION UNDER UNCERTAINTY
in ~ and in L m a y
235
thus be allowed as long as it is consistent with this require-
ment. When the material is also isotropic,
it is well known that the yield surface ve-
rifies some conditions of symmetry with respect to the principal axes of the stress tensor. In particular,
the intersection of the yield surface with the
plane Jl=O should be included, no matter the value of ¢o' between Tresca's and Hill's hexagons.
In view of this result, it may be convenient to exclude uncer-
tainty from the expression of #, and to assume, to within a small spread, that for isotropic behaviour ¢ is a kind of average yield function, expressed for instance by the elastic potential of the stress deviator ¢( g )
(Isotropic uncertainty)
2 42 = Jl - J2 = °o
(5)
A different formulation is obtained assuming that uncertainty involves only the position of the yield surface with respect to the principal axes. Function # depends then on the radius-vector ~ =
( ~i,02,
) of the center C of the yield
surface, assumed to be a random vector. Uncertainty is then
explicitly formulated as follows 2 ¢( z - e I ) = ~
(6)
O
where I is the unit tensor, and o
is, for the present,
assumed to be a deter-
O
ministic constant. This formulation can be named, by analogy with strain-hardening theories, Kinematic uncerta-
FT
inty. It is to be noted that the random parameters ~ cannot be statistically independent,
since the
point @ = 0 must be admissible for any sample volume of the material. m~
%
The inequality ¢( O, @ ) < o --
2
(7)
O
cannot be subjected to uncertainty, i.e. the vector ~ cannot go out of the O-domain drawed in Fig.2 whose boundary is defined by eq.(5) FIG.
I:
Isotropic
Uncertainty
putting ~ = O
o . O
236
A. BARATTA
Vol.3, No.3
A more general formulation can be got by combining isotropic
%
with kinematic uncertainty, thus allowing o
Ly
J
'(~-domain
~."
(8)
0
!
l
I 2
(
J
/"
v
o1
Since,
as b e f o r e ,
¢( O , ~ ) N a
~2
,
0
holds deterministically,
~ and ~ 0
2
t %
uncertainty can then be expressed ¢( ~ ,~ ) = ~2
/
| II
random variations. Iso-kinematic
/
/
in eq.(6) to undergo O
are also mutually
yield surface
the
correlated,
but
O-domain is no more determi-
nistically bounded. More sophisticated formulations of uncertainty can, of course, FIG.
2:
Kinematic
Uncertainty
be derived. In this regard, beca-
use of the analogy pointed out in the previous Section between strain-hardening and uncertainty, any strain-hardening formulation of plasticity can be easily converted into a uncertainty formulation. However, it is to be stressed the prac tical difficulty to get the joint statistics of the random parameters involved both in eqs. (6) and (8), and obviously in more general formulations.
Acknowledgement: Research sponsored by the Italian National Council of Researches (C.N.R.)
References ~] G.Augusti-A.Baratta,Journ. Struct.Mech.l(1):pp.43-62,1972 [2] G.Augusti-A.Baratta,Proc. Int. Symp."Problems of Plasticity",Warsaw,1972 [3] G.Augusti-A.Baratta,Proc.2ndInt.Conf. Struct.Mech.in Reactor Techn.,Berlin, 1973 [4] V.V.Bolotin,Statistical Methods in Structural Mechanics,S.Francisco,1969 [5] A.I.Johnson,Struct.Engrg. Div.,Royal Inst. Technology,Stockholm, 1953. [6] G.A.Alpsten,Int.Conf."Planning and Design of Tall Buildings,Vol. Ib, pp.755-807, Bethlehem, U.S.A.,1972.