Tolerance Analysis of
Compliant Metal Plate
Assemblies Considering
Welding Distortion
Hyun Chung, Ph.D.
Postdoctoral Research Associate
Dept. of Naval Architecture and Marine Engineering
University of Michigan
August 28 2006
Overview




Motivation and Perspective
Research Objective
Background
Current Research






Inherent strain method
Mechanistic variation simulation
Three-bar model
Modified method of influence coefficients
Design Implication
Conclusion and Future Work Recommendations
Motivation and Perspective (1)

Ship block construction processes
 Thick
metal plates ( 10mm~30mm )
 Compliant behavior due to large sizes
 Required accuracy is about 1/100,000
 Joined primarily by heat-flux welding that causes
significant distortions
 Intermediate products that fail to meet the tolerance
requirements are not scrapped, but reworked
 Dimensional quality of intermediate products not only
affects final product quality but also productivity of
shipyard
Motivation and Perspective (2)




Excessive variation due to warpage, shrinkage, etc may
lead to tolerance stack-up that results in increased
assembly stresses
Traditional tolerance analysis usually relies upon a rigid
body assumption, which cannot explain compliant
characteristics of metal plate assembly
Various research work on tolerance analysis and tolerance
synthesis for automotive body and machining production
[S.C.Liu and S.J.Hu 1997]
Assembly sequence can positively affect the welding
distortions during assembly process without any mitigation
processes [ J.K.Roh and J.G.Shin 1999]
Motivation and Perspective (3)

Statistically predict dimensional quality of ship
block assemblies
 Non-nominal
plate geometry
 Compliant metal plates
 Welding distortion
 Fast and efficient calculation
Motivation and Perspective (4)

Ship block construction processes (cont.)
 Research
has provided a means to determine statistical
variation in ship block assembly including distortion of
welded structure in minutes instead of hours with equivalent
accuracy compared to current methods.
 Usually requires many runs and table below shows research
benefit per run
Modified Method of
Method
Direct Monte Carlo
Influence Coefficient
Average distortion
0.003398
0.003746
0.0007
0.0004
No. of FEM runs
1000
3
Real CPU time (sec)
12082
25
Standard deviation of distortion
Motivation and Perspective (5)

Different assembly sequences make different dimensional
qualities


Welding distortions are heavily affected by structural and thermal
boundary conditions
Different sequences make different boundary conditions
Motivation and Perspective (6)
Research Objectives (1)

Combination of tolerance analysis and finite
element analysis to develop compliant assembly
model of ship block construction
 Fast
and efficient calculation
 The effect of heat-flux welding deformation and
residual stresses considered in variation simulation for
compliant assemblies
 Identification of factors that control the deformation of
non-nominal plate assemblies after being welded
Research Objectives (2)

Validation of compliant assembly model through ship
block assembly example
Background (1)
Background (2)

Variation simulation
 Traditional variation analyses
 Mechanistic variation analysis
 Method

of Influential Coefficients
Welding distortion prediction
 Simplified
load
 Inherent strain method
 3D thermo-elasto-plastic FEA

Statistical representation of plate part variations
 B-Spline
Beam/Plate element
Background (3)

Traditional variation analysis
 Worst


case [Greenwood and Chase 1988]
Assumes that all parts are built to its extreme values
Too restrictive
 Statistical analysis


[Chase and Parkinson 1991]
Variations of parts are specified as statistical distribution
Root Sum Square (RSS)
 Monte-Carlo simulation
[Craig 1988, Doepker and
Nies 1989, Early and Thomson 1989]


Straightforward
Computationally expensive
Background (4)

Traditional variation analysis (cont.)
 Individual parts are considered as rigid bodies
 Aggregate behaviors are determined by geometric
and/or kinematic relations
 Assembly sequence does not affect the dimensional
variations of final product

Mechanistic variation simulation [S.C.Liu and S.J.Hu
1995]
 Compliant sheet metal assemblies
 Sequence of assembly can affect the tolerance of final
product [ Liu, Hu and Woo 1995]
Background (5)

Method of Influence Coefficients as the more
efficient method of calculating variations due to
compliant assemblies [Liu and Hu 1995]
 Statistically combining
the sensitivity matrix with root
sum square method to determine mean and standard
deviation, without Monte-Carlo simulation
 Use of sensitivity matrix to interpret geometrical
imperfection to equivalent external forces
 Applied only to simple geometries
 Valid only in elastic deformation range
 Welding distortions not considered
Background (6)

Welding deformation prediction
 Simplified load method
 Simple rule-of-thumb based on previous production data
 Only applicable to existing cases
 Cannot predict transient shape change
 Numerical
simulation method (Inherent strain/load
method)


Assumption of inherent strain distribution
Cannot predict transient shape change
 3-Dimensional thermo-elasto-plastic FEA
 Computationally expensive
 Can predict transient shape, temperature, stress/strain
distribution change
Background (7)

Statistical representation of plate part variations
 Random


Good for plate gap modeling
Not applicable to plate shape
 Finite


Element Mesh
Straightforward
Cannot accurately represent slope and curvature
 B-Spline


Bézier Curve [Merkley K., 1998]
Beam/Plate Element
B-Spline shape functions
Not applicable to statistical shapes
Current Research (1)
Current Research (2)
Current Research (3)
Current Research
Inherent Strain Method (1)





Originally developed to predict residual stress within
structure [Ueda, 1975]
Equivalent to ‘Eigenstrain’ [Mura, 1982]
Hybrid method (partially analytic, partially experimental)
Inelastic and non-compatible strain that accounts for
plastic deformation of the body
Defined as
dS  dS
*
 
2
dS 0
0
Inherent Strain Method (2)


What are the stress/strain states when the temperature is
increased by 1 degree given that the individual bars have
different thermal expansion coefficients?
Sequence


Stress free initial state
 Stress free state after unit temperature increase
 Final state
Total strain = elastic strain + inherent strain
  e *

Residual stress
  E (   * )  E e
Inherent Strain Method (3)

Three-bar model for welding distortion
prediction




Welding distortion is caused by highly nonuniform temperature distribution in the welded
region
Only middle bar undergoes temperature change
Elastic-perfect plastic material assumed
Stress history in the middle bar




OA: elastic compression
AB: plastic compression
BC: elastic tension
CD: plastic tension
Inherent Strain Method (4)
Inherent Strain Method (5)
Inherent Strain Method (6)
Inherent Strain Method (7)

Based on three-bar model, welding
distortion is only dependant on



Material
Maximum temperature
Degree of restraints,


KS
KW  K S
DOR is determined by


Plate thickness
Welding type


Butt or fillet welding
Welding conditions
Inherent Strain Method (8)

Equivalent loading based on Inherent
Strain [Lee, 2002]

1
fy 
bw
my 
1
bw
Equivalent loads are forces and moments
relevant to the deformation modes of welded
plates, which are the integration of inherent
strain distribution
1
E

dA


bw
At
*
y
0
 
bw
h 0
E *y dydz
1
* h
E

(

z
)
dA

 y2
bw
At
Fx   E x* dA  
0

bw
h 0
0
 
bw
h 0
h
E *y (  z )dydz
2
E x* dydz
At
0 bw
h
h
M x   E x* (  z )dA    E x* (  z )dydz
h 0
2
2
At
Inherent Strain Method (9)

Compared to 3D FEA result


~1/500 computation time
Relatively high accuracy

Longer weld line results in higher accuracy
Current Research
Mechanistic Variation Simulation (1)

Method of Influence Coefficients [Liu and
Hu, 1995]



Elastic deformation only
No welding distortion (spot welding)
Linear mapping [S ] between part variation {V }
and assembly deviation {U }
Fw  [ K w ]{U w}  [ Ku ]{Vu }  Fu
{U w }  [ K w ] 1 [ K u ]{Vu }  [ S wu ]{Vu }

The sensitivity matrix
unit force response
[S ]
is obtained by
Mechanistic Variation Simulation (2)

Method of Influence Coefficients [Liu and Hu, 1995]





Small variations are assumed
Unit force is applied at j-th source of variation
Deformation under unit force is calculated by FEM
Obtain the influence of j-th variation to N-th node
Spring-back is calculated by FEM
 V1  N  c1 j 
 c11  c1N   F1 
 
 
 
{V }         F j           [C ]{F }
V  j 1 c 
c N 1  c NN   FN 
 N
 Nj 
[ F ]  [C ]1{V }  [ K ]{V }
 U1  N  s1 j 
 s11  s1N   V1 
 
 
 
{U }         V j           [ S ]{V }
U  j 1 s 
 sM 1  sMN  VN 
 M
 Mj 
Current Research
How part variation
affects welding
distortion?
How welding
distortion affects
variation
simulation?
Three Bar Models (1)

Clamped-clamped 3-bar model with initial
part variation




The structure is clamped to its nominal length
and undergoes temperature changes, and then
clamping is released
Displacement is confined
Model for the region adjacent to clamping
devices
Spring-clamped 3-bar model with initial
variation



External force is applied to the structure and the
structure undergoes temperature changes, and
then the external force is removed
Clamping force is constant
Model for the region far from clamping devices
Three Bar Models (2)

Procedure
 Step1:
initial variation
 Step2: clamping to nominal position

Elastic deformation
 Step3:
welding
Elastic state during heating
 Plastic state during heating
 Elastic state during cooling
 Plastic state during cooling

 Step4:

releasing clampings
Elastic deformation
Three Bar Models (3)
Three Bar Models (4)
Three Bar Models (5)
Three Bar Models (6)
Three Bar Models (7)
Three Bar Models (8)
Modified Variation Simulation (1)

Modified method of influence coefficients
[ K w ]{U w }  [ K u ]{Vu }  Fd 
{U w }  [ K w ] 1 [ K u ]{Vu }  [ K w ] 1 Fd   [ S wu ]{Vu }  [ K w ] 1 Fd 
{U w}  [Swu ]{Vu }  D

Initial part variation affects welding distortion as it slightly changes
degree of restraints

Due to the Linearity between the residual strain and initial variation, the
welding distortion term of non-nominal parts can be replace as the welding
distortion term of nominal parts
Dvar   D   AVu 

The differences could be absorbed in the sensitivity matrix based on the
value found on the inherent strain chart
Current Research
Plate variation representation (1)

Cubic B-Spline finite element


New basis function that require two more control points is adopted
Equally spaced splines
 1
3
 6h3 ( x  xi  2 )

 1 {h3  3h 2 ( x  x )  3h( x  x ) 2  3( x  x )3}
i 1
i 1
i 1
 6h3

i ( x )   1 3
2
2
3
 6h3 {h  3h ( xi 1  x)  3h( xi 1  x)  3( xi 1  x) }

 1 ( x  x )3
 6h3 i  2
0

xi  2  x  xi 1
xi 1  x  xi
xi  x  xi 1
xi 1  x  xi  2
otherwise
Plate variation representation (2)

Cubic B-Spline plate element

Displacement
w
m 1 n 1
 c
j 1 i 1
where
T
 s   c1
c j  c1, j
c0, j
i ( x) j ( y)   N  s 
c0 c1
T
cn 1, j 
   1
0
n1 
    1
0
 m1 
 N       
ij
T
T
cm1 cm
T
T
cm1 
T
T
Plate variation representation (3)

Cubic B-Spline plate element

Strain-displacement relationship
 2w 
 2 
 x        
  2 w  
     2           s    B  s 
 y   2     

 2w  
2

 xy 

Strain-stress relationship

1 
3

Et
 1
  
12(1  2 ) 
0 0


0 

0    D     D  B  s 
(1  ) 

2 
Plate variation representation (4)

Cubic B-Spline plate element

Stiffness matrix with respect to Spline parameters
 K s   y x  B   D  B  dxdy

ym
xn
0
0
T
Displacement relationship
wi , j 
1
{16ci , j  4(ci , j 1  ci 1, j  ci , j 1  ci 1, j )  (ci 1, j 1  ci 1, j 1  ci 1, j 1  ci 1, j 1 )}
36
 w 
1
 w 
1
ix, j    
{4(ci 1, j  ci 1, j )  (ci 1, j 1  ci 1, j 1  ci 1, j 1  ci 1, j 1 )}
 x i , j 12hx
iy, j    
{4(ci , j 1  ci , j 1 )  (ci 1, j 1  ci 1, j 1  ci 1, j 1  ci 1, j 1 )}

y
12
h
 i , j
y

xy
i, j
 2w 
1


(ci 1, j 1  ci 1, j 1  ci 1, j 1  ci 1, j 1 )


x

y
4
h
h

i , j
x y
Plate variation representation (5)

Cubic B-Spline plate element

Stiffness matrix with respect to global coordinate system
 K   T   K s T 
T
where
1
K 
Tc i , j
is (n  3)(m  3)  (n  3)(m  3) square matrix
 1
 36

 1
 12hx

 1
 12h
y

 1
 4h h
 x y
4
36
0

1
36
1
12hx
4
36
4

12hx
16
36
0
4
36
4
12hx
1
36
1

12hx
4
36
1
12hy
4
12hy
4
12hy

1
12hy
0
0
0
0

1
4hx hy
0
0
0

1
4hx hy
0
0






1 
12hy 

1 
4hx hy 
1
36
1
12hx
Plate variation representation (6)

Cubic B-Spline plate element example

Plate deflection under concentrated load
(C-C-C-C) and (S-S-S-S)
Plate variation representation (7)

Cubic B-Spline plate element example
Varmean   wA  0.02 wB  0.01 wC  0.01 wD  0.01
T
Varstdev   0.01 0.01 0.01 0.01
T
Design Implication
Design Implication (1)
Method
Mean value of center distortion
Direct Monte Carlo
Modified Method of Influence Coefficient
0.003398
0.003746
0.0007
0.0004
No. of FEM runs
1000
3
Real CPU time (sec)
12082
25
Standard deviation of center distortion
Design Implication (2)
Design Implication (3)
Design Implication (4)
Design Implication (5)
Design Implication (6)
Design Implication (7)
ANSYS
0.0022
0.0013665
0.0014263
0.00135
0.0019
Modified Method of
Influence Coefficients
0.0028
0.001312
0.0014762
0.001312
0.0028
Conclusions




Inherent strain concept based on three-bar model is well
applied to the prediction of welding distortion since it
captures the core mechanism of welding distortion
generation
Welding distortion is affected by initial part variation as
the force required to clamp the part into their nominal
position is absorbed in the degree of restraints
The modified method of influence coefficients can
predict ship block assembly variation including welding
distortion with equivalent accuracy with FEA in much
less computation time
Statistical B-Spline plate element is developed and it can
be effectively applied to variation simulation when slope
and curvature of final assembly are important
Future Research

Application to free-from plate assemblies



Optimal assembly sequence determination


Including welding distortion mitigation techniques
Further development of inherent strain method


Current equivalent loading method based on inherent
strain only works for flat plate assemblies
Higher order B-Spline element is required
3D Poisson effects
Covariance representation


Material covariance
Geometrical covariance
Thank you for your attention
Hyun Chung

Research Interest

Tolerance Analysis/Optimization for compliant metal plate assembly
considering welding distortion




Simulation based tools to support decision making in ship
production systems




Object-oriented modeling of production systems
Artificial intelligence tools for production planning
Simulation based manufacturing facility design
Education



3D FEM welding distortion simulation and Equivalent loading method
based on eigenstrain concept
Statistical tolerance representation of 3D surface using B-Spline FEM
Sensitivity matrix model for compliant tolerance analysis
2000.8
2006.3
MSE in NAME, University of Michigan
Ph.D in NAME, University of Michigan
Publications
Simulation-based Performance Improvement for Shipbuilding
Processes, JSP Vol. 22, No. 2, 2006
 A Generic Shipyard Computer Model Development – A Tool for
Design for Production, JSP, Vol. 16, No. 3, 2000.
