An integral equation is presented that relates time to equivalent stresses at the interface of two regions in a weldment operating within the creep regime. Taking the creep constitutive equation as the Norton power law, the paper investigates the effects of the stress index and the stress coefficient on the equivalent stresses at the critical interfaces in the weldment

Zarrabi, K.

University of Queensland eSpace

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Effects Of Material Parameters On The Stresses In High Temperature 
Weldments  
K. Zarrabi 
School of Mechanical and Manufacturing Engineering 
The University of New South Wales 
NSW 2052, AUSTRALIA 
 
 
 
ABSTRACT: An integral equation is presented that relates time to equivalent stresses at the interface of two 
regions in a weldment operating within the creep regime. Taking the creep constitutive equation as the 
Norton power law, the paper investigates the effects of the stress index and the stress coefficient on the 
equivalent stresses at the critical interfaces in the weldment. 
 
INTRODUCTION 
Weldments are by far the most common location for creep failures in the thick-walled components 
employed in various industries. In spite of substantial advancement in modelling and understating 
the behaviour of the weldments operating within creep regime, currently there are no generally 
accepted and efficient models to prevent their premature failures. The use of ever more powerful 
computers to perform time dependent finite element analysis (FEA) is now permitting predictions 
of stresses in a weldment more accurately than before [1,2]. However the creep FEA is usually 
expensive and time-consuming. This study describes the development of an integral equation for 
predicting the variation of equivalent stresses at the critical interface of two zones within a high 
temperature weldment; see below. 
 
ANALYSIS 
A weldment typically consists of three zones, viz., the weld material (WM), the heat-affected zone 
(HAZ) and the parent material (PM). There are many factors that make the prediction of the 
behaviour of the weldments at elevated temperatures difficult. For example, when a welded joint is 
subjected to elevated temperature and sustained loading, WM, HAZ and PM each exhibit different 
creep characteristics with time. As a result, complex stress patterns that are time-dependent will set 
up at the interface between these materials, contributing to creep damage, cracking and failure of 
the weldment. Although the literature on welded joints is voluminous [1,2], the published results 
are mainly based on numerical and/or experimental investigations. To the author’s knowledge, the 
analytical and/or semi-analytical solutions that reduce the cost of the analysis and give better 
understanding of variation of stresses with time in a weldment are scarce. The present study 
concentrates on the effects of material mismatch between WM, HAZ and PM on the stresses in the 
weldment only and describes a simple and semi-analytical model for predicting stresses in a butt-
welded plate. 
 Koundy et al [3] have experimentally shown that the equivalent von Mises stress is the main 
stress component that causes creep failure in butt-welded joints. Therefore, the present study 
concentrates in predicting the equivalent von Mises stresses in such weldments. As it was explained 
above, the present study concentrates on material discontinuities at a butt-welded joint and ignores 
the geometrical discontinuities with the objectives to ascertain the significance of various material 
parameters on the stresses. Consider a plate that contains a butt-welded joint and is subjected to a 
uniform and constant traction as shown in Fig. 1. The assumptions used in the analysis are. the 
plate, weld and HAZ have a constant and uniform thickness permitting elimination of the stress 
concentration due to variation in the weld thickness; the plate is subjected to a constant temperature 
(T ) and a constant and uniform traction ( 0σ ) where 0σ  is applied: (a) in the x  – direction, (b) in the 
SIF2004 Structural Integrity and Fracture. http://eprint.uq.edu.au/archive/00000836 
y  – direction, (c) in both the x  and y  - directions simultaneously; the analysis ignores residual 
stresses [3]; the small displacement and strain conditions are assumed to prevail. This may be 
justified as in practice the weldment cracking usually occurs when the material experiences 
relatively small values of strain. 
 Although the HAZ is depicted as a uniform material in Fig. 1, as will become clear shortly, the 
proposed model allows for various grades of material within the HAZ. Let us consider two adjacent 
points at a critical material interface away from the plate boundaries. Note that at the plate 
boundaries the stresses are equal to applied tractions for all values of time ( t ) to satisfy the force 
equilibrium conditions. These two points can, for example, be at the interface of the weld and the 
HAZ, or alternatively at the interface of two different grades of the HAZ, etc. In all cases, these two 
points must refer to the interface region of interest in the weldment. To satisfy the equilibrium 
conditions, initially, at 0=t , the stresses acting on these points must be equal, noting that the loading 
is assumed to be uniform and constant. With time, because of different creep rates, the stresses are 
redistributed and stresses acting at point 1 will differ from those acting at point 2. Fig. 2 shows two 
differential elements of dimensions dx  by dy  by dz  (where z  defines the direction perpendicular to 
the plane of the plate) corresponding with points 1 and 2 at 0=t . One point is inside material 1 (e.g., 
the weld) and the other inside material 2 (e.g., the HAZ). Fig. 2 also shows the acting equivalent 
von Mises stress (which is equal to the applied traction, 0σ ) on points 1 and 2 at 0=t . After a 
period of time, t , the stresses between points 1 and 2 will be redistributed so that the equivalent von 
Mises stress acting at point 1 is 1σ  and that acting at point 2 is 2σ . Normally, 0σ  (the applied 
traction) is known. The objective of the proposed model is to develop expressions to predict 1σ  and 
2σ  and their variations with time. Note that the applied equivalent von Mises stress is 0σ  for both 
the uniaxial and biaxial loading. Now, because of equilibrium the total forces acting on points 1 and 
2 at 0=t  must be equal to those at time t  such that in terms of the stresses depicted in Fig. 2, we 
have: 
021 )( σσσ dzdxdzdxdzdxdzdx +=+  
or, 
021 2σσσ =+           (1) 
Due to compatibility: 
dt
d
dt
d 21 εε =           (2) 
where ε  is the equivalent von Mises total strain at time t  and subscripts 1 and 2 refer to points 1 
and 2. In the absence of plastic deformation, the total strain rate is assumed to be the sum of the 
elastic (superscript e ) and creep (superscript c ) strain rates, viz.: 
dt
d
dt
d
dt
d ce 111 εεε +=           (3) 
dt
d
dt
d
dt
d ce 222 εεε +=           (4) 
with: dt
d
Edt
d e 1
1
1 1 σε =          (5) 
 and dt
d
Edt
d e 2
2
2 1 σε =          (6) 
),,( 111 Ttfdt
d c σε =           (7) 
),,( 222 Ttfdt
d c σε =          (8) 
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where 1E  is the modulus of elasticity of material 1, 2E  is the modulus of elasticity of material 2, 1f  
and 2f  are the functions that describe the creep constitutive relationships for materials 1 and 2 
respectively. 
 Combining and integrating equations (1) to (8) and eliminating 1σ , gives: 
0),,( 02 =σσ tF           (9) 
Eliminating 2σ  between equations (1) and (9) will give 1σ . Note that the specific expression for F  
depends on the specific expressions for 1f  and 2f . For example, for the creep power law where 
1
111
nBf σ=  and 2222 nBf σ=  with 1B  and 1n  are the creep material constant and the creep stress 
index of material 1 respectively and 2B  and 2n  are the creep material constant and the creep stress 
index of material 2 respectively, equations (4) and (5) will change to: 
1
11
1
1
1 1 nB
dt
d
Edt
d σσε +=          (10) 
2
22
22
2
1 nB
dt
d
Edt
d σσε +=          (11) 
Now, by eliminating 1σ  between equations (1) and (10) and then combining the results with 
equations (2) and (11), one will obtain: 
21
22
2
2
201
20
1
1)2(
)2(1 nn B
dt
d
E
B
dt
d
E
σσσσσσ +=−+−  
which may be rearranged to: 
])2([ 21
21
21
22201)(
2
nn
EE
EE BB
ddt
σσσ
σ
−−
=
+
      (12) 
Integrating equation (12) gives: 
∫ −−= +2
0
21
21
21 ])2([ '2
'
01)(
'
σ
σ
σσσ
σ
nn
EE
EE BB
dt     for both the uniaxial and biaxial loading  (13) 
where 'σ  is the integral variable. Having determined 2σ  using equation (13), then the variation of 
1σ  with time is followed from equation (1). Note that equation (13) holds for 0>t . It is obvious that 
at 0=t , 021 σσσ ==  for all load cases (a) to (c). 
 
VALIDATION AND RESULTS 
The following finite element modelling was carried out to verify equations (1) and (13). Finite 
element models were generated using the butt-welded plate shown in Fig. 1. The plate was square 
with a dimension of 150 mm by 150 mm and it had a constant thickness of 20 mm. The weld width 
was 30 mm and the width of the HAZ was 10 mm. Because of symmetry only ¼ of the plate was 
meshed using 792 isoparametric 4-node elements as shown in Fig. 3. Three loading cases were 
modelled, viz., (a) a uniform traction MPa1300 =σ  applied in the x  – direction, (b) a uniform 
traction MPa1300 =σ  applied in the y  – direction, (c) a uniform traction MPa1300 =σ  applied in both 
the x  and y  - directions simultaneously. The plate was made of 18%Cr 11%Ni steel and it was 
subjected to a uniform temperature of 550 o C. The material properties used for the finite element 
analyses were [4]: ,000,15721 MPaEEE PM ===  HrPM xB 132101.1 −= , HrxB 1371 103.2 −= , HrxB 1412 107.1 −= , 
12=PMn , 74.141 =n  and 53.152 =n . The finite element analyses were carried out using 
MSC/NASTRAN [5] and a personal computer Pentium III. The creep finite element analyses were 
performed assuming the strain-hardening model. Also, to be consistent with the proposed model, 
the small displacement and strain conditions were used in the finite element analyses. 
SIF2004 Structural Integrity and Fracture. http://eprint.uq.edu.au/archive/00000836 
To verify the mesh, first a creep finite element analysis was performed assuming that the entire 
plate was made of PM only and subjected to the uniaxial stress MPa1300 =σ . The computed stresses 
were everywhere MPa130  indicating that the mesh was sufficiently fine. Next the creep finite 
element analyses of the butt-welded plate were carried out for load cases (a) to (c). As might be 
expected, the equivalent von Mises stresses at points 1 and 2 were the same for all load cases (a), 
(b) and (c). In all load cases, the maximum von Mises stress was at interface between the weld and 
HAZ and just in the HAZ, i.e., at point 2 (Fig. 4). Fig. 4 shows the comparison of the equivalent 
von Mises stresses at points 1 and 2 for load cases (a), (b) and (c) predicted using equations (1) and 
(13) with those computed using the finite element analyses. The differences were negligible and the 
results depicted in Fig. 4 show that the proposed model can accurately predict the equivalent von 
Mises stresses at the interface of two different materials such as the weld and HAZ. Also, Fig. 4 
shows that the stresses redistribute from the creep soft material (in this case the weld) to creep hard 
material (in this case the HAZ). 
 In above, it was shown that equations (1) and (13) are valid. Next these equations will be used to 
ascertain the effects of the pertinent materials properties on 1σ  (the equivalent von Mises stress at 
point 1, e.g., at the interface between the weld and HAZ and just inside the weld) and 2σ (the 
equivalent von Mises stress at point 2, e.g., at the interface between the weld and HAZ and just 
inside HAZ). To do so, we need to know the explicit creep constitutive equations. For the creep 
power-law relationships, from equation (13) it is clear that the pertinent material properties are: 
212121 ,,,,, EEBBnn . Considering the denominator of the right-hand side of equation (13), it is 
apparent that the first term is positive and the second term is negative noting that 021 2,0 σσσ ≤≤ . 
Therefore, as 1n  or 1B  (e.g., the creep material parameters for the weld) increases, 2σ  increases and 
hence 1σ  decreases (see equation (1)). The reverse is true when 1n  or 1B  decreases. This is 
demonstrated in Figs 5 and 7. Also, as 2n  or 2B  increases, 2σ  decreases and hence 1σ  increases. 
The reverse is true when 2n  or 2B  decreases. This is demonstrated in Figs 6 and 8. 1E  and 2E  are 
multipliers for the first term in the denominator of equation (13), therefore, their increases cause 2σ  
to increase and hence 1σ  to decrease. The reverse is true when 1E  or 2E  decreases. Finally, since the 
applied traction ( 0σ ) is constant and 021 2σσσ =+  (see equation (1)), then an increase in 1σ  causes a 
reduction in 2σ  and vice versa. 
  
CONCLUSIONS 
 A semi-analytical model was developed for predicting the equivalent von Mises stresses at the 
interface of two materials (e.g., the weld and HAZ or two grades of HAZ, or HAZ and parent 
material, etc) in a plate that contains a butt-weld and subject to a uniform and constant temperature 
and traction. It was shown that the explicit relationships between the stresses, time and material 
properties depend on the creep law employed. For power-law creep, the pertinent material 
properties were: 212121 ,,,,, EEBBnn . It was shown that 1σ  (the equivalent von Mises stress at point 
1; this could be the weld stress if the interface between the weld and HAZ is considered for which 
subscript ‘1’ refers to the weld and subscript ‘2’ refers to HAZ) decreases with increases in 1n  or 1B  
or 1E  or 2E  but it increases with increases in 2n  or 2B  and vice versa. On the other hand, 2σ  (the 
equivalent von Mises stress at point 2 or in HAZ) increases with increases in 1n  or 1B  or 1E  or 2E  
but it decrease with increases in 2n  or 2B  and vice versa. Also, it was shown that as 1σ  increases, 
2σ  decreases and vice versa, i.e., the stresses are transferred from one grade of material to the 
adjacent grade material. 
 
SIF2004 Structural Integrity and Fracture. http://eprint.uq.edu.au/archive/00000836 
ACKNOWLEDGMENT 
 The Australian Research Council (ARC) financially supported this study. 
 
REFERENCES 
[1] Zarrabi K, Hosseini-Toudeshky H, and Singh D, "A robust life assessment method for elevated 
temperature components containing stress concentrators", Proceedings of Failure Analysis in Materials 
Engineering Conference, Published by Institute of Metals and Materials Australia Ltd (Melbourne, 
Australia), Melbourne, Australia, 22-23 March, 1994. 
[2] Hyde A, Tang W, “Analytical and computational stress analysis of welded components under creep 
conditions”, Integrity of High Temperature Welds, Institute of Mechanical Engineers, London, pp285-
308, 1998. 
[3] Koundy, V., Allais, L. and Delhaye, M., Creep Behavior of a Large Full-Size Welded Austenitic Steel 
Plate, Transactions of the ASME, Journal of Pressure Vessel Technology, Vol. 120, 1998, pp. 262-269. 
[4] Tu, S. T., Segle, P. and Gong, J. M., Strength Design and Life Assessment of Welded Structures 
Subjected to High Temperature Creep, International Journal of Pressure Vessels and Piping, Vol. 66, 
1996, pp. 171-186. 
[5] MSC/NASTRAN, Handbook for Non-linear Analysis, The MacNeal-Schwendler Corporation, Los 
Angeles, California, USA, 1985. 
SIF2004 Structural Integrity and Fracture. http://eprint.uq.edu.au/archive/00000836 
 
 
        Point 2  Parent Material 
 
 HAZ          Weld 
 
 
 
 
   0σ         0σ  
 
 
 
 
 Parent Material 
 
         y         Point 1 
 
 
           z                      x  
 
Fig. 1 - Uniformly loaded welded plate. 
 
 
 
 
       dx   dy  
 
 
 
 
Point 2 
 
 
          y  
 
 
Point 1 
 
   z          x  
 
 
Fig. 2 - Differential elements corresponding to points 1 and 2 at 0=t . 
0σ0σ  
0σ
0σ  
SIF2004 Structural Integrity and Fracture. http://eprint.uq.edu.au/archive/00000836 
 
 
      y  
 
 
 
 
 
 
      z        x      Point 2 
      
         Point 1 
 
 
 
 
 
 
Fig. 3 - Finite element mesh of welded plate. 
 
50
100
150
200
0 250 500 750 1000
Time (Hour)
vo
n 
M
is
es
 S
tr
es
s 
(M
Pa
)
Eq (13) - Point 2
F.E. - Point 2
Eqs (1) & (13) - Point 1
F.E. - Point 1
 
 
Fig. 4 - Comparison of stresses at Points 1 and 2 predicted by the proposed model and F.E. 
 
75
100
125
150
175
200
0 250 500 750 1000
Time (Hour)
Eq
ui
va
le
nt
 v
on
 M
is
es
 S
tr
es
s 
(M
Pa
)
Point 2 - n1=14.74
Point 2 - n1=15.50
Point 1 - n1=14.74
Point 1 - n1=15.50
 
 
Fig.5 - Variation of stresses at Points 1 and 2 with 1n . 
Y-Symmetry 
X-Symmetry 
Loaded 
Faces 
Weld 
HAZ 
PM
SIF2004 Structural Integrity and Fracture. http://eprint.uq.edu.au/archive/00000836 
 
75
100
125
150
175
200
0 250 500 750 1000
Time (Hour)
Eq
ui
va
le
nt
 v
on
 M
is
es
 S
tr
es
s 
(M
Pa
)
Point 2 - n2=15.53
Point 2 - n2=16.53
Point 1 - n2=15.53
Point 1 - n2=16.53
 
 
Fig. 6 - Variation of stresses at Points 1 and 2 with 2n . 
 
75
100
125
150
175
200
0 250 500 750 1000
Time (Hour)
Eq
ui
va
le
nt
 v
on
 M
is
es
 
St
re
ss
 (M
Pa
)
Point 2 - B1=2.3x10^ -37
Point 2 - B1=2.3x10 -36
Point 1 - B1=2.3x10^ -37
Point 1 - B1=2.3x10 -36
 
 
Fig. 7 - Variation of stresses at Points 1 and 2 with 1B . 
 
75
100
125
150
175
200
0 250 500 750 1000
Time (Hour)
Eq
ui
va
le
nt
 v
on
 M
is
es
 S
tr
es
s 
(M
Pa
)
Point 2 - B2=1.7x10^ -41
Point 2 - B2=1.7x10 -40
Point 1 - B2=1.7x10^ -41
Point 1 - B2=1.7x10 -40
 
 
Fig. 8 - Variation of stresses at Points 1 and 2 with 2B . 


Effects Of Material Parameters On The Stresses In High Temperature Weldments

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Effects Of Material Parameters On The Stresses In High Temperature Weldments

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