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LINING DESIGN FOR VESSELS OF THE STEEL INDUSTRY APPLICATION OF FINITE ELEMENT CALCULATIONS
K. Andreev
H. Harmuth
Christian-Doppler-Laboratory for Building Materials with Optimised Properties
and Department of Ceramics, University of Leoben,
Peter-Tunner-Straße 5, A-8700 Leoben, Austria.
Abstract
Understanding the thermo-mechanical behaviour of lining materials allows an improved
material choice, as well as an optimisation of the lining design and heating-up procedures.
The response of refractory materials to the thermo-mechanical loads typical for a teeming
ladle has been simulated by means of non-linear finite element analysis using several
commercial codes partly with user developed subroutines. For this purpose a number of three
dimensional models representing different parts of the ladle have been analysed. The crack
pattern developing in the case of spalling is determined dependent on the thermal load (the
time/temperature curve) and the lining design. In case of shaped refractories, the effect of
joints and expansion allowances has been taken into account. Special attention is paid to the
influence of thermal expansion of the lining material on the stress in the steel shell and its
deformation.
1. INTRODUCTION AND BACKGROUND
The thermo-mechanical behaviour of refractory lining systems has a very complex nature due
to both a significant number of influencing parameters and a complex material response. In
this investigation the finite element method (FEM) is used for separate modelling of certain
aspects of loading and structural response which provides better understanding of their role in
general behaviour. For this purpose the behaviour of a 150-ton teeming ladle is examined. The
bottom of the ladle is lined with a high alumina mix, different fireclay bricks are used as a
safety lining. The side walls, including the slag zone, are lined with magnesia-carbon bricks as
the working lining, the safety lining is built up by forsterite bricks in the lower part of the wall
and by a high alumina mix in its upper part.
Two features important for a feasible operation of the ladle are addressed in the paper, these
are spalling due to thermo-mechanical reasons and the long term deformation of the shell
caused by thermal expansion of the lining. Elliptical deformation developing in the shell after
some years of use significantly reduce service life. The refractory lining may influence the
shell deformations in two ways. Its cyclic thermal expansion acts in addition to mechanical
service loads and both together strain the shell. The insulating performance of the lining
determines the temperatures developing in the shell, and thus may reduce or accelerate
thermal creep. Therefore thermal and thermo-mechanical analyses have been performed. The
thermo-mechanical investigations presented here are focused on the role of expansion
allowances in developing of spalling and lining deformations.
2. SIMULATION OF MATERIAL BEHAVIOUR
2.1 Mechanical behaviour of refractories
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Refractories as any other brittle disordered materials are characterised by a rather coarse
grain/matrix structure with a maximal grain size of e.g. 5 mm and a defect size in a similar
order of magnitude (fig 1.1). These structural features provide a specific response to
mechanical loading [1]. Linear-elastic material behaviour is exhibited only in the beginning of
loading prior to crack initiation. Initiation and growth of cracks corresponds with a
”softening” behaviour after the maximum stress is reached. The ability to sustain loads even
after crack initiation is favoured by a relatively high resistance against crack propagation due
to processes like grain bridging, microcrack formation, crack branching and irreversible
deformations.
1
2
Fig 1.Typical microstructure of a magnesia-carbon brick (1) and failure curves of the material
under three-axial stress (2) [2].
Refractories demonstrate much higher strength under compression than under tension.
Depending on the compressive stress state, failure may take place either due to tension or
shear. The material strength is sensitive to the hydrostatic pressure, e.g. the strength under
three-axial compression may exceed uni-axial compressive strength by 600 % (fig 1.2). The
compressive failure is followed by plastic deformations characterised by volume change
(dilatancy), materials with greater dilatancy demonstrate more rigid response during pre-peak
loading.
Most material properties are temperature dependent. In general with rising temperature
material response becomes less rigid, and creep may become an important parameter.
2.2 Material models
Simulating material behaviour of refractories under loading involves determination of failure
stresses, and description of linear-elastic material response preceding the failure which can be
followed by strain softening or hardening behaviour. The different material strength observed
under tensile and compressive stresses demand a combined failure criterion. Such a criterion
usually predicts that the failure in tensile quadrant occurs when maximal principal stress
reaches the value of material strength under uni-axial tension. A plastic yield criterion is used
for pure compressive stresses, and for combinations of compressive and tensile stresses the
criterion is either linear (determined by uniaxial compressive and tensile strengthes) or
constant which means it is equal to uni-axial tensile case (Rankine criterion) [3].
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A plastic yield criterion capable to predict material failure under multiaxial compressive stress
was proposed by Drucker and Prager in [4]. It is related to the shear strain energy criterion of
von Mises and can be established using principal stresses (σ1,σ2,σ3) in the following way:
(σ1-σ2)2+(σ2-σ3)2+(σ3-σ1)2=(c+Kσh)2
(1),
Here σh is the hydrostatic pressure, K is a
function of friction angle φ, c is the material
cohesion, and σi are principal stresses. In the
principal stress space this criterion will form a
cone (fig. 2). The cylindrical coordinates of ξ,
r and θ illustrate the position of the failure
surface in terms of hydrostatic pressure (ξ) and
deviatoric stress (r), I1 and J2 are invariants of
stress and deviatoric stress tensors, and the
angle θ defines the position with respect to the
axes. If the friction angle is constant the
cohesion is given by
Fig. 2 Schematic representation of the DruckerPrager model, details are explained in the text.
c = fc
1 − sin φ
2 cos φ
(2)
Here fc is the uni-axial compressive strength.
The hardening hypothesis is specified by the evolution of cohesion c and the intermal state
state variable representing the parameter of hardening k, introduced to allow the models to
describe some of the complexity of the inelastic response of real materials. The relation
between the uni-axial plastic strain rate and the variable k for a strain-hardening hypothesis is
k! = −
1 + 2a g2
1 − ag
ε! p
(3)
The scalar ag is defined by the dilatancy angle ψ as
2 sinψ (k )
ag =
3 − sinψ (k )
The angle of dilatancy is in turn determined from the plastic (index p) strain rates as

ε! p
sinψ = p v p
ε!v − 2ε!1

 ε!v = ε!1 + ε!2 + ε!3
(4)
(5)
The fracture behaviour under tension can be modelled applying the fictitious crack model
according to Hillerborg [5]. The model describes the material behaviour using two
constitutive laws (fig. 3). The elastic stress-strain behaviour describes the behaviour of uncracked material. The non-linear behaviour of the propagating crack is described by
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Fig. 3 Cohesive crack model.
considering a fictitious crack with cohesive forces acting between its crack faces. At the tip of
the cohesive crack the maximum stress is equal to the tensile strength ft. For the process wake
the stress in dependence of the crack opening is given by the softening diagram.
The area under the σ-x curve is equal to the specific fracture energy (Gf). Two ways are
common to apply the fictitious crack model for finite element calculations. One of them is so
called smeared crack model. In this case the position of the crack within an element is not
localised, the total strain of the element is composed of the crack strain and the elastic strain.
If the bandwidth h is a length characterising the element size perpendicular to the crack faces,
the mode I specific fracture energy can be calculated by
Gf = h
ε ncr = ∞
∫σ
cr
n
dε ncr
(6)
ε ncr = 0
Here σ ncr and ε ncr have the meaning of the stress and the crack strain perpendicular (subscript
n) to the crack faces. The crack strain ε ncr is equal to the ratio of the crack opening x to the
length h.
It is common to use dimensionless strain-softening models which result e.g. from the
following standardisation:
σ cr
ε cr
σ ncr, r (ε ncr, r ) = n ; ε ncr, r = crn
(7)
ε n , ult
ft
In equation (7) the standardized quantities are identified by an additional subscript r; ft is the
tensile strength and ε n,crult the ultimate strain, where σ n,crr approaches zero. Several e.g. bilinear
or exponential functions are used to describe ε n,crr . Depending on the model they are defined
by material properties like ft and Gf. The experimental procedures allowing to obtain
necessary material properties are reported elsewhere [6, 7].
3. ANALYSIS
Two commercial FEM codes were tried for the simulation, ABAQUS and DIANA. The code
DIANA was chosen for the greater part of the investigations as it showed to be especially
suitable for the models applied here. In this code following analysis options were chosen.
Material failure was predicted by a linear tension cut-off criterion combined with DruckerPrager plasticity model. The hypothesis of Moelands [3] was used to model the softening
under tension, and a user specified hardening/softening based on the results of uni-axial
compression tests was specified for compression. The case of associated plasticity (φ=ψ) was
considered. Material properties were described as temperature dependent. For the steel shell
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the plastic flow criterion of von Mises with perfect plasticity was applied. A special contact
algorithm was utilised to model contact between two expanding bricks. The smeared multidirectional fixed crack concept was used to describe cracking.
The study was performed in two steps:
- calculation of steady state and transient temperature distribution for the whole ladle
- calculations of stresses in the lining and the shell and failure by crack formation.
3.1 Thermal calculations
The thermal problem was solved by means of an axisymmetrical model applying steady state
and transient analysis. The model featured all major lining layers and the steel shell with its
support. The thermal boundary conditions with the temperatures of steel of 1580 °C and of the
ambient air of 15 °C were modelled in steady state analysis. The heat transfer due to radiation
and convection was taken into account. The obtained results have been checked by
comparison with measurements by an infrared camera (fig 4).
Fig. 4 Results of the steady state thermal analysis.
A quite good accuracy of the temperature filed representation follows from the comparison of
the calculated and measured profiles. The only significant deviation is at the point #6 lying on
the support. This can be explained by the fact that the heat transfer by the air between the
support and the shell was not taken into account for the sake of simplicity. The steel shell had
its highest temperature in the middle of the ladle wall near the support.
The heating-up of the cold ladle with several operation cycles has been simulated by transient
calculations. The cold ladle is warmed-up form 15 °C to 1100 °C, a holding time of 3 hours
precedes the first charging. Operation cycles were considered 150 min. long, including a
period of 60 min. after the end of teeming when the ladle is idle. Temperature-time profiles at
certain points on the outer shell surface and on the hot face are presented in fig 5.
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Fig 5. Some results of the transient thermal analysis.
During the operation the highest shell temperatures are observed in the area under the support
construction (point #2 in fig. 5) and therefore the greatest heat losses through the lining takes
place in this area. This is explained by the fact that lining has its smallest thickness in this part
of the ladle. Here the lining is also least thermally inert, which follows from the fact that
under the support the difference between highest and lowest shell temperatures in the cycle
showed the largest value (50-75 °C); this difference almost vanished at the bottom.
3.2 Calculation of stresses and prediction of crack formation and spalling
Calculation of stresses developing in a brick lining with expansion allowances in two
directions is only possible by means of a three dimensional (3-D) analysis. Due to the fact that
the simulation of non-linear material behaviour demands too much calculation time the 3-D
modelling of the whole ladle structure seems not to be realistic. Therefore a unit-cell
modelling concept was applied for this study. It allows to reduce the complex structure of the
cylindrical part of the vessel to a much smaller domain which is still representative for the
whole structure. This model (fig. 6) features a quarter of a refractory brick, layers of the
refractory mix and the steel shell. At the planes of symmetry shown in fig. 6 the model is
constrained perpendicular to their direction. The mix and the shell are also constrained on
both vertical faces. Two special unbreakable blocks are introduced to model the expansion
allowances; they play the role of neighbouring bricks. Varying the distance between the
blocks and the brick allows modelling joints of different size. Cubic elements with an edge
length of 7x7x7 mm have been used.
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Fig. 6 The unit-cell model.
The expansion allowances are important for reducing compressive stresses. They allow
unconstrained expansion of the brick at the beginning of heating up and compressive stresses
start evolving only when bricks come into the contact to each other. Different ratios between
allowances in axial and circumferential directions may provide different stress and crack
patterns. The stresses and the failure by crack formation due to the heating-up procedure and
the thermal shock caused by pouring the liquid steel into the ladle have been investigated for
models with expansion allowances of different sizes. Fig. 7 shows stresses and crack patterns
for different sizes of the axial (horizontal) and circumferential (vertical) joints.
In spite of the fact that only the size of the axial joint was varied all stresses are different in all
cases. For an axial joint of 0,25 and 0,5 mm (0,2 and 0,4 % of the brick height, resp.) the
closure of the joint can be seen from the diagrams of fig. 7 by the arising compressive stresses
in vertical direction (σzz). The differences of the principal stresses in circumferential direction
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Fig. 7 Time-stress profiles and resulting crack patterns for models with different sizes of
expansion allowances (Aj – axial joint, Cj – circumferential joint; values in mm). The symbol
x in the diagrams shows the time for which the crack pattern is calculated.
(σxx) in these models result from the fact that growing axial constraints provide deformation
in orthogonal (circumferential) direction, which in turn caurses earlier closure of the
expansion joint and larger circumferential strains. Therefore a smaller axial expansion
allowance causes higher stresses in circumferential direction. The model with the smaller joint
of 0,25 mm predicted a crack formation that leads to spalling of an up to 50 mm thick part of
the brick. The model with the joint of 0,5 mm showed no spalling, but some crack initiation.
The model with the thickest axial joint (0,8 mm or 0,64 % of the brick height) shows lowest
stresses. In this model the circumferential joint closes, unlike other cases of fig. 7, before the
axial joint. Due to the temperature gradient in radial direction, this closure occurs only in the
vicinity of the hot face, where it can cause disintegration and spalling of the brick.
Almost all models predicted crack formation in the mix, the cracks were initiated on the outer
(cold end) surface of the mortar layer and propagated in radial direction.
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3.3 Influence of lining expansion on the shell deformation
The influence of the expansion of the lining on the deformations of the steel shell becomes
evident when time-displacement curves obtained for a shell node for the models discussed
above are compared with volumetric stresses of the bricks (fig. 8). The stresses shown in the
diagram of fig. 8 are valid fir the element 1610 shown in fig. 7.
Fig. 8 Influence of lining expansion on the shell deformation.
The volumetric stress is composed of the three principal stresses σxx, σyy and σzz, in this way it
is influenced by the action of both joints. In the beginning of the heating up the displacement
curves for different expansion allowances coincide, but with the increase of temperature they
start to diverge due to differences in stress fields caused by closing of the joints. The intensity
of the shell deformation is greater in models with smaller allowances and therefore greater
volumetric stresses.
An attempt was undertaken to establish which of the joints is more significant for decreasing
the shell deformation. For this purpose the first derivative of the radial stress in the shell σxx, s
with respect to the relative thickness of the allowance δj, r was calculated: A=dσxx, s/dδj,r
[MPa]. Here δj, r is the ratio of the joint thickness to the brick dimension perpendicular to it.
The calculation is performed for the end of the period when the liquid steel is being casted
into the ladle. The following results have been obtained.
Table 1.
Influence of the expansion allowances on the stress in the shell in circumferential direction
Ratio A, MPa
4,73.E+04
8,43.E+04
Axial joint
Circumferential joint
The table shows that the increase of the relative thickness of the circumferential joint
produces a greater reduction of the shell stresses than that of the axial joint.
SUMMARRY AND CONCLUSIONS
The calculations showed to be a suitable approach to answer the following questions related to
ladle operation and lining design:
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-
Which expansion allowance can be recommended with respect to brick failure and
shall deformation?
- What are the optimal thermal regimes to avoid spalling as far as possible?
The answers to these questions are influenced by material selection. The calculation procedure
presented here should be a more physically justified alternative to in most cases still applied
phenomenological design criteria for refractory linings.
ACKNOWLEDGEMENT
The presented investigations have been financed by the Christian Doppler Society (Vienna,
Austria). They were part of cooperation with VOEST Alpine Stahl Linz GesmbH (Linz,
Austria). This support is thankfully acknowledged.
REFERENCES
1.
H. Harmuth, Fracture mechanical characterisation of concrete and refractories –
development and application of a new approach based on the strain softening behaviour.
World cement research and development, 10, 71-78 (1995)
2.
Ch. A. Schacht, Thermomechanical behaviour of refractories. Key Engineering
Materials, 88, 193-218 (1993)
3.
DIANA-User’s Manuel release 7. TNO Building and Construction Research, third
edition, 1999
4.
D.C. Drucker, W.Prager, Soil mechanics and plastic analysis or limit design. Q. Applied
Math, 10 (2), 157-165 (1952)
5.
A. Hillerborg, Analysis of one single crack. In: F.H. Wittmann (Ed.), Fracture
Mechanics of Concrete, Elsevier Science Publishers, 223-249 (1983)
6.
S. Schrempf, H. Harmuth, Bestimmung der elastischen Konstanten von
Feuerfestmaterialien. In: H.Harmuth, H.Sandtner (Eds.), Gesteinshüttenkolloqium 2000,
Leoben/Austria 2001, 91-97
7.
H. Harmuth, E.K. Tschegg, A fracture mechanics approach for the development of
refractory materials with reduced brittleness. Fatigue Fract. Engng Mater. Struct., 20
(11), 1585-1603 (1997)
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