Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

3, 39, 2001

THREE-DIMENSIONAL ELASTIC AND ELASTO-PLASTIC

FRICTIONAL CONTACT ANALYSIS OF TURBOMACHINERY

BLADE ATTACHMENTS

Grzegorz Zboiński

Wiesław Ostachowicz

Institute of Fluid Flow Machinery, Polish Academy of Sciences

e-mail: zboi@imppan.imp.pg.gda.pl

Thepaper presents a summary of the researchon three-dimensional con-
tact analysis of turbomachinery blade attachments in the elastic and
elasto-plastic range. In that context the paper deals with theoretical,
methodological and phenomenological description of the problem. The
paper focuses on the applied variational formulation of contact problems
of elasticity and elasto-plasticity and the corresponding finite element
methods, utilizes contact mechanics algorithms as well as describes di-
splacement, stress, contact and slip states within real turbomachinery
blade attachments. Non-linear character of contact mechanics procedu-
res and influence of these nonlinearities on stress anddeformationwithin
the attachments are treated with special care.

Keywords:finite elements, contactmechanics, elasticity, elasto-plasticity

1. Introduction

We will start this paper with presentation of three different aspects dealt
with in the topic. Firstly we will express some general remarks concerning
complexity of of the analysis under consideration.Thendevelopment and state
of the art will be presented.We will end up with research issues and authors’
contribution to the subject.

1.1. General remarks

Contact analysis of turbomachinery blade attachments constitutes a very
good example of how recent advances of contemporary mechanical and com-



770 G.Zboiński, W.Ostachowicz

puter science can influence our approach to the solution of complex technologi-
cal problems. The analysis of turbomachinery blade attachments is not a new
task for turbomachinery engineeres. The numerical tools, however, that are at
their disposal are still in progress. That is why the quality of finite element
(FE) analysis of turbomachinery blade attachments is still being improved.
Note that the main difficulties in modeling and analysis of such objects ari-
se from their complexity, i.e. highly complicated geometrical form as well as
strong non-linearity of the related phenomena caused by plasticity and fric-
tion (physical non-linearity) as well as the changing contact zone (geometrical
non-linearity) of two parts of the attachment, i.e. blade root and grooved disc
sector. It is clear that analysis of such problems can be a challange from both
the engineering and scientific points of view.

Defining precisely our requirements for proper modeling and sufficiently
accurate FE analysis of blade attachments, we should mention the following
factors affecting the numerical solution:

• three-dimensional character of the attachment geometry and loading as
well as of the resulting strain and stress states,

• elastic or elasto-plastic properties of the bodies in contact (blade root
and disc sector),

• geometrically non-linear character of contact (generally unknowncontact
zone) due to unilateral contact constraints,

• friction occuring between contacting bodies,
• three-dimensional character of cyclo-symetric and support constraints
on the outer boundaries of the disc sector of the attachment.

1.2. State of the art

There were (or should we say, there are) four main steps in the develop-
ment of application of the finite element methods to the analysis of blade
attachments. The first step started in early 1970s with plane elastic models
loaded by concentrated forces and subject to simplified boundary and sta-
ble frictionless contact conditions (Gontarovskǐı et al., 1978; Nigin, 1976). At
the same time the plane elastic models were enriched by inclusion of friction
(Chan and Tuba, 1971). The next step which began in early 1980s accoun-
ted for elasto-plastic planemodels, firstlywithout (Gontarowskǐı andKirkach,
1982) and later with friction (Bloch and Orobinskǐı, 1983). The third stage
started in late 80s andwas caused by advances in computer technology (incre-
ase of internal memories of the available machines) and resulted in transition



Three-dimensional elastic and elasto-plastic frictional... 771

to three-dimensional attachment models. Again, elastic models without fric-
tion for very simple geometry (Alderson et al., 1967) and for complicated ones
(Ikeda et al., 1989; Teper andMoore, 1989; Robertson andWalton, 1990)were
analysed first, and then friction was included (Zboiński, 1993d). At the pre-
sent stage, three-dimensional elasto-plastic models without friction are often
employed (Rao et al., 2000; Meguid et al., 2000). Inclusion of friction to such
models has become a fact (Zboiński andOstachowicz, 1997a,b).

Two importatant comments concerning the past and contemporary FE
calculations are necessary. Firstly, there has been a great deal of work done in
this area by variousmanufacturers, usually bymeans of commercial computer
codes. Much of this work is of considerably high quality. However, manufac-
turers have generally not decided to present the results of their work in the
accessible literature. And secondly, the nowadays available very efficient and
general commercial tools for technological FE calculations may not fulfill the
specific needs of the user when non-standard, non-linear models of materials,
friction and contact are of interest. For example, comparing friction modeling
capabilities of NASTRAN (1995), ABAQUS (1995) and ADINA (1999) with
the capabilities of the code utilized in this paper (Zboiński and Ostachowicz,
1997b), one can notice that only the latter one accounts for changing friction
coefficient.

1.3. Contribution of the authors

Contribution of the authors to the subjects concerning the FE contact
analysis of turbomachinery elements can be described as follows. The gene-
ral incremental local and variational formulations for elastic contact problems
which canbeapplied toderivation of the correspondingfinite elementmethods
was presented by Zboiński (1993c, 1995a). The specific variational principle
and the basis of the corresponding finite element method applied to contact
problems of turbomachinery blade attachments can be found in the paper
by the same author (1993e). The algorithms and computer programs corre-
sponding to this method are described by Zboiński in (1993a) and (1993b),
respectively.Various aspects of stress, strain andcontact analysis of real turbo-
machinery blade attachments in the elastic range are included in the following
papers: Zboiński (1992, 1993d, 1995b). The methods, algorithms and com-
puter programs were then extended by Zboiński and Ostachowicz (1997a,b)
to elasto-plastic contact problems. This paper completes the presentation of
the research on application of elasto-plastic contact analysis to turbomachi-
nery elements. The main objective here is to show some spectacular results
of elasto-plastic stress and contact states within real turbomachinery blade



772 G.Zboiński, W.Ostachowicz

attachments. Some qualitative comparisons of these results with the case of
attachments operating in the elastic range will also be given.

2. The method and algorithm of the analysis

2.1. Local and variational formulation

Weconsider the incremental contact problemof two elasto-plastic or elastic
bodies in common non-inertial reference frame with global Cartesian coordi-
nates X. Themotion in the reference system is assumed tobe causedby small
deformations. The localmatrix formulation of the problemconsists of the equ-
ation of motion, constitutive relations of elasto-plasticity (or elasticity), and

kinematic relations (X ∈
a

V )

∆σ
j
,j +∆f−

a
ρ [∆a+(Ω⊤Ω+E)∆q+(∆Ω⊤∆Ω+∆E)X] =0

∆σ=
a

D (∆ε− aα∆Tg) (2.1)

∆ε=Γ∆q

which hold in volumes
a

V of each body a=1,2. Thematrix of elasto-plastic
properties (seeZboińskiandOstachowicz, 1997a) and the six-component stress

and strain increment vectors are denoted
a

D,∆σ,∆ε, respectively. The term
∆σj is a global vector component (j=1,2,3) of the stress increment vector,
while Γ stands for the matrix derivative operator transforming the displace-
ment increment vector ∆q into the strain vector. Increments ∆f and ∆a are
due to the body force and translation acceleration, respectively. The known
skew-symmetric matrices of angular velocity and acceleration increments are
denoted ∆Ω and ∆Ewhile their total values from the previous incremental
step are denotedwith Ω and E. The initial incremental strains due to the sca-
lar stationary incremental temperature field ∆T are defined by the coefficient

of thermal expansion
a
α and six-component Cartesian metric vector g.

The stress boundary conditions holding on the part
a

P of the body surface
can be written in the following matrix form

∆σν =∆p X ∈
a

P (2.2)

where the quantities ∆σ,∆p and ν are the stress incrementmatrix, surface
traction increment vector and the unit exterior normal vector, respectively.



Three-dimensional elastic and elasto-plastic frictional... 773

Additionally, on the part
a

Q of the body surface the incremental kinematic
boundary conditions and stress relations hold

∆q=∆d ∆σν =∆r X ∈
a

Q (2.3)

Terms ∆d and ∆r in the above relations stand for the known values of
displacement increments and the unknown values of surface stress reaction
vector increments, respectively.

The incremental contactmechanics equations valid on the common contact
surface K of the two bodies contain vectors n and t: unit exterior vector
normal to K and the matrix of vectors of two unit tangents of the lower
numbered body (a = 1). The one-component vector of gap hn is expressed
in terms of the initial gap vector h0n and normal displacement increments
∆qn and total values qn. Note that both the latter vectors are defined as
differences of the proper quantities of the first and second body, for example:

∆qn = ∆
1
qn −∆

2
qn. Four terms: ∆rn, rn, ∆rt, rt denote incremental

and total (known from the previous incremental step) values of normal and
tangential components of the surface reaction vectors. The two-component
vector of slip increment ∆st is expressed in terms of tangential displacement
increments ∆qt defined as a difference of the displacements of two bodies

in contact: ∆qt = ∆
1
qt −∆

2
qt. The function ∆g represents increment

of the tangential traction bound, while C is a general surface constitutive
matrix of either Coulomb or elasto-Coulombian slip rule. Using such notions
the mentioned equations of contact mechanics can be written as (X ∈K)

n⊤∆σν=∆rn t
⊤∆σν =∆rt

∆qn+qn+h0n =hn hn ·∆rn =0
hn ·rn =0 ∆rn ¬ 0
rn ¬ 0 ∆st =∆qt
√
∆st ·∆st(

√
∆rt ·∆rt+∆g)= 0

√
∆st ·∆st(

√
rt ·rt+µrn)= 0

√
∆rt ·∆rt+∆g+ b¬ 0 b=

√
rt ·rt+µrn ¬ 0

√
∆st ·∆st ­ 0 ∆st =C∆rt

(2.4)
One shoud notice that the above set of relations correspond to the case of
Coulombian slip rule for which one can distinguish between adhesion, slip and
gap states. For the case of the model based on elasto-Coulombian slips, for
which adhesion state does not exist four last but one equations have to be



774 G.Zboiński, W.Ostachowicz

rejected. The specific form of the surface constitutive laws for bothmentioned
cases can be found in (Zboiński and Ostachowicz, 1997a).
A variational formulation corresponding with the above local realtions

(2.1)-(2.4) takes the incremental version of the Hamilton principle as a ba-
sis. The formulation presented here assumes that two last equations of (2.1),
first equality (2.3) and all except the first, second, sixth, thirteenth and fifte-
enth relations (2.4) hold. Thus the proper variational inequality (see Zboiński
andOstachowicz, 1997a) can be written as

2
∑

a=1

{

∫

a

V

{δ(∆ε⊤)
a

D (∆ε− aα∆Tg)− δ(∆q⊤)∆f−

−δ(∆q⊤)
a
ρ [∆a+(Ω⊤Ω+E)∆q+(∆Ω⊤∆Ω+∆E)X]} d

a

V −

−
∫

a

P

δ(∆q⊤)∆p d
a

P −
∫

a

Q

δ(∆q⊤)∆r d
a

Q
}

− (2.5)

−
∫

K

[δ(
√

∆st ·∆st)∆g+ δ(∆q⊤n +q⊤+h⊤0n−h⊤n)∆rn+

+δ(∆q⊤t −∆s⊤t )∆rt] dK ¬ 0

The inequality form of the above principle is caused by unilateral normal
contact conditions and existence of tangential traction bound in the case of
Coulombian slips. For the case of elasto-Coulombian slips, one should replace
in (2.5) the term δ(

√
∆st ·∆st)∆g accounting for either adhesion or the slip

states with δ(∆s⊤t )∆rt which corresponds to the slip state only. After such a
change, the inequality form of the principle is still preserved due to unilateral
normal contact constraints.
In order to derive one general method of solution of the Coulombian and

elasto-Coulombian problemsunder considerationwe canpropose the introduc-
tion of the proper reduced contact problem at each incremental step, as one
of the possibilities. The suggested method is based on the assumption that
the potential contact area which is divided either into adhesion, slip or gap
(Coulombianmodel) or slip and gap (elasto-Coulombian one) parts, does not
changewithin the incremental or iterative step of the solution procedure.Note
that the three parts of the potential contact area are denoted by letters A,
S and G, respectively. The mentioned divisions need employing the proper
criteria based on inequalities from the set (2.4). In particular, we use the sixth
and eighth relations to choose between the gap and contact states, while the
twelvth and fourteenth relations are used to distinguish between the adhesion



Three-dimensional elastic and elasto-plastic frictional... 775

and slip states. Additionally, in our formulation we have to take into account
that: the work of tangential tractions is done only on the part S, for parts
A and S the relation hn = 0 holds, whereas on A we have ∆st = 0.
Moreover, we assume that the tangential traction bound increment and slip
increments are known in each iterative step of the solution process.We denote
the known values of these quantities by ∆g ≡ µ∆pn and ∆st ≡ ∆zt. Use
of the above notions, expression of the slip increments δ(∆st) through the
tangential displacement increments δ(∆qt), and change of surface degrees of
freedom (DOFs) ∆qn,∆qt into global DOFs ∆q in X-directions bymeans
of relations: ∆qn =n∆q,∆qt = t∆q allows to write the proper stationarity
principle corresponding to Coulomb slips in the following form

2
∑

a=1

{

∫

a

V

{δ(∆ε⊤)
a

D (∆ε− aα∆Tg)− δ(∆q⊤)∆f−

−δ(∆q⊤)
a
ρ [∆a+(Ω⊤Ω+E)∆q+(∆Ω⊤∆Ω+∆E)X]} d

a

V −

−
∫

a

P

δ(∆q⊤)∆p d
a

P −
∫

a

Q

δ(∆q⊤)∆r d
a

Q− (2.6)

−(−1)a+1
[

∫

S

δ(∆q⊤)
t∆ztµ∆pn√
∆zt∆zt

dS+

∫

S

δ(∆q⊤)n∆rn dS+

+

∫

A

δ(∆q⊤)∆r dA
]}

=0

while for the elasto-Coulombian case we can write

2
∑

a=1

{

∫

a

V

{δ(∆ε⊤)
a

D (∆ε− aα∆Tg)− δ(∆q⊤)∆f−

−δ(∆q⊤)
a
ρ [∆a+(Ω⊤Ω+E)∆q+(∆Ω⊤∆Ω+∆E)X]} d

a

V

−
∫

a

P

δ(∆q⊤)∆p d
a

P −
∫

a

Q

δ(∆q⊤)∆r d
a

Q− (2.7)

−(−1)a+1
[

∫

S

δ(∆q⊤)t∆tt dS+

∫

S

δ(∆q⊤)n∆rn dS
]}

−

−
∫

S

δ(∆q⊤)tktt
⊤∆q dS=0



776 G.Zboiński, W.Ostachowicz

It is worth noticing that we replaced differences ∆q with increments ∆q
of each of two bodies in contact in (2.5) and (2.6), when it was posssible. Note
also that in the last relation, kt and ∆tt represent the surface constitutive
matrix corresponding to slip increments and tangential traction increment
vector due to normal traction increment (compare Zboiński andOstachowicz,
1997a). We should underline that two above relations constitute a starting
point for introduction of one general algorithm for two finite elementmethods
of contact problems of elastoplasticity. Elucidation of these methods can be
found in the paper cited above.

2.2. Some details of the algorithm

A full algorithm for the FE solution of the problems under consideration
consists of generation of the element stiffness matrices and force increments
vectors, structure stiffness and force increment formation, solution of the set
of nonlinear algebraic equations and frictional contact algorithms. The for-
mer two parts of the procedure have the property that they correspond with
the standard FE algorithms for non-linear elastoplastic problems, while the
latter two include the non-standard, non-linear aspects of the contact analy-
sis. The simplified flow diagram of the algorithm is presented in Fig.1. Note
that twodifferentprocedures corresponding toCoulomb (C-model) and elasto-
Coulombian (EC-model) models are combined in one general algorithm with
the proposed approach. This algorithm is utilized in the computer program
PLAST presented in (Zboiński and Ostachowicz, 1997b) which is applied to
the research conducted in this paper.

Details of the above algorithmwere presented in the previous work by the
authors (Zboiński and Ostachowicz, 1997a). Here we restrict ourselves to so-
me original aspects concerning the calculation of friction stiffnesses and forces
which both reflect the physical non-linearity of the contact. The first impor-
tant feature of the proposed approach is introduction of all friction terms on
the global level, i.e. through nodal surface stiffness and nodal friction forces of
the structure instead of defining them on the element level. We should stress
that the global nodal description of friction forces has been succesfully ap-
plied to problemswith Coulombianmodels accounting only for non-reversible
(Coulombian slips) and neglecting elastic (reversible) slips. On the contrary,
introduction of the friction terms to elasto-Coulombian models has been per-
formed on the element level through element vectors of frictional stresses and
element surface stiffnesses, so far. The reason for this is dependence of the ela-
stic part of the surface stiffness on discretization of the contact area. Hence,
in order to introduce the friction terms on the global nodal level, we have to



Three-dimensional elastic and elasto-plastic frictional... 777

Fig. 1. General algorithm for frictional elastic and elasto-plastic contact analysis



778 G.Zboiński, W.Ostachowicz

overcome the problem of discretization, or shouldwe say, of assignment of the
the contact area to a global (structure) contact node. The proper method is
given below.
The second feature of the proposed approach deals with the solution me-

thod of the contact problems, which assumes a reduced problemat each incre-
mental and iterative step of the solution process. Such an assumption leads to
linearization of theFEelement equilibriumequations. Inpractice itmeans that

the global surface stiffness
i

Kt (includedonly for the case of elasto-Coulombian
friction) defined for any pair of nodes I and J of the first and second body,

respectively, as well as the global incremental friction forces ∆
i

T t at node I,
which both depend on the unknown displacement increments, are prescribed
at any step i of the solution procedure. The assumed to be known values

of the nodal vectors of the normal traction increments ∆
i

Pn, total surface

tractions
i

P= col[
i

P t,
i

Pn] and slip increments ∆
i

Zt are used for the purpose
of friction terms determination and are assumed to be equal to the proper
resultant values from the previous step i−1, i.e.

∆
i

Pn=∆
i−1

R n
i

P t=
i−1

T t

i

Pn=
i−1

R n ∆
i

Zt=∆
i−1

S t

(2.8)

For the elasto-Coulombian model of friction, contribution of the surface
constitutive stiffness of the pair of nodes I and J to the global equilibrium
equation is

δ(∆q⊤t )
i

Kt ∆qt =
[

δ(
1

∆q ⊤t ),δ(
2

∆q ⊤t )
]







i

Kt −
i

Kt

−
i

Kt

i

Kt











∆
1
qt

∆
2
qt



 (2.9)

where each matrix block
i

Kt is given by

i

Kt=







i

Kt11
i

Kt12
i

Kt21
i

Kt22






(2.10)

and terms
i

Ktkl (k,l,m=1,2) are defined as follows

i

Ktkl=−kAI
(

δkl−β
kAI
i

P

i

Ptk
i

Ptl
i

Ptm
i

Ptm

)

(2.11)



Three-dimensional elastic and elasto-plastic frictional... 779

with δkl being the Kronecker symbol, while contribution of the increments
of friction forces at node I to the same global equilibrium equations can be
defined by

δ(
1

∆q ⊤t )∆
i

T t=
[

δ(
1

∆qt1),δ(
1

∆qt2)
]







∆
i

T t1

∆
i

T t2






(2.12)

with terms ∆
i

Ttk given by the following formula

∆
i

T tk=−β
kAI
i

P

i

Ptk
√

i

Ptm
i

Ptm

µ∆
i

Pn (2.13)

In the above relations β=0defines the case of elastic slipswhereas for β=1
the slips are elasto-Coulombian. Constant k characterizes the surface elastic

properties while the auxiliary term
i

P can be calculated according to

i

P= kAI +
∂µ

∂
√

∆Sctm∆S
c
tm

∆
i

Ztm
i

Ptm
√

∆
i

Ztm ∆
i

Ztm

√

i

Ptm
i

Ptm

i

Pn (2.14)

Note that the term ∂µ/∂
√

∆Sctm∆S
c
tm
represents a known relation between

the friction coefficient and the Coulombian part of slip. This relation corre-
sponds to elasto-Coulombian slip rule with hardening. The term AI stands
for the contact area assigned for node I or J (AI ≡AJ) and can be obtained
through summation of contact areas aeI assigned to nodes of the elements eI
meeting at node I

AI =
EI
∑

eI=1

aeI (2.15)

where EI stands for total number of elements forming the contact area under
consideration. The elemental part of the contact area can bedefined as follows

aeI =

∫

Ae

NINI dAe (2.16)

with Ae standing for the element face within contact surface and NI being
the shape function assigned to element node I.
Subsequently, for the Coulombian model of friction we have to introduce

only the vector of increments of friction forces updatedwithin each step of the



780 G.Zboiński, W.Ostachowicz

solution procedure. The components of this vector are

∆
i

T tk=−
i

Ptk
√

i

Ptm
i

Ptm

µ∆
i

Pn (2.17)

Note that for idealCoulombian slip rule also another equivalent representation
is possible

∆
i

Ttk=−
∆
i

Ztk
√

∆
i

Ztm ∆
i

Ztm

µ∆
i

Pn (2.18)

Note thatwhennecessary (change fromadhesion to slip), in order to stabi-
lize and thus to speedup the iterationprocess for the latter case ofCoulombian
model, one can use the relaxation method which assumes

∆
i

T tk=(1−α)∆
i−1

T tk +α∆
i

Ttk (2.19)

with α standing for the relaxation factor.

3. Elastic and elasto-plastic contact analysis of real blade

attachments

3.1. Applied FE models of real attachments

The applied examples of real turbomachinery blade attachments (see e.g.
Fig.2) as utilized in this research are the same as in our previous attempts
(Zboiński, 1993d, 1995b; Zboiński and Ostachowicz, 1997b). The correspon-
dingFEmodels of a single blade airfoil and root aswell as a disc sector with a
single groove consist of either about 40 thousandDOFs and five thousand iso-
parametric solid elements or about thertis thousandDOFs and four thousand
elements. These two cases correspond to the attachments operating within
elastic or elasto-plastic ranges, respectively. In bothmodels the first-order and
second-order solid elements as well as mixed-order transition elements (from
first- to second-order solid elements) are employed. The second-order elements
are applied in the most-stressed regions of the attachment (minimum neck
sections of the root and disc as well as contact surfaces of hooks). For other
less-stressed parts of the attachment, first-order elements are applied. One
layer of the transition elements is utilized in order to join the first-order and



Three-dimensional elastic and elasto-plastic frictional... 781

Fig. 2. Exemplarymesh of the blade, root and disc sector of the attachment



782 G.Zboiński, W.Ostachowicz

second-order regions together. The elements are put into 11 layers parallel to
the attachment discmid-plane. TheFEmesh details are those shown in Fig.3
for the attachment blade root and disc sector.

Fig. 3. Mesh details within cross-section of a single contact surface

3.2. Elastic analysis of frictional contact problem

Thefirst example deals with the incremental elastic frictional contact ana-
lysis of an axial entry, curved, fir-tree attachment of the blade, with four pairs
of root and groove hooks on each side of the attachment. The attachment cu-
rvature radius is 0.125m, a shift of the curvature center from the disc middle
plane is 0.025m. The attachment is loaded by the blade 0.39m long of the
first stage of the low pressure part of the 200MW steam turbine. The bla-
de rotating at the angular velocity of 3.1415× 1021/s (3000rpm) produces
the centrifugal force equal to 109.146kN acting on the root platform 0.0078m
thick. The outer radius of the disc is 0.519m, the inner one is 0.250m, the disc
(and attachment) thickness is 0.085m, while the number of the blade attach-
ments on the disc circumference is 82. Young’s modulus of the root and blade
material equals 1.94×1011N/m2, whereas the modulus of the disc material
is 1.88×1011N/m2. Thematerial densities and Poisson’s ratios are assumed
to be equal to the values of 0.78×104kg/m3 and 0.3, respectively. Four cases



Three-dimensional elastic and elasto-plastic frictional... 783

corresponding to geometrically nonlinear contact problem (unknown geome-
try of contact) are included. The first case deals with the physically linear
(frictionless) contact problem, while the rest three cases refer to physically
non-linear (frictional) contact problems. The constant and uniform Coulomb
friction coefficient µ for the latter three cases equals 0.05, 0.10 and 0.15,
respectively.

An example of effective stress distribution within the attachment for the
case 1 is shown in Fig.4. It corresponds to the middle cross-section of the at-
tachment.Effective stress isolines from1to8correspondto thevalues changing
proportionally from 0.25× 108 to 2.00× 108N/m2. Note that for the rest
three cases stress distribution is qualitatively the same. For all four cases the
maximal normal contact stresseswithin themaximally loaded contact surfaces
as a function of friction coefficient are presented below in Table 1. For cases 1
and 4, additionally, the initial and resultant contact statuses of contact nodes
within maximally stressed contact surfaces are shown in Table 2. Note that
the equivalent initial and resultant statuses corresponding to adhesion, slip
and gap are denoted with symbols: A, S and G, respectively, while transition
from adhesion to slip, from adhesion to gap and from slip to gap are denoted
as: A/S, A/G, S/G, respectively.

Table 1.Maximal normal contact stresses for the elastic cases 1-4

Normal contact stresses [108N/m2]
Case Location (lowest left hooks) Location (highest right hooks)

Left edge Middle Right edge Left edge Middle Right edge

1 1.931 0.726 0.557 1.380 0.604 0.458

2 1.685 0.577 0.666 1.176 0.493 0.525

3 1.586 0.495 0.707 1.043 0.431 0.564

4 1.532 0.460 0.719 0.974 0.406 0.583

Table 2. Map of initial and resultant contact status for elastic cases 1
and 4

Node Initial and resultant status of contact nodes
Case loca- Node layer (lowest left hooks) Node layer (highest right hooks)

tion 1 2 3 4-10 11 12 1 2 3 4-10 11 12

Left S S S S S S S/G S/G S/G S S S
1 Middle S S S S S S S S S S S S
Right S/G S S S S S S S S S S S
Left A/S A/S A/S A/S A/S A/S A/G A/G A/G A/S A/S A/S

4 Middle A/S A/S A/S A/S A/S A/S A/S A/S A/S A/S A/S A/S
Right A/G A/S A/S A/S A/S A/S A/S A/S A/S A/S A/S A/S



784 G.Zboiński, W.Ostachowicz

Fig. 4. Elastic effective stress in the cross-section of the attachment



Three-dimensional elastic and elasto-plastic frictional... 785

It is worth noting that the stress distribution within the attachment is
typical. It means that high values of effective stresses appear at theminimum
neck sections or the adjacent contact surfaces of the root and disc, and the
maximal values correspond to the first left root and fourth right disc hooks.
It is interesting to notice that inclusion of friction (µ = 0 in comparison
to µ = 0.05) changes the values of contact stresses much more than further
increase of friction coefficient (µ = 0.10 and µ = 0.15). Note also that
inclusion of friction diminishes the contact stresses and does not affect the
tendency to gap appearance.

3.3. Elasto-plastic analysis of frictional contact problem

The second example deals with the elasto-plastic frictionless contact ana-
lysis of an axial entry, curved, fir-tree attachment of the blade, with three
pairs of root and groove hooks on each side of the attachment. The attach-
ment curvature radius is 0.185m, the shift of the curvature center from the
disc middle plane is 0.024m. The attachment is loaded by the blade 0.71m
long of the penultimate stage of the low pressure part of a steam turbine. The
blade rotating at the angular velocity of 3.1415×102 1/s (3000rpm) produces
the centrifugal force equal to 110.503kN acting on the root platform 0.0075m
thick. The outer radius of the disc is 0.705m, the inner one is 0.250m, the disc
(and attachment) thickness is 0.190m, while the number of the blade attach-
ments on the disc circumference is 53. Young’s modulus of the root and blade
material equals 1.94×1011N/m2, whereas the modulus of the disc material
is 1.88×1011N/m2. Thematerial densities and Poisson’s ratios are assumed
to be equal to 0.78× 104kg/m3 and 0.3, respectively. The yield points of
the blade root and disc materials are 530×106N/m2 and 540×106N/m2,
respectively. The common value of the plastic modulus is 0.01×1011N/m2.
Again four cases are analysed, the first one of which corresponds to geome-
trically non-linear contact problem with µ = 0, while the next three cases
to geometrically and physically nonlinear contact with µ equal to: 0.05, 0.10
and 0.15.

An example of effective stress distribution within the attachment for the
case is shown in Fig.5. It corresponds to the middle cross-section of the at-
tachment. Effective stress isolines from 1 to 5 correspond with values chan-
ging proportionally from 1.00× 108 to 3.00× 108N/m2. The last isoline
(denoted with number 6) corresponds to the value of effective stress equal to
5.35×108N/m2. Hence the regions of elasto-plastic deformation can be easily
recognized as those neighboring the last isoline. As before, for all four cases
themaximal normal contact stresseswithin themaximally loaded contact sur-



786 G.Zboiński, W.Ostachowicz

Fig. 5. Elasto-plastic effective stress in the cross-section of the attachment



Three-dimensional elastic and elasto-plastic frictional... 787

faces as a function of friction coefficient are presented (see Table 3 below). For
cases 1 and 4 also the initial and resultant contact statuses of contact nodes
within maximally stressed contact surfaces are shown in Table 4.

Table 3.Maximal normal contact stresses for the elasto-plastic cases 1-4

Normal contact stresses [108N/m2]
Case Location (lowest right hooks) Location (highest left hooks)

Right edge Middle Left edge Right edge Middle Left edge

1 4.384 1.562 2.832 2.891 1.826 2.833

2 3.989 1.250 2.889 2.602 1.589 2.861

3 3.858 1.172 3.002 2.457 1.479 2.918

4 3.814 1.125 3.036 2.371 1.443 2.946

Table 4. Map of initial and resultant contact status for elasto-plastic
cases 1 and 4

Node Initial and resultant status of contact nodes
Case loca- Node layer (lowest right hooks) Node layer (highest left hooks)

tion 1 2 3 4-10 11 12 1 2 3 4-10 11 12

Right S S S S S S S/G S S S S S/G
1 Middle S S S S S S S S S S S S

Left S S S S S S S S S S S S
Right A/S A/S A/S A/S A/S A/S A/G A/S A/S A/S A/S A/G

4 Middle A/S A/S A/S A/S A/S A/S A/S A/S A/S A/S A/S A/S
Left A/S A/S A/S A/S A/S A/S A/S A/S A/S A/S A/S A/S

The stress distribution within the attachment is typical, that is high va-
lues of effective stresses appear at the minimumneck sections or the adjacent
contact surfaces of the root and disc, and the maximal values correspond to
the first root and third disc hooks. Note, however, that the elasto-plastic de-
formations occur only for the latter hooks. It is interesting to notice again that
inclusion of friction (µ = 0 in comparison to µ = 0.05) changes the values
of contact stresses much more than further increase of the friction coefficient
(µ = 0.10 and µ= 0.15). Again, inclusion of friction diminishes the contact
stresses and does not affect the tendency to gap appearance.

Comparing the results of the values of the normal contact stresses from
Tables 1 and 3 one can notice that inclusion of friction changes these values
more significantly in the elastic case. Also, a tendency to gap appearance is
stronger for that case (compare Tables 2 and 4). These two observations can
be atributted to smaller effective stresses within the attachment operating in
the elastic range.



788 G.Zboiński, W.Ostachowicz

4. Conclusions

It is possible to generate one general FE algorithm conforming both to the
Coulombianandelasto-Coulombian slip rules for incremental contact problems
of elasticity and elasto-plasticity as applied to three-dimensional turbomachi-
nery blade attachment analysis.

Physical non-linearity of contact problems due to friction plays an impor-
tant role in the stress analysis both in the elastic and elasto-plastic range. In
the case of the attachments under consideration, inclusion of friction changes
the effective stresses at the hook notches and a tendency to reduce them pre-
vails. Also a tendency to change the distribution and to reduce the normal
contact stresses appears.

On the contrary, inclusion of friction does not seem to affect the gap appe-
arance within contact surfaces for both the cases of elastic and elasto-plastic
analysis.

In the case of highly stressed attachments (which may include also elasto-
plastic deformation), changes of the normal contact stresses due to inclusion
of friction areweaker than in the case of less-stressed attachments. A tendency
to gap appearance is weaker for highly stressed attachments.

References

1. Alderson R.G., Tani M.A., Tree D.J., 1967, Three-Dimensional Optimi-
zation of a Gas Turbine Disk and Blade Attachment, J. Aircraft, 13, 994-998

2. Bloch M.V., Orobinskǐı A.V., 1983, O Modifikacii Metoda Konechnykh
ElementovdljaResheniyaDvumernykhUprugikh iPlasticheskikhKontaktnykh
Zadach,Problemy Prochnosti, 5, 21-27

3. Chan S.K., Tuba I.S., 1971,AFinite ElementMethod for ContactProblems
ofSolidBodies.Part II –Application toTurbineBladeFastenings, Int. J.Mech.
Sciences, 13, 627-639

4. Gontarovskǐı P.P., Kirkač B.N., 1982, Issledovanie Napryazhenno-
Deformirovannogo Sostoyaniya Zamkovykh Soedinenǐı Lopatok Turbomashin
MetodomKonechnykh Elementov,Problemy Prochnosti, 8, 37-40

5. Gontarovskǐı P.P., Kirkač B.N., Marcenko G.A., 1978, Priblizhenny̌ı
Metod Rascheta Zamkovykh Soedinenǐı Lopatok Turbomashin,Problemy Ma-
shinostroeniya, 6, 52-55



Three-dimensional elastic and elasto-plastic frictional... 789

6. Ikeda T., Hisa S., Matsuura T., 1989, The Development of a Titanium
Last-StageBlade for 3600 rpmSteamTurbines,Latest Advances in SteamTur-
bine Design, Blading, Repairs, Condition Assessment, and Condenser Interac-
tion, PWR-7, 69-76, ASME, NewYork

7. Meguid S.A., Kanath P.S., Czekanski A., 2000, Finite Element Analysis
of Fir-Tree Region in Turbine Discs, Finite Elements in Analysis and Design,
35, 305-317

8. NiginA.A., 1976,RashchëtElochnogoZamka, Izv. Vysshykh Uceb. Zavedenǐı.
Mashinostroenie, 2, 17-21

9. Rao J.S., Ramakrishnan C.V., Gupta K., Singh A.K., 2000, Elasto-
Plastic StressAnalysis of LPSteamTurbineBlades underCentrifugal Loading,
J. Turbomachinery (to appear)

10. RobertsonM.D.,WaltonD., 1990,DesignAnalysis of SteamTurbineBlade
Roots under Centrifugal Loading, Journal of Strain Analysis for Engineering
Design, 25, 185-195

11. Teper B.,MooreC.T., 1989,TheRedesign of a Fir Tree BladeRoot for the
Penultimate Stage of a 500MW Turbine, Latest Advances in Steam Turbine
Design, Blading, Repairs, Condition Assesment, and Condenser Interaction,
PWR-7, 17-22, ASME, NewYork

12. Zboiński G., 1992, Numerical ContactAnalysis of TurbomachineryBladeAt-
tachments,Archives of Mechanical Engineering, 32, 317-331

13. Zboiński G., 1993a, FEAlgorithm for Incremental Analysis of Large 3DFric-
tional Contact Problems of Linear Elasticity, Computers and Structures, 46,
669-677

14. Zboiński G., 1993b, FEComputer Program for IncrementalAnalysis of Large
3DFrictionalContactProblemsofLinearElasticity,Computers and Structures,
46, 679-687

15. Zboiński G., 1993c, Incremental Variational Principles for Frictional Contact
Problems of Linear Elasticity,ASME J. Appl. Mech., 60, 982-985

16. Zboiński G., 1993d, Numerical Research on 3D Contact Problems of Turbo-
machineryBladeAttachments in the ElasticRange, Int. Journal of Mechanical
Sciences, 35 (1993), 141-165

17. Zboiński G., 1993e, The Incremental Variational Principle and Finite Ele-
ment Displacement Approximation for Frictional Contact Problem of Linear
Elasticity, Journal of Non-Linear Mechanics, 28, 13-28

18. ZboińskiG., 1995a,Derivation of theVariational Inequalities of the Incremen-
tal Frictional Elastic Contact Problems,Archives of Mechanics, 47, 725-743



790 G.Zboiński, W.Ostachowicz

19. ZboińskiG., 1995b,Physical andGeometricalNon-Linearities inContactPro-
blems of Elastic Turbine Blade Attachments,Proc. of the Instn. of Mechanical
Engineers. Part C: Journal of Mechanical Engineering Sciences, 209, 273-286

20. Zboiński G., Ostachowicz W., 1997a, A General FE Algorithm for 3D In-
cremental Analysis of Frictional Contact Problems of Elasto-Plasticity, Finite
Elements in Analysis and Design, 27, 289-305

21. Zboiński G., Ostachowicz W., 1997b, A General FE Computer Program
for 3DIncrementalAnalysis ofFrictionalContactProblemsofElasto-Plasticity,
Finite Elements in Analysis and Design, 27, 307-322

22. 1995,ABAQUS. Theory Manual, Version 5.5, Hibbit, Karlsson and Sorensen,
Inc., Pawtucket, RI (USA)

23. 1995, MSC/NASTRAN. Release Notes for Version 68, The MacNeal-
Schwendler Corporation, Los Angeles, CA (USA)

24. 1999, Automatic Dynamic Incremental Nonlinear Analysis. Theory and Mo-
deling Guide. Volume I: ADINA, Report ARD 99-7, ADINA R & D, Inc.,
Watertown,MA (USA)

Trójwymiarowa analiza kontaktowa z tarciem zamocowań łopatek

maszyn wirnikowych w zakresie sprężystym i sprężysto-plastycznym

Streszczenie

Niniejsza praca przedstawia podsumowanie wyników badań nad trójwymiarową
analizą kontaktu w zamocowaniach łopatek maszyn wirnikowych w zakresie spręży-
stymi sprężystoplastycznym.Wtymkontekściezaprezentowanezostały istotneaspek-
ty analizy związane z teoretycznym, metodologicznym i fenomenologicznym opisem
problemu.Pracakoncentruje sięna zastosowanychsformułowaniachwariacyjnychpro-
blemówkontaktowychsprężystości i sprężysto-plastyczności,odpowiadających imme-
todach elementów skończonych, zaproponowanych algorytmach kontaktowych, a tak-
że opisach stanów przemieszczeń, naprężeń, kontaktu i poślizgów w rzeczywistych
zamocowaniach łopatek turbinowych. Ze szczególną uwagą potraktowano problemy
nieliniowego charakteru algorytmówkontaktowychorazwpływu tych nieliniowości na
naprężenia i odkształcenia w zamocowaniu.

Manuscript received December 11, 2000; accepted for print February 20, 2001