Finite element models of spot welds in structural dynamics

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.

mailto:[email protected]

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


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


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,



• SH1: TH4 plus FP1,



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


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



• 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.


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


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


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


(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


(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


(Hz)


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



(Hz)


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


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



(Hz)


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


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.

References

[1] Salvini P, Scardecchia E, Demofonti G. A procedure for

fatigue life prediction of spot welded joints. Fatigue Fract

Eng Mater 1997;20(8):1117–28.

[2] Salvini P, Scardecchia E, Vivio F. Fatigue life predic-

tion on complex spot welded joints. SAE 1997 Trans, J

Mater Manuf 1997;106, paper no. 971744:967–975

[section 5].

[3] Radaj D. Stress singularity, notch stress and structural

stress at spot welded joints. Eng Fract Mech 1989;34(2):

495–506.

[4] Radaj D, Sonsino CM. Fatigue assessment of welded joints

by local approaches. Cambridge: Abington Publishing;

1998.

[5] Fermer M, Svensson H. Industrial experiences of FE-based

fatigue life predictions of welded automotive structures.

Fatigue Fract Eng Mater Struct 2001;24:489–500.

[6] Fermer M, McInally G, Sandin G. Fatigue life analysis of

Volvo S80 Bi-Fuel. In: 1st worldwide MSC automotive

conf, Munich, Germany, 20–22 September 1999.


[7] Heyes P, Fermer M. A spot-weld fatigue analysis module

in the MSC/Fatigue environment. In: MSC 1996 world

users� conf proc, vol. III. paper no. 30. June 1996.

[8] Henrysson HF. Fatigue life predictions of spot welds using

coarse FE meshes. Fatigue Fract Eng Mater Struct 2000;

23:737–46.

[9] Zhang G, Richter B. A new approach to the numerical

fatigue-life prediction of spot-welded structures. Fatigue

Fract Eng Mater Struct 2000;23:499–508.

[10] Chang B, Shi Y, Dong S. Studies on a computational

model and the stresses field characteristics of weld-bonded

joints for a car body steel sheet. J Mater Process Technol

2000;100:171–8.

[11] Chang B, Shi Y, Dong S. Comparative studies on stresses

in weld-bonded, spot-welded and adhesive-bonded joints. J

Mater Process Technol 1999;87:230–6.

[12] Deng X, Chen W, Shi G. Three-dimensional finite elements

analysis of the mechanical behaviour of spot welds. Finite

Elem Anal Des 2000;35:17–39.

[13] Radaj D, Zhang S. Geometrically nonlinear behaviour of

spot welded joints in tensile and compressive shear loading.

Eng Fract Mech 1995;51(2):281–94.

[14] Zhang Y, Taylor D. Fatigue life prediction of spot welded

components. In: Proc of the seventh int fatigue conf, 8–12

June 1999, Beijing.

[15] Chen W, Deng X. Performance of shell elements in

modelling spot-welded joints. Finite Elem Anal Des 2000;

35:41–57.

[16] Heiserer D, Charging M, Sielaft J. High performance,

process oriented, weld spot approach. In: 1st MSC

worldwide automotive user conf, Munich, Germany,

September 1999.

[17] Fang J, Hoff C, Holman B, Mueller F, Wallerstein D. Weld

modelling with MSC. Nastran. In: 2nd MSC worldwide

automotive user conf, Dearborn, MI, October 2000.

[18] Pal K, Cronin DL. Static and dynamic characteristics of

spot welded sheet metal beams. J Eng Indus—Trans

ASME 1995;117(3):316–22.

[19] Rupp A, Storzel K, Grubisic V. Computer aided dimen-

sioning of spot-welded automotive structures. In: SAE

technical paper 950711, Int congress and exposition,

Detroit (USA), 1995.

[20] Backhans J, Cedas A. A finite element model of spot welds

between non-congruent shell meshes—calculation of stres-

ses for fatigue life prediction. MSc thesis, Analysis report,

93841-2000-106. Volvo Car Corporation, Gothenburg,

Sweden, 2000.

[21] Salvini P, Vivio F, Vullo V. A spot weld finite element for

structural modelling. Int J Fatigue 2000;22:645–56.

[22] Di Fant-Jaeckels H, Galtier A. Fatigue life prediction

model for spot welded structures. In: Fatigue design 1998

symp, vol. I. Espoo (Finland), May 1998.

[23] Zhang Y, Taylor D. Optimisation of spot-welded struc-

tures. Fin Elem Anal Des 2001;37:1013–22.

[24] Zhang S. Recovery of notch stress and stress intensity

factors in finite element modelling of spot welds. In: Proc

of Nafems world congress 99, Newport (USA), April 1999.

[25] Vivio F, Ferrari G, Salvini P, Vullo V. Enforcing of an

analytical solution of spot welds into finite elements

analysis for fatigue life estimation. Int J Comput Appl

Technol 2002;15:218–29.

[26] Friswell MI, Mottershead JE. Finite element model

updating in structural dynamics. Kluwer Academic Pub-

lisher; 1995.

[27] Komzsik L. MSC/NASTRAN numerical methods user�sguide. The MacNeal-Schwendler Corporation, 1990.

[28] Palmonella M. Improving finite element models of spot

welds in structural dynamics. PhD thesis, University of

Wales Swansea, 2003.

[29] Ahmadian H, Gladwell GML, Ismail F. Parameter selec-

tion strategies in finite element model updating. J Vib

Acoust 1997;119:37–45.

[30] Friswell MI, Mottershead JE, Ahmadian H. Finite element

model updating using experimental test data: parameteri-

sation and regularisation. Trans R Soc Lond, Ser A 2001;

359:169–86.

Documents

Finite element models of spot welds in structural dynamics