273_kir
Numerical simulation of combined effects of heat treatment and
crack length on tensile panels integrity
M. Kiric1, A. Sedmak2, J. Lozanovic1
1
The Innovation Centre of the Faculty of Mechanical Engineering
2
The Faculty of Mechanical Engineering
Kraljice Marije 16, Belgrade
[email protected]
ABSTRACT
This paper describes numerical modelling of the post weld heat treatment (PWHT) effects as well crack length influence
on integrity of modelled specimen subjected to tension load. The two-parameters fracture mechanics approach is applied
to the estimation of integrity of the welded structure with a flaw in the field of residual stresses using a software FADrs
made for the calculation. The FAD diagram constructed includes the R-6 curve and also boundary curves determined
using J integral for three values of deformation hardening exponent n=5, 10 and 20 according to the EPRI engineering
approach. Their comparison with SINTAP procedure is given, too. The results of integrity analysis according to the
PD 6493, EPRI and SINTAP approaches show significant effects of the PWHT and the crack length.
1. Introduction
One-parameter linear fracture mechanics is not applicable to flawed structures with large scale yielding at the crack tip, as
it is the crack in the field of large welding stresses, because they may achieve yield strength of the material. In order to
include real situations which are mostly between two limits – the brittle fracture and the plastic colapse, it is applied twoparameters approach of fracture mechanics and the failure assessment diagram (FAD). The failure assessment curve
(FAC), defined by the functional dependence of the relative stress intensity factor (SIF), Kr, on the relative stress Sr, is the
boundary line between potentially unsafe region above the curve and safe region below the curve in which fracture is not
expected [1,2].
Thermal and residual stresses (RS) known as secondary stresses or loadings, can arise from several causes and can be
defined by a set of imposed displacements or by a set of strains imposed throughout the volume of the structure. In many
cases the loading from secondary stresses is a combination of both these categories. RS are not significant if the section
contains no cracks, but the presence of a crack in the same cross section requires that the analysis takes the loading of
this component into the account at the assessment of the integrity.
The differences between strengths of the parent material (BM) and weld metal (WM) produce significant effects on failure
of flawed welded joint [2-4]. It is shown in [5] that crack driving force calculated using King’s model is sensitive to the
difference. This missmatching is not significant if strains are elastic. However, if the strains are plastic, the crack tip
loading is additionally influenced by the deformation properties of the material as given by its stress-strain curve.
Many papers are devoted to the treatment of residual stresses in the defect assessment of welded structures, especially
on ECF 16 in 2006. In this paper it is considered the integrity of cracked panels with weld after heat treatment for relief of
residual stresses (PWHT) using FAD diagram. The analysis uses boundary lines for the failure assessment based on
different approaches, such as R–6, the J-controlled growth and SINTAP.
2. FAD and secondary stresses
The failure assessment procedure of the CEGB and SINTAP assume that secondary stresses do not influence fracture
when the structure fails by plastic colapse, thus they are not taken into account in determining Sr nor the limit load, or the
degree of ligament plasticity Lr, but are taken into account in determining Kr. The total SIF is obtained by superposition of
the contributions of primary and secondary stresses, K
p
I
and
K s respectively, around the crack tip [3,4,6,7]
I
Kr =
KI
K Ic
K
=
p
I
+Ks
I
(1)
K Ic
In structures in the as-welded condition and if the actual residual stress distribution is not known, it may be assumed that it
is uniform and of magnitude equal to the appropriate material yield strength σy for levels 1 and 2, [7]. RS assumed in the
analysis for level 2, applied in this work, can be reduced to the lower of
σ ⎞
⎛
σ or ⎜1,4 − n ⎟ σ
⎜
Y
σ ⎟ Y
⎝
⎠
where
σn
is the effective net section stress and
(2)
σ
is the flow stress. When a structure is loaded by primary stresses, a part of
RS is relieved by plastic strain. A simple model of this mechanical stress relief is to assume that the sum of the primary and RS
cannot exceed the flow stress. For yield magnitude of RS in the unloaded state, the PD 6493 permits the use of the equ. (2) to
compute
s
K . When secondary stresses are present, PD 6493 introduces a correction term for level 2 and 3 assessments to
I
allow for plasticity interactions of the primary and secondary stresses by introducing the plasticity correction factor
K
Kr =
I
K
+ρ
ρ:
(3)
Ic
The level of RS remaining after the PWHT for the lower alloy steel are 30% of the room temperature WM yield strength
when stresses are parallel to the weld, i.e. if the crack is transverse to the weld. Transverse weld flaws are normally
located within the zone of tensile stress, whose width is greater than the weld width. Even for flaws whose tips are located
in the zone of compressive stresses, the net effect of the stress distribution is to produce a positive SIF.
For the level 1 and 2 estimation, the parameter Sr is given by
Sr = σ / σ
(4)
n
If flow stress is above 1,2 σy, it is assumed σ =1,2 σy in the calculation of Sr. For the level 1, Sr is limited to 0,8. Sr is
proportional to the applied load for the constant crack size, but Kr given by equ. (1) is proportional to load only if the
second term on the right side of equ. (1), corresponding to secondary stress, can be neglected, or if it is proportional to the
load, too. Thus, the points (Sr, Kr) for increasing load and the constant crack size in general case cannot be displaced
along the ray through the origin, as it is noted in [6].
For a through thickness crack the conservative estimate of applied SIF is calculated using the equation
KI = σ
where
πa
1
(5)
σ
1 is the maximum value of tensile stress in cross section and 2a is the crack length. For a surface crack of depth
c, the Newman and Raju stress intensity solutions are used
K
p
=
(P + H
I
m
s
+H
K = (S
I
m
P ) πc Q F
(6a)
NR b
S ) πc Q F
NR b
(6b)
where Pm and Pb are primary membrane and bending stresses, Sm and Sb are the secondary stresses and F and HNR are
constants obtained by approximating 3D FE calculation results [3].
The Duhamel-Neumann analogy, as it is known, states that initial – thermal or residual strains can be simulated by an
appropriate distribution of surface tractions and body forces.
As a consequence, Heaton [8] and Chell [9] have shown that the change in strain energy resulting from the introduction of a
crack is independent of the origin of the stress i.e. whether it originates from mechanical loading, imposed displacement, body
forces, initial strains etc. Analogously, SIF for an arbitrary stress distribution, depends only on the form of stress distribution
and not on whether it results from mechanical and/or thermal and residual stresses.
3. The calculation of FAD
Tensile specimens of size 300 x 80 x 20 mm are welded by the SAW process. Cracks of length 2a=24 mm and depth
c=9 mm, are made by the EDM machining with tips located in the heat affected zone (HAZ). The influence of PWHT on
RS is estimated for the centre cracked specimen (MT panel) made of the HSLA steel T.StE 460 with σy=460 MPa,
σ =542,5 MPa, ε0=0,002 , α=1,12, [10]. The differences in yield strength of the BM and WM are neglected here. It is
assumed according to PD 6493, that RS=σy in the as-welded condition and RS=0,3 σy after the PWHT, as more conservative
criterion than the equ. (2). It is applied the correction for finite specimen width, [3,6]
K =σ
I
[
π a sec(π RAW / 2 )
1
] (1 − 0,025RAW 2 + 0,06 RAW 4 )
1/ 2
(7)
where RAW is the ratio of crack length and specimen width W=80 mm.
in the specimen. Srt values are
It is introduced an external force P producing a remote membrane tensile stress σ
calculated for given P-values using the formula, derived from the equ. (4), which takes into the account the ratio RAW:
P
σP
t
Sr =
(8)
σ (1 − RAW )
Since the derivation of the FAC postulates the small scale yielding (where RS have a significant influence on fracture), it is
also used FAD based on J-controlled crack growth with the ratio
S
EPRI
r
=
P
P
(9)
0
where P and P0 are forces per unit thickness and P0 is the limit load
P0 = 4dσ 0 / 3
(10)
and Kr dependent on J integral elastic solution, Je, and on J integral as a function of the stable crack extension JR( ∆ a), [4,6]:
K =
r
J =
r
J ( a , P ) J ( ∆a )
e
(11)
R
The Fig. 1 gives FAD diagram calculated using the computer program FADrs for eight values of P and two levels of RS for
the surface crack. Solid circles represent points for residual stress level RS=σy appended to eight P-values given in Fig. 1,
so that the first point on the left hand side corresponds to the lowest load (0,5 MN) and the last point on the right hand side
corresponds to the maximum load (0,85 MN). Corresponding linear regression is given in Fig. 1 (the correlation coefficient is
R=1 and the standard deviation is D=3,89427 10-6) as well as FAC based on J-controlled crack growth for n=5, 10 and 20:
1.1
ZN=σ :
1.0
0.9
0.8
0.7
0.5
K
r
0.6
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
y
Y=0,29976+0,30579 X
R=1 SD=3,89427E-6
R - 6 kriva
EPRI, n=5
EPRI, n=10
EPRI, n=20
P, MN ZN=σ ZN=0,3σ
y
ZN=0,3 σ :
y
y
Y =0,08992+0,3058 X
R=1 SD=3,142E-6
0,50
0,55
0,60
0,65
0,70
0.75
0,80
0,85
CET=0,45 RAW=0,3 RCA=0,75
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
S
r
Figure 1. The influence of PWHT and the tensile load P on stability of the specimen
with a surface crack (CET=c/d, RAW=2a/W, RCA=c/a)
Representative points for the state after the PWHT with RS=0,3 σy are plotted for eight values of P, too, so that when P
increases, points move along a straight from the left hand side to the right hand side of the diagram. These points labeled
with open circles, define the linear regression which is also given in Fig. 1 and is characterized with R=1, too.
Upper linear regression, the dashed line, gives Kr for the crack without the PWHT. The critical P-value, i.e. the cross
section of the linear regression line with the R-6 curve determines the value P1=735 kN, while the plane stress EPRI
boundary curve for n=10 (the n value for the steel is about 10), determines critical value P2=775 kN. The difference is
40 kN. For the state after PWHT the critical P-value determined by the R-6 curve is equal to P3=750 kN and by the EPRI
curve for n=10 it is P4=863 kN. The effect of the PWHT can be estimated by the difference in P-values: for the R-6 curve it
is P3 – P1 =15 kPa, while for n=10, it is P4 – P2 = 88 kPa.
1.2
ZN=σ
1.1
Y =0,47656+0,34869 X
R=0,99998 SD=1,91337E-4
y
1.0
0.9
0.8
0.7
K
r
0.6
0.5
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
R - 6 kriva
EPRI, n=5
EPRI, n=10
EPRI, n=20
P(MN) ZN=σ
ZN=0,3 σ
y
Y =0,13284+0,35374 X
R=1 SD=1,8303E-5
y
0,50
0,55
0,60
0,65
P(MN) ZN=0,3 σ
y
0,50
0,55
0,60
0,65
0.0
RAW=0,3
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
S
r
Figure 2. The influence of PWHT and the tensile load P on stability of the specimen
with the through thickness centre crack (RAW=0,3)
For comparison it is analysed the centre through-thickness crack of the same length and in the same specimen, Fig. 2. The
cross sections of the upper regression line for the as-welded state with the R-6 curve and the EPRI curve for n=10 determine
critical P-values of 530 kN and 550 kN respectively, while after the PWHT the critical values are 586 kN and 664 kN.
4. Failure analysis according to SINTAP
The values of the loading parameter Lr for the basic level of SINTAP
L =σ
r
/σ =σ /σ
ref
Y
n
(12)
Y
and the values of the correction function f(Lr)
⎛ 1 ⎞
f ( L ) = ⎜ 1 + L2 ⎟
⎝ 2 r⎠
r
−1/ 2
⎡
− µ L6 ⎤
r ⎥
⎢ 0 , 3 + 0, 7 e
⎢⎣
⎥⎦
( N − 1) / 2 N
r
0< L < L
r
r
(13a)
r
max
f ( L ) = f ( L = 1) L
r
0≤ L ≤1
(13b)
r
are calculated for the BM and for the P-values given in Fig. 1 with
N = 0,3(1 − σ / R )
Y
and
=
L
r max
m
1 σ Y + Rm
2 σ
(14)
Y
For the through crack of length 2a = 24 mm, it is calculated N=0,0792 and Lr max=1,179. The results are illustrated in Fig. 3
for as welded condition (solid circles) with RS=σy and the state after the PWHT (open circles) with RS=0,3 σy. Points
shown in diagram correspond to the same P-values as given in Fig. 1.
f(L )
r
1,1
1,0
0,9
0,8
0,7
f(Lr)
RS=0,3 Sy
RS=Sy
0,6
0,5
0,4
0,3
0,2
0,1
0,0
0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
L = σ /σ
r
ref
y
Figure 3. FAD analysis of the tensile specimen containing the through thickness crack of length 2a=24 mm
The critical P-value for as welded state is P=645 kN and for the condition after the PWHT it is P=695 kN.
5. The influence of crack length and yield strength
If the crack increases at constant load, Sr is reciprocal to the effective net cross section and depends linearly on crack
area only if the crack area is small compared to the total cross section. Sr and Kr are not proportional to the crack size at
the same time, since KI given by equ. (5), is a nonlinear function of crack size. Thus, in general case, the path to failure for
increasing crack size is not a straight line and cannot be the same as the path for increasing load for a crack of fixed
length, as it is noted in [2].
The analysis is here illustrated by Fig. 4 (left). The diagram given left compares paths in FAD for given four tensile load
1/2
values and for the same through crack in the same BM (KJc=220 MPa m is an experimental value for the BM) and in the
WM ( σ =650 MPa). RS are neglected here. For BM plastic colapse is at 600 kPa when using R-6 curve, while the
specimen made of WM would be safe for all given P-values. Linear regressions are given in the diagram.
1.1
1.1
RAW = 0,3
1.0
0.9
K
r
0.6
0.5
0.4
0.3
0.2
Y =-5,58226E-4+0,35322 X
0.7
0.6
r
0.7
0.8
K
0.8
1.0
0.9
R-6
EPRI, n=5
EPRI, n=10
EPRI, n=20
P=0,50 MN (BM)
0,55
0,60
0,65
P=0,50 MN (WM)
0,55
0,60
0,65
Y =0,00857+0,29114 X
0.5
0.4
0.3
0.2
0.1
R-6
RAW=0,1 (BM)
0,20
0,30
0,40
0,50
BM
RAW=0,1 (WM)
0,20
0,30
0,40
0,50
WM
EPRI, n=5
EPRI, n=10
0.1
0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
S
0.0
0.0
EPRI, n=20
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
S
r
r
Figure 4. The plane stress paths to failure for the BM and WM for the centre cracked specimen
when changes the load (left) and the crack size (right)
The diagram on the right hand side of Fig. 4 shows plane stress paths to failure for BM and WM drawn for five through
thickness crack sizes given as RAW values. For the BM the critical crack size for plastic colapse is RAW=0,5 (2a=40 mm).
The specimen made of WM is safe for all given crack sizes. The curves are calculated by the condition that the ratio of
applied load and the load at fracture is
σ
P
σ P /σ =1/ 4 and
that the applied stress and crack size fulfil the condition
π W / K = 3/ 4.
Ic
6. Conclusion
From given results, it is clear that the presence of the residual stress significantly reduces the load bearing capacity of
cracked welded structures. The FAD concept enables the analysis of crack size and thermal stresses influence on integrity
as well a sensitivity analysis with respect to fracture toughness and tensile properties. The analysis using boundary Jcurves enables to take into the account the hardening exponent, while both, SINTAP and FAD approach based on Jcontrolled growth, take advantage of the material’s full load carrying capabilities. The assessment of integrity by using the
approach based on J-controlled growth is more conservative than SINTAP, except possibly, when materials hardening is
pronounced.
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