Residual welding stresses and distortions _____________________________________________________________________________________ When steel structures are constructed by welding, deformations and welding residual stresses could occur as a result of the high heat input and subsequent cooling (Boley & Weiner 1960). The welding process can create significant locked-in stresses and deformations in fabricated steel structures (Rykalin 1951). The residual stresses and initial imperfections can have an important influence upon the behaviour of the structure under the variable loading (Gurney 1979). It is well known that these initial imperfections due to welding reduce the ultimate strength of the structure. Even though various efforts have been made in the past to express the deflection of panels from experimental aspects and measurements of actual structures, it may be said that there are few investigations from the theoretical point of view. In order to find out a practical estimation method for the welding distortion of a panel, the following analyses have been carried out by Okerblom (1955). To determine the thermal effect on the structure it is advisable to investigate some simple structures (Chang Doo Jang & Seung Il Seo 1995). Finite element calculations can help to determine these residual stresses (Josefson 1993, Wikander et al. 1994). 1.1 Simple examples of thermoelasticity We assume the following: - the coefficient of thermal expansion and the Young modulus are independent from the temperature, - the deflections are in the elastic range, the Hooke-law is valid, - the cross sections of the beam will be planar after deflection, - the cross section is uniform, - the beam is made of one material grade, - the thermal distribution is uniform along the length of the beam and steady state. The change in length ( ) and deflection ( ) of a simply supported beam, as shown in Fig. 1.1 due to linear heat distribution is as follows: ΔL w max The strain in the centre of gravity is ε α α G o es o e e T T T h = = − ( / 1 1 ) Δ e . Here Δ is the temperature difference and T T e e e = - T 1 2 α o is the coefficient of thermal expansion. The change in length ( ΔL ) caused
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Residual welding stresses and distortions _____________________________________________________________________________________
When steel structures are constructed by welding, deformations and welding residual stresses could occur
as a result of the high heat input and subsequent cooling (Boley & Weiner 1960). The welding process
can create significant locked-in stresses and deformations in fabricated steel structures (Rykalin 1951).
The residual stresses and initial imperfections can have an important influence upon the behaviour of the
structure under the variable loading (Gurney 1979). It is well known that these initial imperfections due
to welding reduce the ultimate strength of the structure.
Even though various efforts have been made in the past to express the deflection of panels from
experimental aspects and measurements of actual structures, it may be said that there are few
investigations from the theoretical point of view. In order to find out a practical estimation method for the
welding distortion of a panel, the following analyses have been carried out by Okerblom (1955).
To determine the thermal effect on the structure it is advisable to investigate some simple structures
(Chang Doo Jang & Seung Il Seo 1995). Finite element calculations can help to determine these residual
stresses (Josefson 1993, Wikander et al. 1994).
1.1 Simple examples of thermoelasticity
We assume the following:
- the coefficient of thermal expansion and the Young modulus are independent from the
temperature,
- the deflections are in the elastic range, the Hooke-law is valid,
- the cross sections of the beam will be planar after deflection,
- the cross section is uniform,
- the beam is made of one material grade,
- the thermal distribution is uniform along the length of the beam and steady state.
The change in length ( ) and deflection ( ) of a simply supported beam, as shown in Fig. 1.1 due to
linear heat distribution is as follows:
ΔL wmax
The strain in the centre of gravity is ε α αG o es o e eT T T h= = −( /1 1 )Δ e . Here Δ is the
temperature difference and
T Te e e= - T 1 2
α o is the coefficient of thermal expansion. The change in length (ΔL ) caused
by heat expansion is ΔL G= Lε . The heat distribution is non-uniform at the cross section, so a curvature
occurs at the beam. The radius of curvature is ρ o .
Fig. 1.1 Deflections of a beam with linear temperature distribution
The curvature is C Tho
o e= =1ρ
α Δ. There is a relation between radius of curvature, bending moment and
bending stiffness as 1ρ o x
MEI
= . So the curvature can be considered as the effect of a uniform bending
moment across the length as M T EIh
o e x=α Δ .
The rotation of the angle is ϕ( )z MEI
L zx
= −⎛⎝⎜
⎞⎠⎟2
, so the maximum value is ϕ max =CL2
.
The deflection is (w z MEI
Lz zx
( ) = −2
2 ) , so the maximum value is w CLmax =
2
8.
Another problem is, when the thermal distribution is nonlinear as shown in Fig. 1.2. The thermal strain
would be different at different points of cross section if they were independent form each other:
ε α= o eT y( ) . Because they are connected to each other, we assume that the cross section remains planar,
only a linear strain can occur in the cross section. This linear strain is characterized by the strain of the
gravity centre and the curvature of the beam: ε ε= +G Cy . The differences between the theoretical
thermal strain and the linear strain cause the stresses:
σ ε ε α= = + −E E Cy T yG o e{ ( )} (1.1)
There is no external loading on the beam, so the internal stresses caused by thermal difference are in
equilibrium,
and (1.2) σdAA
=∫ 0 σydAA
=∫ 0
Fig. 1.2. Beam with nonlinear temperature distribution
By inserting Eq. (1.1) to (1.2), we get
ε αG o ee
e
AT y t y dy= ∫
1
1
2
( ) ( ) and CI
T y yt y dyx
o ee
e
= ∫1
1
2
α ( ) ( ) . (1.3)
If the thickness is constant, i.e. t(y)=t we can define the thermal shrinkage impulse AT as
(1.4) A T yT o ee
e
= ∫α ( )1
2
dy
The thermal impulsive moment is defined as follows
(1.5) A y T y ydyT T o ee
e
= ∫α ( )1
2
Using these definitions the strain at the center of gravity and the curvature are as follows
ε GTA tA
= (1.6)
CA t y
IT T
x
= (1.7)
1.2 The Okerblom’s analysis
When a structural section is welded, it undergoes distortion as a result of thermal shrinkage along the axis
of the weld. For example, an edge-welded bar section shortens (ΔL ) and deflects ( ). Experiments
indicate that the Okerblom’s analysis provides excellent prediction for longitudinal deflections caused by
thermal shrinkage along the weld (Okerblom 1955, Okerblom et al 1963).
wmax
The analytical heat-transfer theory of welding was developed by Rykalin (1951). Essentially, Okerblom
utilises the analytical heat-transfer theory of moving heat sources to establish the thermal strain and stress
distributions around the weld. The primary objective of the Okerblom’s analysis is to predict the beam
shrinkage (Δ ) and deflection (w) as shown in Fig. 1.1. The analysis is used to generate a series of
temperature isotherms that are stationary with respect to the heat source (Fig. 1.3).
L
Fig. 1.3 Strain distribution during and after welding
In the Okerblom’s analysis the material is linearly elastic and ideally plastic. The yield stress is constant
till 500 Co, and between 500 and 600 Co it decreases to zero. If the temperature is larger than 600 Co,
there is no measurable stress in material (Fig. 1.4).
Fig. 1.4 The yield stress in the function of the temperature and strain
Fig. 1.5 Distribution of thermal strains during and after welding
The approximation of the Te temperature suggested by Okerblom is as follows
TQ
c t yeT
o
=0 4840
2.ρ
(1.8)
where c0 is the specific heat, ρ is the material density, t is the thickness of the plate.
The thermal impulse can be calculated according to Fig. 1.5. The diagram determined by points 1-10
shows the stress distribution during welding. It can be obtained by projection of points B and C to the line
ε A which occurs due to elastic deformation of the structure during welding. Points B and C are
determined by the line 600o α , so between points 5 and 6 no stresses occur. Points 3-4 and 7-8 are
obtained by the line parallel to line ε A in a distance εy , with projection to this line the points D and E
determined by the line 500o α . It can be seen that plastic strains occur during welding between points 3
and 8. These retrained strains cause residual stresses after cooling.
The residual stress diagram after cooling can be obtained projecting points 3 and 8 onto the basic line 1’-
10’. Considering the elastic deformation during cooling and the line of ′ ≈ε εA A εy , one obtains the
residual stress diagram 1’-3’-F’-G’-8’-10’. The area 3’-F’-G’-8’ characterizes the thermal shrinkage
impulse AT which causes the residual stresses and deformations in the structure.
Since the line parts of 3’-F’ and 8’-G’ are the same as parts 3-F and 8-G, the AT can be calculated by
investigating the area 3-F-G-8 in the diagram drawn for the state during welding.
A b dTQ
c tdTTT o e
o T
o
e
eT
T T
A y
A y
e A y o
e e
= =+
+
= +
=
∫ ααρε ε
ε ε
ε ε α
2 20 4840
1
2 1( )
( )/
.∫ (1.9)
AQ
c tQ
c tTo T
o
o T
o
= =0 4840
20 3355.
ln.α
ραρ
(1.10)
where Q UIv
q AT ow
o w= =η , U arc voltage, I arc current, vw speed of welding, co specific heat, ηo
coefficient of efficiency, q0 is the specific heat for the unit welded joint area (1 mm2), Aw is the welded
joint area.
For a mild or low alloy steels, where α o =12*10-6 [1/Co], coρ = 4.77*10-3 [J/mm3/Co], the thermal
impulse is
. A t QT T [mm ] [J / mm]2 = −0844 10 3. *
Inserting this into Eqs. (1.6) and (1.7), we get the basic Okerblom formulae
ε GTA tA
QA
= = − −0 844 10 3. * T (1.11)
The minus sign means shrinkage.
CA t y
IQ y
IT T
x
T T
x
= = − −0844 10 3. * (1.12)
Note that the distorted form can be determined by view. yT and C have opposite signs (Fig. 1.15).
The elastic strain in the weld can be calculated using the previous two expressions
ε εA G Cy= + T (1.13)
The average width of the plastic tension zone around the weld is
bA
yT
A y
=+ε ε
(1.14)
At the region of weld the residual tensile stress after welding reaches the yield stress (Fig. 1.5). The area
of the plastic zone is
A b tA t
y yT
A y
= =+ε ε
(1.15)
By using Eqs. (1.6), (1.7) and (1.13) one obtains
1 1 2
A AyI Ay
T
x
y
T
= + +t
ε (1.16)
If no crookedness is developed in beam during welding, as for example in the case of a symmetrical weld
arrangement, Eq. (1.16) takes the form
1 1A A Ay
y
T
= +t
ε (1.17)
For steels
1 1 142
A AyI Qy
T
x T
= + +.3 [J, mm] (1.18)
If the structure can be regarded as a very stiff one, when ε A = 0 , area of plastic zone is
1A Ay
y
T
=t
ε (1.19)
For steels
A Qy
T=143.
(1.20)
The equilibrium equation for a section with tension and compression stresses is according to Fig. 1.6
( )b b b fy c y− = yσ (1.21)
Fig. 1.6 Approximate stress distribution for a plate with a single weld at the middle
Using Eq. (1.17) one can compute the residual compressive stress,
σε
α ηρc
T y
y
T o
o w
AA
A tA
EUIE
c v= = =
t f b t
0 3355. o (1.22)
With data , α o =−11 10 6* coρ = ⎡
⎣⎢⎤⎦⎥
−353 10 3. * Jmm C3 o , E = 2.05*105 [MPa], vw is the welding speed,
used by White, the Okerblom formula is
ση
co
w
UIv bt
=0 214.
, (1.23)
White (1977a,b) proposed an approximate formula based on own experiments
ση
co
w
UIv bt
=0 2.
, (1.24)
It can be seen that Okerblom’s formula is in agreement with White’s experimental results.
The formulae above are valid for symmetrically arranged welds yA = 0
QA
T ≤ ⎡⎣⎢
⎤⎦⎥
2 50. Jmm3 , (1.25)
for eccentric welds ( )yA ≠ 0
QA
T ≤ ⎡⎣⎢
⎤⎦⎥
0 63. Jmm3 . (1.26)
For approximate calculations one can use the simple formula
ε yyf
E= (1.27)
where fy is the yield stress of the parent material. The weld metal may have a different yield stress. This
discrepancy arises due to the electrode material. In this case it is important to measure the yield stress of
the weld metal. Using high strength steels, the yield stress of the weld metal can be smaller that of the
parent material. Therefore the residual stresses are relatively smaller, than in the case of mild steel.
For some simple cross sections the residual stress distribution can be seen on Fig. 1.7.
Fig. 1.7 Residual stresses in different cross sections and weld positions
1.3 Multi-pass welding
The basic Okerblom formulae are valid for single pass-welding. For multi-pass welding it is necessary to
modify Eq. (1.10.), because the new weld pass resolves the plastic zone, made by the previous weld pass.
For the value of residual stresses that weld pass is governing, which causes the largest plastic zone.
Introducing a parameter for the correction of thermal shrinkage impulse
A mQ
c tT to T
o
=0 3355. α
ρ (1.28)
where mAAt
yt
y
=1
, (1.29)
Ay1 , Ayt the areas of plastic zone due to single- and multi-pass welding.
For example at a two-pass (equal passes) butt joint mt = 1. For a double fillet weld for thin plates, where
the welds are welded one after the other mt = 12 13. - . . For intermittent fillet welds m LLt
w
u
= , where Lw
and Lu are the distances of the welded and unwelded part at intermittent fillet weld.
White (1977c) suggested to calculate the tendon force from the parameters of that pass, which has the
greatest section area.
The effect of preheating can be taken account with a correction parameter
FT
Fp' ( . )= −111000
(1.30)
where the temperature of preheat is T Co. p > 100
1.4 Effect of initial strains
In the previous calculations it was assumed, that there are no initial strains and stresses in the structure. In
practice there are usually some strains and stresses before welding, or previous welds cause initial strains
and stresses for the next weld(s). Preheating, flame cutting and pre-stressing have the same effect.
The strain diagram is similar to Fig. 1.5 except of the initial tensile strain ε I . Fig. 1.8 shows the strain
distribution during welding and after welding. The final deformations after welding are caused by the
difference of ε εy − I . The effective zone is between ABCD point. The thermal impulse can be computed
as follows:
′ = =+ =
=
∫A b dTQ
c tdTTT o e
o T
o
e
eT
T
I y
y
e I y o
e y
ααρε ε
ε
ε ε α
ε α2 20 4840
1
2 0.
( )/
/
+∫ (1.31)
′ =+
AQ
c tTo T
o
y
y I
0 4840 2.ln
αρ
εε ε
(1.32)
To consider the effect of initial deformation we introduce a modifying parameter ν m , which is the ratio of
the thermal impulse with and without initial strain.
ν
εε ε
εmT
T
I
y I
y
AA
=′
= −
+
≈ −11
21
ln( )
ln (1.33)
The approximate formulae is valid when εε
I
y
≥ 0.
Fig. 1.8 Distribution of thermal strains during and after welding due to initial strains
Fig. 1.9 shows ν m in the function of εε
I
y
. Without initial deformation no modification is necessary, so if
ε I = 0, then ν m =1. If there is a tension in the elastic zone, 0 < <ε εI y 0, then 1> >ν m . If the initial
strain is equal to the yield strain, ε εI y= , ν m = 0 , there is no residual stresses and deformations after
welding. If the initial strain is negative (compression), ε I < 0, ν m > 1, this strain increases the
deformation, but the approximate formula can not be used.
Fig. 1.9 Modifying parameter in the function of initial strain
The thermal shrinkage impulse is according to Fig. 1.8
Ab
Ty
y I
''
=−ε ε
(1.34)
the area of plastic zone is according to Fig. 1.10
A b tA t A t
y yT
y I
m T
y I
' ''
= =−
=−ε ε
νε ε
(1.35)
if ε i > 0 then AA t
yT
y
' =ε
and for a normal grade steel AQ
yT'
.=
143 (J, mm).
Fig. 1.10 The area of plastic zone
By using the modifying parameter it is possible to deternine the right welding sequence. If there are more
welds on a structure, the welding sequence can be very important, its effect on the final strain is
significant. Calculating ν m , one can compute the effect of the welds on each other, how large is the initial
deformation at the place of the other weld, what is this effect on the total deformation, what is the effect
of changing the welding sequence (Fig.1.11).
Fig. 1.11 Initial strain in the place of the second weld due to the first weld
The strain at the gravity centre line and the curvature are as follows
ε GTA tA1
1= ; CA ty
IT T
x1
1= , (1.36)
ε εI G Tx
C y A tA
y yI12 1 1 2 11 21
= + = +⎛⎝⎜
⎞⎠⎟ , (1.37)
the modifying factor
ν
εε ε
εm
I
y I
y12
12
1211
21= −
+
≈ −
ln( )
ln , (1.38)
expresses the effect of the first weld at the place of the second weld.
The final strain and curvature caused by two welds is
ε ε ν ε ε νG G m G G mT
T
QQ( )1 2 1 12 2 1 12
2
1
1+ = + = +⎛⎝⎜
⎞⎠⎟ , (1.39)
C C C C Q yQ ym m
T
T1 2 1 12 2 1 12
2 2
1 1
1+ = + = +⎛⎝⎜
⎞⎠⎟ν ν . (1.40)
Changing the welding sequence ε G( )2 1+ and can be calculated using Eqs (1.39) and (1.40), changing
the subscripts:
C2 1+
ε ε ν ε ε νG G m G G mT
T
QQ( )2 1 2 21 1 2 21
1
2
1+ = + = +⎛⎝⎜
⎞⎠⎟ , (1.41)
C C C C Q yQ ym m
T
T2 1 2 21 1 2 21
1 1
2 2
1+ = + = +⎛⎝⎜
⎞⎠⎟ν ν . (1.42)
Comparing the two strains and curvatures, the smallest absolute value gives the better welding sequence.
If there are two longitudinal welds in an asymmetric I-beam and the 1st weld is closer to the gravity
center, y y1 < 2 , it means C C1 < 2
1
, so the 2nd weld has greater effect that the 1st one. The modifying
parameter is always less then 1 in this case, 0 < <ν m . The conclusion is, that the closer weld should be
made first, because the final deformation will be less, C C1 2 2 1+ +< .
The maximum deflection caused by welding is
w C Lmax =
2
8, w C L
1 2 1 2
2
8+ += , w C L2 1 2 1
2
8+ += . (1.43)
1.5 The effect of external loading on the welding residual stresses
To investigate the effect of static tension forces on residual stresses, we simplify the distribution of
residual stresses according to Fig. 1.12.
The stresses in the tension field are σ b , in the compression field σ a . If the sum of tension stresses due to
the external force and the residual stress is less than the yield stress, , the resulting
stress on the width part (b) of the plate is , on the width part (a) of the plate .
If the tension stresses due to the external force are larger , in this case on the width part (b)
of the plate , on the width part (a) of the plate
σ σ σe yf< = −'b
e e
y
y
σ σ σb b' = + σ σ σa a
' = +
σ σ' ≤ ≤e f
σ b f'' =
σ σ σ σ σa a ea b
a'' ' '( )= + + −
+22
(1.44)
During unloading, all fibres deform elastically
σ σ σ σ σ σa a e a e y bf ba
''' '' (= − = + − +2
)
e
(1.45)
(1.46) σ σ σ σb b e yf''' ''= − = −
It can be seen, that the residual stresses decrease. If the external force were , then there
no residual stresses remain (Fig. 1.12) and if the material is ideally plastic, then the residual stresses will
cut down to zero but residual deformations will occur.
F f t a by y= +(2 )
Fig. 1.12 Simplified residual stress distribution and the cutting down of initial stresses due to static
loading
1.6 Reduction of residual stresses
There are several ways to reduce the deformation and the residual stresses in the welded structures. They
are as follows:
Reduction techniques in the design stage:
cross-section symmetrical to the gravity center,
symmetrical welded joints,
suitable choice of welding sequence,
suitable choice of welding parameters,
welding in clamping device,
welding in prebent state in clamping device.
The deformation is much larger, if the cross-section is asymmetrical, or the welded joint is only on one
side of the cross section. An opposite weld can reduce the deformation. The welding sequence can also be
important for the final deformation of the structure. The best welding sequence is to start with the welded
joint closest to the gravity center of the cross-section and continue with an opposite joint to reduce the
final deformation. There is a choice of welding parameters, such as voltage, current and welding speed
among the technological limits. The use of different heat input for different welded joints can decrease
the final deformation.
Welding in a clamping device
The production sequence: tacking, clamping, welding, loosening (Fig. 1.13).
During welding the deformation w occurs, but it is restrained by clamping moments M. The bending
moment necessary to keep the beam straight against the welding deformations in
M = Iξ EC (1.47)
where Iξ is the moment of inertia for the elastic part of the cross-section area, calculated without the
plastic zone Ay , C is the curvature of the beam caused be welding in free state.
It is assumed, that the beam material is ideally elasto-plastic, that means that the tensile stress in the
plastic zone cannot be larger then the yield stress, so this zone cannot be loaded beyond this limit.
The loosening of the clamped state acts as the bending moments M with opposite sign. These
moments cause compressive stresses in the plastic zone which behaves elastically during thus unloading,
this one can calculate with the moment of inertia of the whole cross-section Ix. Thus, after the loosening
of the clamped state the following curvature occurs
C MEI
CIIx x
'= = ξ (1.48)
where Ix is the moment of inertia for the total elastic section area.
Fig. 1.13 Welding in a clamping device
It can be concluded, that using a clamped state the residual welding deformations cannot be totally
eliminated, they can be decreased only in a measure of II x
ξ , . The ratio between the two
curvatures depends on the area of the plastic zone.
C C C' ; '< ≠ 0
Welding in elastically prebent state in clamping device
The production sequence: tacking, prebending, clamping, welding, loosening (Fig. 1.14)
To prevent very large deformations and cracks, it is advisable to use prebending moments not larger than
Mf Iyy
y x=max
(1.49)
The curvature and deformation caused by My are
CMEIy
y
x
= , w Lyy y= ε
2
8 max
(1.50)
The prebending wp < wy causes a tensile prestrain in the place of the longitudinal weld
ε P p T pTC y w y
L= =
82 (1.51)
the corresponding modifying factor is
νεεm
P
y
= −1 (1.52)
The bending moment necessary to keep straight the beam after welding consists of two parts as follows:
the moment which is necessary for prebending
M I EC wEILp P'= =ξ
ξ8 2 (1.53)
and the moment which is necessary to eliminate the residual welding deformations
M I EC wEILm m' '= =ν νξ
ξ8 2 (1.54)
These moments act opposite after the loosening and decrease the prebending deformations,
M M M I EC I ECp m= + = +' ' ' ξ ξν (1.55)
so that the remaining final deformations can be expressed as
w w w M MEI
L wf px
p= − =+
−' ''
82 (1.56)
w w wII
wf p mx
p= + −( )ν ξ (1.57)
where νεmp T
y
w yL
= −18
2
Ix is the moment of inertia for the elastic section area.
Iξ is the moment of inertia for the elastic section area, reduced by the plastic zone,
C is the curvature of the beam caused be welding in free state,
ν m is the correction parameter according to Eq. 1.52.
The prebending wP necessary to totally eliminate the residual welding deformations can be calculated
from the condition wf = 0.
w wII
y wL
px T
y
=+ −
ξ ε8
12
. (1.58)
Fig. 1.14 Welding in prebent state in clamping device
Reduction techniques after production:
straightening welded plates,
pressing different welded shapes,
vibration (Wozney & Crawmer 1968),
heat treatment,
weld geometry modification methods (see Section 1.8.1),
residual stress methods (see Section 1.8.2).
1.7. Numerical examples
1.7.1 Suitable welding sequence in the case of a welded asymmetric I-beam
Find the best welding sequence due to two welded joints in an I-beam (Fig. 1.15)
Fig. 1.15 Welding sequence for an I-beam
Section dimensions
t = 10 mm, h = 600 mm, L = 6 m, steel grade Fe 360
Welding parameters
QT1 = QT2 = 60.7 Aw1 J
mm⎡⎣⎢
⎤⎦⎥
,
Aw1 = Aw2 = 100 mm2.
Determination of the center of gravity
, ydAA( )∫ = 0
082
0 42
00 0 0. .ht h t y hty ht h t y+−⎛
⎝⎜⎞⎠⎟− −
++⎛
⎝⎜⎞⎠⎟= ,
y0 = 55.5 mm.
Determination of the moment of inertia
I y dA h t hty ht ht h t y ht ht h t yxa
= = + + ++
−⎛⎝⎜
⎞⎠⎟
+ ++
+⎛⎝⎜
⎞⎠⎟∫ 2
3
0
3
0
2 3
0
2
1208
1208
20 4
120 4
2( )
. . . . ,
Ix = 8.881*10-4 mm4,
ε GTQA1
3 1 40 844 10 3881 10= − = −− −. * . * ,
A = 0.8*600*10 + 600*10 + 0.4*600*10 = 1.32*104 mm2,