finite element models of spot welds in structural dynamics
TRANSCRIPT
Computers and Structures 83 (2005) 648–661
www.elsevier.com/locate/compstruc
Finite element models of spot welds in structuraldynamics: review and updating
Matteo Palmonella a, Michael I. Friswell b,*,John E. Mottershead c, Arthur W. Lees a
a School of Engineering, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, UKb Department of Aerospace Engineering, Queen’s Building, University Walk, University of Bristol, Bristol BS8 1TR, UK
c Department of Engineering, University of Liverpool, Liverpool L69 3GH, UK
Received 17 October 2003; accepted 7 November 2004
Available online 11 January 2005
Abstract
Spot welds are used extensively in the automotive industry to join panels, and car bodies contain many thousands of
spot welds. Different finite element models of spot welds have been created for various types of analysis. When struc-
tures with many spot welds are analysed, these detailed models have too many degrees of freedom to be used in practice.
Simple models that use few elements must be used instead. This paper reviews the spot weld models available in the
literature. Model updating based on the measured vibration characteristics is then used to improve the accuracy of
the most common coarse models of spot welds.
� 2004 Elsevier Ltd. All rights reserved.
Keywords: Spot welds; Finite element analysis; Structural dynamics; Model updating; Automotive structures
1. Introduction
There are many models of spot welds used for static
and dynamic structural analysis. Modelling spot welds is
difficult, mainly because there are many local effects such
as geometrical irregularities, residual stresses, material
inhomogeneities and defects due to the welding process,
that are not taken into account by finite element model-
ling. Furthermore, models with a small number of de-
grees of freedom must be used since real structures
usually contain many spot welds and modelling each
of them in detail would require a major computational
0045-7949/$ - see front matter � 2004 Elsevier Ltd. All rights reserv
doi:10.1016/j.compstruc.2004.11.003
* Corresponding author. Fax: +44 117 927 2771.
E-mail address: [email protected] (M.I. Friswell).
effort. Two main types of spot weld models exist, namely
those that require the stress within the weld spot to be
calculated and those that do not. In the first case very
detailed models are necessary to compute a smooth
stress field at the spot weld. In the second case the only
requirement from the model is to simulate, as closely as
possible, the stiffness (and mass) characteristics of the
real spot welds and their influence on the rest of the
structure. This allows the use of much simpler models
with far fewer degrees of freedom.
A very detailed model produces a detailed and
smooth stress field, but it will not necessarily accurately
predict the stiffness of real spot welds and their effect on
the rest of the structure. Detailed models will produce
apparently reliable stress fields, whereas they may poorly
estimate the forces that are interchanged between the
ed.
M. Palmonella et al. / Computers and Structures 83 (2005) 648–661 649
spot weld and the rest of the structure. For fatigue life
estimation, efforts have been made to obtain stresses at
the spot welds that are significant for fatigue estimation
(i.e. to be used in S–N curves) from coarse finite element
models. Simple models can, if updated, lead to a good
estimation of the forces interchanged between the spot
weld and the rest of the structure, and these forces can
be related to the local stresses at the spot weld by analy-
tical approximations [1–9]. In this paper, an overview
of spot weld models in literature is given, comprising
of both detailed and coarse models. The updating of
appropriate models is then examined.
2. Models for stress analysis
Generally, models of spot welds for stress analysis
use brick or plate elements to model the welded plates
and brick elements to model the nugget. The following
descriptions of the models will be presented in order of
refinement from the most detailed to the least.
Chang et al. [10] used a very detailed model to study
weld-bonded joints. The only difference between weld-
bonded and spot welded joints is the presence of an
adhesive layer between the sheets. Hence the proposed
model can be regarded as a spot weld model if the adhe-
sive layer is not taken into account. Four regions were
modelled in the joint zone, namely the base metal, the
heat affected zone, the nugget and the adhesive layer.
An annular contact zone was located between the upper
and lower parts of the heat affected zone, which was di-
vided into five parts whose yield strength decreased with
the increasing distance from the nugget. The finite ele-
ment meshes for the specimens consisted of three dimen-
sional brick elements with eight nodes and the plates
were divided into two layers. The edges of the spot were
divided into smaller elements than the other parts, and
the minimum dimension of the element was 0.15 mm
(for a spot weld diameter of 5 mm, plate thicknesses of
1 mm and a distance between the plates of 0.3 mm).
The elastoplastic properties of the materials were de-
scribed by linear hardening functions, and the hardening
modulus of the material was assumed to be known. The
electrode indentation (i.e. the plastic deformation caused
by the welding process), the heterogeneity of the materi-
als and the separation of the plates were all taken into
account. The joints were assumed to undergo small
deformations, and geometrical non-linearity was ne-
glected (small displacements).
A simplified version of this model was presented by
the same authors [11]. The indentation geometry was
not considered and the number of different material
properties was limited to the definition of one set of
material properties for the base metal and one set for
the spot weld. In fact, this model was an early model
that the authors improved in Ref. [10].
Deng et al. [12] used a three dimensional mesh for the
stress analysis of spot welded specimens under tensile-
shear and coach-peel loading conditions. Small finite
elements were used within and around the nugget and
larger elements were used in the outer regions. The ele-
ment size inside the nugget was set to one tenth of the
nugget radius. Along the thickness direction, the plate
was cut into four equal layers of elements. The material
properties of the nugget and base plate were taken to be
the same. Convergence studies demonstrated that a finer
mesh than the one described did not lead to significant
improvements in the stress distribution.
Radaj and Sonsino [4] and Radaj and Zhang [13]
used finite element models to study the geometric non-
linear behaviour of spot welded joints and also used
the same models for fatigue calculations. The mesh con-
sisted of plate elements outside of the weld spot and
solid elements within the spot. The plate and solid ele-
ments were connected by pin-jointed rigid bars and the
material was assumed to be the same in the plate and
brick elements.
Zhang and Richter [9] proposed a model for fatigue
analysis that used solid elements for the nugget and shell
elements for the remaining structure with a much coar-
ser mesh than the model used by Radaj and co-workers
[4,13]. Using kinematic constraints, for example RSS-
CON in NASTRAN, the kinematic compatibility be-
tween the solid and shell elements was ensured. The
curvature of the boundary of the spot weld was approx-
imated by using elements with mid nodes.
Zhang and Taylor [14] used a model of the spot weld
for fatigue analysis that used shell elements both within
and outside the nugget. They also suggested a model of
the spot weld that used shell elements within the nugget
area connected by vertical beams around the perimeter
of the nugget.
Chen and Deng [15] proposed a mesh composed of
shell elements for the plates and considered the nugget
to be rigid. Comparison with models with solid ele-
ments, such as those suggested by Deng et al. [12],
showed that converged shell element solutions provided
an overall good approximation to converged solid ele-
ment solutions, even near the nugget boundary. Further-
more, shell element solutions generally predicted a
higher maximum stress than solid element solutions with
rigid nuggets by about 9%. It was remarked that the
rigid nugget assumption had a minimal effect on the
shape of the stress distribution but did lead to a higher
stress level than when a flexible nugget was assumed.
3. Models for stiffness simulation
In real structures the number of spot welds is such
that a detailed model of every weld would lead to an
overwhelming computational effort. For example, in
650 M. Palmonella et al. / Computers and Structures 83 (2005) 648–661
automotive bodies there are typically between three and
five thousand spot welds. The only practical approach is
to model the spot welds with very coarse meshes, taking
care to verify that the model used accurately represents
the stiffness (and mass) characteristics of the real welds.
Apart from accuracy, the other major requirement of
spot weld models is a short modelling time. This is
mainly affected by any requirement to remesh the welded
plates around the spot weld. This is necessary for models
that require a congruent mesh in the two plates and for
this reason spot weld models have been developed that
can be implemented without any remeshing of the plates
models. In fact these latter models can be more accurate
than those that require congruent meshes. An overview
of the coarse mesh models available in literature is
now given.
3.1. Single beam models
Very simple models of spot welds have been used
extensively in the automotive industry for many years,
consisting of elastic or rigid beams, or even coincident
nodes [16–22]. All of these models require the upper
and lower plates to be meshed congruently, to allow
the nodes that are joined to lie on a line perpendicular
to both of the plates. They represent the behaviour of
the real spot weld inadequately and generally tend to
underestimate its stiffness.
3.2. Brick models
Pal and Cronin [18] proposed the use of a single brick
element to characterise the spot weld nugget for static
stiffness analysis and the calculation of the modal behav-
iour. The connection between the solid elements and the
shell elements was achieved using rigid connections to
transfer the moments from the shell to the solid ele-
ments. The nodes of the brick element are co-incident
with nodes of the shell elements. This model requires
the two plates to be meshed congruently and gives a
good estimation of the local stiffness at the spot weld.
Unfortunately the model does not have suitable para-
meters for updating (i.e. parameters to which the
response is sensitive).
3.3. Umbrella and ACM1 models
Zhang and Taylor [14,23] proposed a model of
spot welds for fatigue analysis that used a vertical beam
plus more beams in the planes of the two con-
nected plates, forming an umbrella shape. Although
the plate mesh may be a fine one, this model can also
be used with much coarser meshes, becoming simple
and accurate.
Zhang and Taylor [24] proposed a model similar to
the umbrella model where the radial beams instead of
being elastic are rigid. This is known in the literature
as ACM1 (area contact model 1) [20] and gives far fewer
possibilities for updating than the umbrella model, since
the radial beam stiffnesses are fixed.
3.4. First Salvini model
Salvini et al. [21] developed a spot weld model based
on an analytical model of a spot weld often used in fati-
gue estimation to relate forces and stresses. The analy-
tical model consists of a circular clamped plate that
simulates one of the welded plates, and a rigid core that
simulates the spot weld. The plate radius is taken as ten
times greater than the core radius. Forces and moments
that the spot weld transmits to the plate are applied at
the rigid core, and the stiffness of the system is calculated
analytically. The finite element model consists of a cen-
tral beam element with six degrees of freedom per node
representing the spot weld nugget. Each node of the
beam is connected to the chosen shell nodes of the two
plates by beams that have a rigid part and a flexible part.
The stiffness characteristics of these beams are calculated
from the stiffness of the analytical model of the spot
weld area. Every radial beam is associated with an angu-
lar sector of the circular plate and the beam is designed
to reproduce its stiffness.
3.5. ACM2 model
Heiserer et al. [16] proposed the model known as
ACM2 (area contact model 2). The model consists of a
brick element connecting the upper and lower plates
via RBE3 elements that are available in NASTRAN.
The RBE3 element distributes the applied loads
throughout the model. Forces and moments applied to
the brick nodes are distributed to the shell nodes in a
way that depends on the RBE3 geometry and weight
factors assigned to the shell nodes. The weights are the
values assumed by the shape function corresponding
to each shell node at the location of the brick node.
The force acting on the brick node can be transferred
to the weighted centre of gravity of the shell nodes
along with the moment produced by the force offset.
The force is distributed to the shell nodes in proportion
to the weighting factors. The moment is distributed as
forces, whose magnitude are proportional to their dis-
tance from the centre of gravity times their weighting
factors.
3.6. CWELD model
The spot weld model proposed by Fang et al. [17] is
implemented in NASTRAN as the CWELD element.
The CWELD element is modelled with a special shear
flexible beam-type element with two nodes, and 12 de-
grees of freedom. Every node of the beam is connected
M. Palmonella et al. / Computers and Structures 83 (2005) 648–661 651
to a chosen set of nodes of the plate it belongs to. The
area enclosed by the nodes of each plate that are
attached to the beam element are called ‘‘patches’’.
The six degrees of freedom of each beam node are con-
nected to the three translational degrees of freedom of
each plate node with constraints from Kirchoff shell
theory.
3.7. Second Salvini model
Vivio et al. [25] recently proposed another model of
spot welds that consists of a central beam that connects
the two plates and two sets of radial beams: one pinned
to the rest of the structure and one connected through
two offsets to the central beam and the rest of the struc-
ture. The internal nodes of pinned beams are connected
to the central beams by simple links. The development
of the model stems from the same approach as the first
Salvini model, and the stiffness that arises from the ana-
lytical model of the spot welds is similar. The difference
between the first and second Salvini models is that in the
first case, the stiffness of the in-plane rotation and the
vertical displacements were coupled leading to complica-
tions in the definition of the beam stiffness. In the second
Salvini model the external beams bear all the normal
load, while the internal beams react only against in-
plane moment and in-plane force. This allows the defini-
tion of a simpler way to characterise the radial beams.
Fig. 1. The double hat (DH) benchmark.
Fig. 2. The single hat (SH) benchmark.
4. The benchmark structures
The discussion thus far has given an overview of the
spot weld models available in the literature, but nothing
has been said about the accuracy of each model. In the
rest of the paper the accuracy issue will be considered
and six models chosen from those presented above will
be updated. The models considered are:
• CWELD (referred to in the reminder of the paper as
CW),
• ACM2 (referred to as AC),
• First Salvini (referred to as SA),
• Brick (referred to as BR),
• Beam (referred to as BAR),
• Radaj (referred to as DET, being a detailed model).
The rigid beam model has not been investigated be-
cause it is a particular case of the beam model when
the beam rigidity is very high. The umbrella, ACM1
and the second Salvini model have not been analysed be-
cause they are very similar to the SA model. The Radaj
model has been taken as representative of all of the
detailed models presented above, and this model is the
most widely used in fatigue analysis when the knowledge
of stress field is required.
In order to perform the validation and updating of
the spot weld models, two benchmark structures have
been built:
• a ‘‘double hat’’ structure (DH), shown in Fig. 1 and
• a ‘‘single hat’’ structure (SH), shown in Fig. 2.
The benchmarks consist of hat section steel plates
joined together by spot welds at the flanges, and are de-
signed to represent simplified models of the beams used
in the construction of car bodies, for example the roof
pillars.
Figs. 3 and 4 show the section of the hat plates that
form the DH benchmark. They are a tall hat (TH) and
a short hat (SSH). Fig. 5 shows the tall hat (TH) plate
constituting the SH benchmark. The flat plates for the
SH benchmarks are denoted by FP. Both structures
are 565 mm long and the metal sheets are 1.5 mm thick.
Both the DH and SH structures had 20 spot welds
60 mm apart (Fig. 6).
Each structure was joined by spot welds having the
same dimensions, although the spot weld diameter can
vary from structure to structure. Three DH and three
SH plates were analysed and these are denoted DH1,
Fig. 3. Section view of the tall hat plate (TH) with long flanges.
Fig. 4. Section view of the short hat plate (SSH).
Fig. 5. Section view of the tall hat plate (TH) with short
flanges.
Fig. 6. Top view of the do
Table 1
Average error in the first 10 natural frequencies of the plates constitu
TH1 TH2 TH3 TH4 TH5
Average error (%) 0.6 0.4 0.3 1.1 1.2
652 M. Palmonella et al. / Computers and Structures 83 (2005) 648–661
DH2, DH3, SH1, SH2, SH3. The DH1 plate was welded
with 5 mm spot welds, the DH2 plate with 7 mm spot
welds, and the rest of the benchmarks were welded with
a spot weld diameter of 6 mm. In order to isolate the
error in the spot weld models from those in the rest of
the benchmark structures, the plates belonging to the
benchmark structures were tested before being welded
together. The plates were updated separately before
updating the complete benchmark structures so that
the error in the welded structures should arise mainly
from the spot weld models. Hence the benchmark struc-
tures were updated using only parameters that characte-
rise the spot weld models. Table 1 shows the average
error in the first ten frequencies of the hat and flat plates.
The benchmark structures consisted of the following
parts:
• DH1: TH1 plus SHH1,
• DH2: TH2 plus SHH2,
• DH3: TH3 plus SHH3,
• SH1: TH4 plus FP1,
• SH2: TH5 plus FP2,
• SH3: TH6 plus FP3,
where THi denotes the ith tall hat plate, SHHi denotes
the ith short hat plate and FPi denotes the ith flat plate.
DHi denotes the ith double hat benchmark structure
and SHi denotes the ith single hat benchmark structure.
5. Updating of the models
5.1. Updating with MSC/NASTRAN
The optimisation algorithm of the finite element code
NASTRAN has been used to perform the updating. The
uble hat plate (DH).
ting the benchmark structures
TH6 SSH1 SSH2 SSH3 FP1 FP2 FP3
0.7 0.4 0.6 0.6 0.5 0.4 0.6
M. Palmonella et al. / Computers and Structures 83 (2005) 648–661 653
code allows the specification of the objective function to
be minimised using the output variables of the finite ele-
ment analysis. In the present study only the natural fre-
quencies are used, and the objective function defined for
updating is [26],
J ¼Xn
i¼1
W ixi
xexpi � 1
� �2
ð1Þ
where
• xi is the ith computed natural frequency,
• xexpi is the ith experimental natural frequency,
• Wi is the weight assigned to the ith natural frequency,
• n is the number of measured frequencies.
The weighting criterion is such that all natural fre-
quencies have W = 1, except those corresponding to
the first torsion and first bending of the beam. For these
frequencies extra weight is assigned (W = 2) since these
modes of the benchmark structure are most similar to
the deformation of the beams within global modes of
automotive bodies. Ref. [27] gives further details of the
optimisation algorithm.
Ensuring the correct pairing of mode shapes between
the experimental and analytical modes is vital to ensure
that the updating results in a physically meaningful
model. The Modal Assurance Criterion (MAC) matrix
[26] is used for the initial correlation, and the optimisa-
tion algorithm in NASTRAN allows the modes to be
tracked to ensure this pairing is maintained during the
optimisation procedure.
5.2. Parameter sensitivities
The six spot weld models analysed can be divided into
‘‘patch-like’’ models and non-patch-like models. CW,
AC and SA are patch like models, while BR, BAR and
DET are not. For non-patch-like models, the only
parameters eligible for updating are the dimensions and
material properties of the spot element. For patch-like
models, the dimensions and material properties of both
the patch and spot element are eligible parameters for
updating. Young�s modulus may be considered as an
equivalent stiffness parameter that reduces the differences
in the response between the finite element model and the
measured data, and accounts for unmodelled stiffness
variation. The structure�s response is very sensitive to
parameters related to the patch because these parameters
greatly affect the bending and peeling stiffness, although
they do not influence the stiffness in the shear direction.
The response is, in general, very insensitive to parameters
related to the spot element, with the notable exception of
the brick dimension in the AC and BR models.
Depending on the type of spot element used, the six
spot weld models analysed can be classified as brick-like
models or beam-like models. CW, SA and BAR are
beam-like models, while AC and BR are brick-like mod-
els. DET falls outside of this classification. When using
beam-like models, the natural frequencies of the structure
are not sensitive to the dimensions and material proper-
ties of the spot element. For brick-like models, the re-
sponse is insensitive to spot element material properties
but sensitive to spot element dimensions. The spot ele-
ment, being either a brick or a beam, is very short and
wide; in the examples presented here the spot element is
1.5 mm high and 6 mmwide. This makes the spot element
very stiff so that, in practice, it may be considered rigid.
Varying the stiffness of this almost rigid element produces
only a small variation in the peeling and shear forces
transmitted. However, changing the brick dimensions sig-
nificantly affects the rotational stiffness because the arm
of the moments interchanged between the plates is larger.
For real spot welds the experimental evidence [28]
shows that the diameter does not influence the dynamics
of the structure. The likely reason for this phenomenon
is that the contact between the welded plates greatly
influences the local stiffness in bending moment transfer.
From this point of view, as long as the spot weld creates
the contact, the nugget dimensions do not influence the
rotational stiffness. Shear and peeling actions are not
influenced by the spot diameter because the spot weld
is practically rigid. Varying the diameter does not pro-
duce significant changes of stiffness. From the modelling
point of view this allows the use of the same spot weld
model for all nugget diameters.
The DET model cannot be classified as above. This
model is able to closely represent the spot weld geome-
try, and it would be meaningless to then use a parameter
that modifies this geometry. Only the Young�s modulus
of the elements that constitute the DET model can be
considered as candidate parameter for the updating,
but this is not a sensitive parameter for the reasons dis-
cussed above. The only other possibility would be the
use of generic elements, where the eigenvalues and eigen-
vectors of the stiffness matrices are adjusted [29,30].
However generic elements are not easily implemented
into finite element codes, and would lead to increased
computational effort.
Table 2 shows the sensitivities associated with the
most sensitive spot weld parameters for the CW, AC,
SA and BR models applied to the DH structures. The
sensitivities of the spot diameter for the CW and SA
models and the sensitivities associated with the spot ele-
ment Young�s modulus are shown in Table 3. Table 4
shows the sensitivities of the parameters used to update
the SH plates. The acronyms used in Tables 2–4 have the
following meaning,
• SD: spot diameter,
• SE: spot Young�s modulus,
• PA: patch area,
Table 2
Double hat structures: sensitivities associated with sensitive spot weld parameters for the CW, AC, SA and BR models (Hz)
Model CW AC SA BR
Parameter PA PE SD PA PE PA BE SD
Mode Description
1 A1 3.9 2.4 1.5 3.50 1.8 �3.5 2.3 6.3
2 A2 3.7 2.4 1.5 3.30 1.9 �3.4 2.2 6.2
3 B2 0.43 0.31 0.22 0.36 0.24 �0.48 0.28 0.81
4 B1 0.52 0.34 0.17 0.44 0.26 �0.53 0.33 0.88
5 BOTTOM 2.0 1.9 1.0 1.60 1.5 �1.7 1.1 2.8
6 TOP 0.99 1.2 0.60 0.65 0.79 �1.8 0.63 1.9
7 B3 0.57 0.41 0.90 0.28 0.18 �0.48 0.29 0.82
8 FLANGES 3.5 2.4 0.58 0.57 0.95 �3.8 1.7 3.1
9 C3 3.3 3.6 1.9 2.30 2.2 �5.3 2.6 5.9
10 C1 3.6 2.2 0.68 2.50 1.5 �4.0 2.4 5.4
11 2ND BEND. 2.4 2.2 2.3 1.90 1.6 �5.7 4.7 4.7
Table 3
Double hat structures: sensitivities associated with non-sensitive spot weld parameters for the CW, AC, SA and BR models (Hz)
Model CW SA CW AC SA BR
Parameter SD SD SE SE SE SE
Mode Description
1 A1 0.56 0.30 0.19 0.12 0.16 0.00
2 A2 0.53 0.24 0.18 0.13 0.15 0.00
3 B2 0.10 0.01 0.03 0.05 0.02 0.00
4 B1 0.08 0.04 0.03 0.02 0.03 0.00
5 BOTTOM 0.34 0.06 0.14 0.22 0.14 0.17
6 TOP 0.28 �0.30 0.14 0.21 0.14 0.11
7 B3 0.30 0.11 0.11 0.21 0.08 0.03
8 FLANGES 0.67 �0.07 0.37 0.24 0.43 0.05
9 C3 0.67 �0.38 0.23 0.35 0.18 0.22
10 C1 0.50 0.20 0.17 0.09 0.17 0.01
11 2ND BEND. 0.76 0.77 0.30 0.50 0.40 0.10
Table 4
Single hat structures: sensitivities associated with the spot weld parameters used in the updating for the CW, AC, SA and BR models
(Hz)
Model CW AC SA BR
Parameter PE POS PE POS BE POS SD POS
Mode Description
1 A2 1.8 �4.8 1.5 �4.7 0.62 �5.0 3.3 �5.5
2 A1 2.2 �6.1 2.1 �6.1 0.83 �6.3 4.2 �6.6
3 1ST BASE 4.5 �18 5.0 �19 2.3 �19 8.8 �19
4 2ND BASE 5.8 �26 6.3 �27 3.0 �26 13 �27
5 B1 5.1 �27 5.5 �27 2.5 �27 11 �27
6 B2 5.2 �27 5.8 �27 2.6 �27 11 �27
7 3RD BASE 6.1 �27 1.2 �1.5 3.2 �27 14 �27
8 A3 1.8 �1.4 6.8 �28 0.57 �1.9 2.0 �1.9
9 4TH BASE 6.4 �28 7.0 �29 3.4 �28 15 �28
10 1ST BEND. 5.9 �26 4.1 �21 3.1 �25 4.9 �26
654 M. Palmonella et al. / Computers and Structures 83 (2005) 648–661
M. Palmonella et al. / Computers and Structures 83 (2005) 648–661 655
• PE: patch Young�s modulus,
• BE: bar Young�s modulus (refers to the elastic bars
that connect the spot element to the patch in the
SA model),
• POS: spot welds transverse change of position (in the
direction perpendicular to the axis of the flanges,
positive when the two spot weld rows move apart
from each other).
The sensitivity values quoted in Tables 2–4 are the
change in the ith natural frequency, Dxi, due to a change
in the parameter, given by Dh. This sensitivity is quoted
in Hz. For most parameters, the variation Dh was taken
as a 10% change in the parameter value. In the case of
the PA parameter, the percentage variation was in the
edge dimension of the patch (which is the square root
of the area). For the POS parameter, Dh was taken as
a millimetre change in the transverse position of all of
the spot welds. The parameter variations used were con-
sidered as realistic variations that might be expected
in the parameter, using engineering judgement, and al-
low the sensitive parameters to be identified. Fig. 7
shows the first twelve mode shapes of the DH bench-
marks and Fig. 8 shows the first ten mode shapes of
the SH structure.
From Tables 2–4, the only sensitive parameters for
each model are:
• CW: PA, PE.
• AC: SD, PA, PE.
• SA: PA, PE.
• BR: SD.
• BAR: none.
• DET: none.
The BAR and DET models have not been updated
due to the absence of sensitive parameters. The sensi-
tivities associated with the PA parameter for the
CW and AC models are positive, because increasing
the PA extends the effect of the spot weld to a larger por-
tion of the structure which increases the structure�sstiffness. However, the sensitivities associated with the
PA parameter for the SA model are negative, because
the rigid beams in the element have a fixed length,
and increasing the patch area causes the length of the
elastic beams to increase, leading to a more flexible
structure.
Table 3 shows that the PA, PE and BE parameters
for the SA model and the SD parameter for the AC
model affect the local stiffness in a similar way. In partic-
ular they mainly influence the peeling and rotational
stiffness. If two of these parameters were used simulta-
neously in updating, an ill-conditioned system of equa-
tions would result [26]. Thus, in the updating of the
DH plates, it has been decided to use only the PE
parameter.
5.3. Parameter estimation
The values for the geometric and material parameters
of the spot weld models before updating were as follows:
• patch edge for the DH benchmarks = 12 mm,
• patch edge for the SH benchmarks = 10 mm,
• patch (or bar in SA) Young�s modulus = 210 GPa,
• spot beam diameter or brick edge = 6 mm,
• spot beam or brick Young�s modulus = 210 GPa.
Table 5 shows the percentage parameter change after
updating of the DH and SH structures. For the DH
structures only the PE parameter has been used. For
the SH structures, the PE and POS parameters have
been used. There are two effects that are immediately
apparent from the results. The first effect is that the po-
sition of the spot welds for the SH benchmarks seems to
have moved significantly. The initial finite element
model placed the spot welds in the centre of the flange.
However, accurately positioning of the electrodes during
spot welding was difficult for the SH benchmarks be-
cause of the small flanges, leading to significant errors
in the weld position. This was confirmed by measure-
ments on the benchmarks after the updating analysis.
The other issue is the difference in the estimated PE
parameters between the SH and DH benchmarks for
the CW, AC and SA models. This difference is due to
the different flange lengths, which mean that the patch
edge length for the DH structures is 12 mm, while for
the SH structures it is 10 mm. This 2 mm difference be-
tween the two patches lengths is responsible for the per-
centage variation of the PE parameter estimates. For
example, for the CW and AC models, a smaller PA
parameter is compensated by a larger PE parameter to
reach approximately the same level of local stiffness.
The opposite is true for the SA model. Overall, Table
5 shows that updating the spot weld models yields very
similar parameter values for both benchmark structures
and demonstrates that the results achieved are physically
meaningful.
5.4. Forces acting at the spot welds
Before analysing the updating results, it is interesting
to consider the forces and moments acting at the spot
welds. Tables 6 and 7 show a comparison between bend-
ing moments, shear and peel forces in the first eleven
mode shapes of the DH structures and in the first ten
mode shapes of the SH structures. These forces and mo-
ments are the internal quantities for the nodes of the
beam element in the CWELD model and calculated by
NASTRAN. The shear forces and bending moments
are the magnitude of the resultant force or moment in
the plane of the two plate structures. The peel force is
normal to the plates, and the torsional moment is the
Fig. 7. Mode shapes of the DH benchmark structure.
656 M. Palmonella et al. / Computers and Structures 83 (2005) 648–661
torsion of the beam element. The quoted values in the
tables are an average of all of the twenty spot welds pres-
ent in the structures. The bending moment is the average
between the values at the two ends of the spot element.
The torsion moment is usually one order of magnitude
smaller than the bending moment, and therefore is not
considered in Tables 6 and 7.
Tables 6 and 7 show that the bending moment is lar-
ger in the mode shapes where the shear force is small
compared to the peel force. Comparing these forces
and moments with the spot weld parameter sensitivities
(Tables 2–4) shows that the spot weld parameters used
for updating have much more influence on those modes
where the peeling forces and bending moments are high
Fig. 8. Mode shapes of the SH benchmark structure.
Table 5
Parameter changes due to updating of the double and single hat
structures
Model BR AC CW SA
Parameter SD (%) PE (%) PE (%) BE (%)
DH1 �40 �76 �26 44
DH2 �30 �70 �17 41
DH3 �29 �67 �8 100
SH1 �43 �55 �7 �12
SH2 �36 �49 �3 30
SH3 �37 �50 �10 0.0
Parameter POS (mm) POS (mm) POS (mm) POS (mm)
SH1 1.20 1.80 0.90 1.30
SH2 1.00 1.60 0.50 1.18
SH3 1.41 2.11 1.01 1.32
Table 6
Comparison between the shear and peel forces in the first twelve
mode shapes of the double hat benchmark
Mode Description Moment
(Nm)
Shear
(kN)
Peel
(kN)
Shear/peel
1 A1 276 2.1 5.5 0.38
2 A2 242 2.0 4.9 0.41
3 B2 88 12 0.23 55
4 B1 109 7.6 0.71 11
5 BOTTOM 202 52 4.6 11
6 TOP 125 73 9.0 8.1
7 B3 80 59 0.60 100
8 FLANGES 235 134 16 8.4
9 C3 325 46 28 1.7
10 C1 393 5.9 3.3 1.8
11 2ND BEND. 261 103 15 6.7
M. Palmonella et al. / Computers and Structures 83 (2005) 648–661 657
Table 7
Comparison between the shear and peel forces in the first ten
mode shapes of the single hat benchmark
Mode Description Moment
(Nm)
Shear
(kN)
Peel
(kN)
Shear/peel
1 A2 137 29 1.6 18
2 A1 192 9.5 0.64 15
3 1ST BASE 287 38 15 2.5
4 2ND BASE 352 16 18 0.87
5 B1 384 3.8 17 0.23
6 B2 319 9.1 15 0.58
7 3RD BASE 393 13 21 0.63
8 A3 116 61 6.8 9.0
9 4TH BASE 453 13 26 0.52
10 1ST BEND. 380 44 21 2.1
658 M. Palmonella et al. / Computers and Structures 83 (2005) 648–661
compared to the shear forces. In particular, modes A1
and A2 for the DH structures and all the local base mo-
Table 8
DH1 updating: natural frequency errors in the initial and updated FE
Mode Description Experimental
(Hz)
CW model AC mod
Initial
(%)
Updated
(%)
Initial
(%)
1 A1 474.6 3.6 2.1 7.1
2 A2 497.3 0.8 �0.7 3.9
3 B2 525.4 2.3 2.1 2.7
4 B1 536.3 0.6 0.5 1.0
5 BOTTOM 768.4 �0.3 �1.0 1.4
6 TOP 899.1 0.1 �0.3 1.6
7 B3 913.3 1.1 0.9 1.9
8 FLANGES 1060.0 1.4 0.7 4.1
9 C3 1110.0 0.1 �0.9 2.5
10 C1 1170.0 �0.3 �0.8 0.6
11 2ND BEND. 1190.0 �0.8 �1.3 1.6
Average % error 1.0 1.0 2.6
Table 9
DH2 updating: natural frequency errors in the initial and updated FE
Mode Description Experimental
(Hz)
CW model AC mod
Initial
(%)
Updated
(%)
Initial
(%)
1 A1 475.1 3.0 2.0 6.4
2 A2 503.3 �0.9 �1.7 2.2
3 B2 527.8 1.5 1.4 1.9
4 B1 535.8 0.5 0.4 0.9
5 BOTTOM 776.3 �0.9 �1.4 0.7
6 B3 907.4 1.2 1.1 2.1
7 TOP 919.9 �2.5 �2.8 �1.0
8 C3 1060.0 4.3 3.7 6.8
9 2ND BEND. 1180.0 0.1 �0.3 2.5
Average % error 1.7 1.6 2.7
tion modes (modes 3, 4, 5, 6, 7, 9) for the SH structures
are very sensitive to the spot weld parameters. The brick
area in the AC and BR models have a greater stiffening
effect for the bending action than for the in-plane defor-
mations. In the same way, the patch parameters affect
the rotational and peeling stiffness much more than the
shear stiffness.
5.5. Updating results
Tables 8–10 show the experimental natural fre-
quencies for the DH benchmarks and the percentage
frequency error in the initial and updated finite ele-
ment models. Tables 11–13 show the same quantities
for the SH structures. Tables 8–10 show that the modes
that are affected by the greatest errors in the initial
models of all of the benchmark structures are the
same modes that are sensitive to the spot weld para-
meters. The analysed spot weld models do not have
models
el SA model BR model Initial models
Updated
(%)
Initial
(%)
Updated
(%)
Initial
(%)
Updated
(%)
DET
(%)
BAR
(%)
1.7 �0.7 0.7 5.9 0.5 �10.7 �14.5
�1.3 �3.0 �1.7 3.2 �1.9 �5.2 �15.6
2.1 1.9 2.0 2.7 2.1 1.5 1.5
0.3 0.1 0.3 1.0 0.3 �0.6 �1.8
�1.4 �1.0 �0.5 1.1 �0.4 �3.1 �0.2 �0.2 0.0 2.0 1.1 �0.8 �1.6 1.3 1.4 1.9 1.5 0.0 �10.6
2.5 1.4 1.8 4.8 3.5 � ��0.4 �0.9 �0.3 2.9 0.5 �2.5 �9.9
�1.2 �2.1 �1.5 0.6 �1.4 � �9.8
�0.3 �1.1 �0.7 1.8 �0.4 �1.9 �10.1
1.2 1.2 1.0 2.5 1.2 2.8 9.2
models
el SA model BR model Initial models
Updated
(%)
Initial
(%)
Updated
(%)
Initial
(%)
Updated
(%)
DET
(%)
BAR
(%)
2.0 �1.2 0.1 5.3 1.3 �10.3 �15.0
�2.1 �4.5 �3.4 1.5 �2.2 �4.1 �16.9
1.4 1.1 1.2 1.9 1.5 1.0 0.7
0.3 0.0 0.1 0.8 0.3 0.3 �2.0
�1.7 �1.6 �1.2 0.4 �0.7 �0.7 �1.8 1.4 1.5 2.0 0.0 1.5 �10.7
�2.2 �2.8 �2.6 �0.7 0.4 1.7 �4.2 3.2 3.8 7.1 5.4 5.8 �6.2
0.8 �0.3 0.0 2.6 1.4 0.9 �9.3
1.8 1.8 1.5 2.5 1.5 2.9 8.7
Table 10
DH3 updating: natural frequency errors in the initial and updated FE models
Mode Description Experimental
(Hz)
CW model AC model SA model BR model Initial models
Initial
(%)
Updated
(%)
Initial
(%)
Updated
(%)
Initial
(%)
Updated
(%)
Initial
(%)
Updated
(%)
DET
(%)
BAR
(%)
1 A1 478.5 3.0 2.5 6.5 2.3 �1.3 1.3 5.3 1.4 �10.8 �15.1
2 A2 506.8 �0.9 �1.3 2.2 �1.8 �4.6 �2.3 1.5 �2.1 �6.1 �17.1
3 B2 538.3 0.7 0.6 1.0 0.5 0.2 0.5 1.0 0.6 �0.1 0.0
4 BOTTOM 780.1 �1.5 �1.7 0.2 �2.0 �2.2 �1.3 �0.1 �1.1 �3.4 �22.0
5 B3 906.4 1.6 1.6 2.5 2.3 1.8 2.0 2.5 2.2 �3.9 �10.2
6 TOP 925.8 �2.9 �3.0 �1.5 �2.5 �3.2 �2.8 �1.2 �1.7 �3.3 �9.9
7 FLANGES 1080.0 �0.2 �0.4 2.5 1.3 �0.2 0.6 3.2 2.3 5.2 �8 C1 1150.0 1.7 1.6 2.7 1.2 �0.2 0.9 2.7 1.2 �5.3 �8.2
9 C2 1220.0 �1.1 �1.4 3.5 �0.4 �3.7 �2.5 2.2 �1.4 � �12.2
Average % error 1.5 1.6 2.5 1.6 1.9 1.6 2.2 1.6 4.2 10.5
Table 11
SH1 updating: natural frequency errors in the initial and updated FE models
Mode Description Experimental
(Hz)
CW model AC model SA model BR model Initial models
Initial
(%)
Updated
(%)
Initial
(%)
Updated
(%)
Initial
(%)
Updated
(%)
Initial
(%)
Updated
(%)
DET
(%)
BAR
(%)
1 A2 562.6 0.5 �0.5 2.6 �0.6 1.3 �0.1 4.2 0.5 2.1 6.5
2 A1 603.6 3.3 2.1 5.4 1.7 3.9 2.4 7.0 2.7 �4.9 �1.8
3 1ST BASE 625.7 3.4 0.3 10.2 1.2 4.9 0.7 10.4 0.6 3.9 �16.8
4 2ND BASE 706.1 7.0 3.0 14.6 3.3 8.2 2.8 15.3 2.7 5.5 �8.8
5 B1 739.8 4.7 0.9 10.3 �0.2 5.3 0.1 11.5 0.5 4.1 �9.0
6 B2 762.1 4.2 0.5 10.0 �0.2 5.0 0.0 11.0 0.2 � �8.9
7 3RD BASE 793.6 4.3 0.6 11.9 1.5 5.6 0.7 12.2 0.4 2.7 �10.2
8 A3 889.2 �1.7 �2.0 0.0 �1.0 �0.7 �1.1 0.5 �0.7 �6.3 �19.9
9 4TH BASE 910.2 �0.3 �3.6 7.0 �2.3 1.0 �3.4 7.0 �3.7 �1.3 �13.0
10 1ST BEND. 1000.0 1.3 �0.6 1.7 �0.5 1.9 �0.7 2.4 0.1 �2.6 �14.6
Average % error 3.1 1.4 7.4 1.3 3.8 1.2 8.1 1.2 3.7 10.9
Table 12
SH2 updating: natural frequency errors in the initial and updated FE models
Mode Description Experimental
(Hz)
CW model AC model SA model BR model Initial models
Initial
(%)
Updated
(%)
Initial
(%)
Updated
(%)
Initial
(%)
Updated
(%)
Initial
(%)
Updated
(%)
DET
(%)
BAR
(%)
1 A2 569.3 �1.2 �1.8 0.9 �1.9 �0.5 �1.3 2.4 �0.7 0.7 4.2
2 A1 623.6 �1.0 �1.6 1.3 �2.0 �0.4 �1.3 2.6 �1.0 � �5.9
3 1ST BASE 630.9 2.8 1.2 10.0 2.0 4.3 1.7 9.8 1.6 0.7 �17.2
4 2ND BASE 718.1 5.2 3.1 13.1 3.3 6.4 3.0 13.3 3.0 3.6 �10.3
5 B1 738.7 4.8 2.7 10.6 1.4 5.3 1.8 11.5 2.3 �1.3 �8.9
6 B2 774.5 2.5 0.6 8.5 �0.4 3.2 �0.1 9.2 0.3 �0.3 �10.5
7 3RD BASE 837.8 �1.2 �3.1 6.5 �2.4 0.0 �2.9 6.3 �3.1 �2.4 �14.9
8 A3 895.2 �2.2 �2.4 �0.5 �1.5 �1.3 �1.4 �0.1 �1.1 �11.3 �20.2
9 4TH BASE 914.8 �0.8 �2.5 7.0 �1.5 0.5 �2.3 6.5 �2.5 �2.1 �13.4
10 1ST BEND. 1010.0 0.3 �0.6 1.0 �0.6 0.9 �1.0 1.4 �0.2 �3.6 �15.3
Average % error 2.2 2.0 5.9 1.7 2.3 1.7 6.3 1.6 2.9 12.1
M. Palmonella et al. / Computers and Structures 83 (2005) 648–661 659
parameters that significantly influence the in-plane stiff-
ness and thus the in-plane stiffness is already sufficiently
well simulated by these models without the need for
updating. The spot elements are short and wide, and
Table 13
SH3 updating: natural frequency errors in the initial and updated FE models
Mode Description Experimental
(Hz)
CW model AC model SA model BR model Initial models
Initial
(%)
Updated
(%)
Initial
(%)
Updated
(%)
Initial
(%)
Updated
(%)
Initial
(%)
Updated
(%)
DET
(%)
BAR
(%)
1 A2 554.3 1.9 0.6 4.1 0.7 2.7 1.4 5.6 2.0 3.9 7.2
2 A1 600.6 3.0 1.6 5.4 1.4 3.7 2.2 6.7 2.6 – �2.1
3 1ST BASE 623.0 4.6 0.9 11.9 2.0 6.1 2.1 11.7 2.0 2.3 �15.6
4 2ND BASE 709.1 6.7 2.1 14.7 2.7 7.9 2.9 15.0 2.8 5.1 �9.0
5 B1 757.9 2.2 �2.1 7.9 �2.9 2.8 �1.9 8.9 �1.7 �3.7 �11.2
6 B2 787.6 0.9 �3.2 6.9 �3.7 1.6 �2.9 7.5 �2.7 �1.8 �11.9
7 3RD BASE 812.6 2.0 �2.2 10.0 �1.0 3.2 �1.2 9.7 �1.4 0.8 �12.2
8 A3 887.7 �0.6 �0.9 1.2 0.2 0.4 0.1 1.6 0.5 �9.9 �18.8
9 1ST BEND. 1000.0 1.7 �0.2 2.4 �0.5 2.2 �0.2 2.9 0.7 �2.4 �14.1
Average % error 2.6 1.5 7.2 1.7 3.4 1.7 7.7 1.8 3.3 11.3
660 M. Palmonella et al. / Computers and Structures 83 (2005) 648–661
therefore very stiff, just as the spot weld nuggets are in
reality.
Tables 8–10 show that the first natural frequency of
the DH structures has the greatest error in the initial
models, in particular for the AC model where the error
reaches 7.1%. For the SA model the second natural fre-
quency is the least accurate, predicted in the DH3 struc-
ture with an error of 4.6%. The error in these natural
frequencies is considerably reduced by updating using
the PE parameter for the CW and AC models, the BE
parameter for the SA model and the SD parameter for
the BR model. The DET model is very inaccurate in
the prediction of the first and second natural frequen-
cies. The error in the first natural frequency reaches al-
most 11% for all of the three DH structures. The BAR
element is inaccurate for the whole frequency range con-
sidered, and the average error for the DH3 structure
reaches 10.5%. These models have not been updated
due to the absence of sensitive parameters.
Tables 11–13 show that the initial models of the SH
structures have large errors in the modes that involve
the flat plate motion alone (i.e. modes 3, 4, 5, 6, 7 and
9). The error reaches 15% for the fourth mode for the
AC and BR models, while for the CW and SA models
the error is always less than 8%. The updating process
has produced very good results, and has greatly improved
the initial models. The average error in the AC model for
the SH1 structure, for example, has been reduced from
7.4% to 1.3%, using just two parameters. The average
error for the DET model reaches 3.7% with a peak of
11% in the eighth mode. The BAR model is very inaccu-
rate, with an average error of 12% for the SH2 structure.
6. Conclusions
In this paper a review of the spot weld models present
in the literature has been given and six of these models
have been updated, including the industry standard
CWELD and ACM2 models. The results show that
the detailed or the single beam models are relatively
inaccurate, mainly because they cannot be updated sat-
isfactorily using material or geometrical parameters.
The other models reach similar level of accuracy after
updating, as they are all able to approximate the local
stiffness due to the spot weld. The results achieved have
given a good insight on the stiffness characteristics of the
models.
Acknowledgement
The authors acknowledge the support of the Engi-
neering and Physical Sciences Research Council through
grants GR/R34936 and GR/R26818. Prof. Friswell
acknowledges the support of a Royal Society-Wolfson
Research Merit Award.
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