CRASH BEHAVIOR OF THREE DIMENSIONAL
THIN-WALLED STRUCTURES
UNDER COMBINED LOADING
by
Heung-Soo Kim
B.S. Mechanical Design and Production Engineering, Seoul National University, 1994
M.S. Mechanical Design and Production Engineering, Seoul National University, 1996
Submitted to the Department of Ocean Engineering
in Partial Fulfillment of the Requirement for the Degree of
Doctor of Philosophy in Applied Mechanics
at the
MASSACHUSETTS INSTITUE OF TECHNOLOGY
June 2001
@ Massachusetts Institute of Technology 2001. All rights ri
IsUT
HUSETTS INSTITUTE
OF TECHNOLOGY
JUL 11 ?001
LIBRARIES
BARKER
Author .............
Heung-Soo Kim
April 2001
Certified by ..
/
'Tomasz Wierzbicki
Professor of Applied Mechanics
Thesis Supervisor
Accepted by .........
Henrik Schmidt
Professor of Ocean Engineering
Chairman, Department Committee on Graduate Studies
1
CRASH BEHAVIOR OF THREE DIMENSIONAL
THIN-WALLED STRUCTURES UNDER COMBINED LOADING
Heung-Soo Kim
Submitted to the Department of Ocean Engineering
in April 2001, in partial fulfillment of the requirement for the degree of
Doctor of Philosophy in Applied Mechanics
ABSTRACT
For the weight efficient and crashworthy design of the structural body of a transportation
system, a thorough understanding of crushing behavior of thin-walled structural members
such as spot-welded sheet metal beams or extruded aluminum beams must be gained. In
the present thesis, the complex crushing process of three-dimensional thin-walled
structures subject to combined loading is solved analytically and numerically. Also,
several new design concepts of strengthening "S" shaped frame with regard to weight
efficiency and energy absorption are proposed.
The mechanics of biaxial bending collapse and the collapse under combined bending
and compression of thin-walled prismatic member are formulated and initial and
subsequent shrinking interaction curves between the loading components are constructed.
All the analytical derivations show close correlations with the results of the
accompanying finite element analysis. Based on these two complex crushing
mechanisms, the analytical derivation of the crushing resistance of three-dimensional "S"
shaped frame is presented.
Extensive study on the strengthening of the three-dimensional "S" shaped frame is
performed with two types of internal reinforcing member, diagonally positioned sheet
metal stiffener and ultralight metallic foam-filler. The optimization process involving
varying the cross-sectional shape and the type of reinforcing member for both aluminumextruded member and spot-welded hat-type cross-section member is developed. Using
the analytical closed form expression of the crushing force of "S" shaped frame, the
optimization process was performed based on Sequential Quadratic Programming. As a
more realistic application, a front side rail and subframe structure of a mid size passenger
car is analyzed. The combinational optimization process of "Design of Experiment" and
"Response Surface Method" is carried out with the objective of weight minimization
while maintaining the same or higher level of crash energy absorption. Both methods of
internal reinforcement show high increase in the energy absorption and weight efficiency.
The gain in terms of the specific energy absorption varies from 37% to 267% depending
on the method. The proposed theoretical understanding and the design methodologies
could be used as crash oriented early-stage component design tools.
Thesis Supervisor: Tomasz Wierzbicki
Title : Professor of Applied Mechanics
2
Acknowledgements
First and foremost, I would like to express my deepest gratitude to my advisor, Professor
Tomasz Wierzbicki. His guidance, encouragement, and inspiration throughout my studies
were invaluable. Not only did I earn my doctoral degree from MIT, but I also obtained an
example from him for how I should live my life. Furthermore, I would like to thank the
members of my doctoral thesis committee, Professor Mary Boyce and Professor Nicholas
Patrikalakis, for their comments and advice in completing this thesis. I am also indebted
to Professor Frank McClintock for his valuable comments on my research.
I would like to express my gratitude to the Safety Optimization and Robustness Group at
Ford Motor Company - especially Dr. Ren-Jye Yang and Mr. Cheng-Ho Tho - for their
support and kindness during my time there as a summer intern and for their continued
assistance in the optimization part of my work. Also, I greatly appreciate the financial
support provided by the Joint MIT/Industry Consortium on the Ultralight Metal Body
Project.
Many thanks are due to the members of the MIT Impact & Crashworthiness Laboratory. I
have been very fortunate to work with such intelligent and cooperative members. In
particular, I would like to thank Dr. Sigit Santosa, Dr. Weigang Chen, Dr. Mulalo
Doyoyo, Dr. Osamu Muragishi, Mr. Yingbin Bao, and Mr. Dirk Mohr.
I am also indebted to my former group members in the Advanced Machine Element
Design Laboratory at Seoul National University. I will never forget their willingness to
assist with my research and constant encouragement. I would like to express thanks to my
former advisor, Professor Dong-Chul Han, and the Structure Analysis Group - Professor
Shin-You Kang, Dr. Shin-Hee Park, Mr. Hong-Wook Kim. My special gratitude goes to
Mr. In-Hyuk Lee at ESI Korea for his kind teaching and help with my research.
I wish to thank my friends for their continuous encouragement and for providing
refreshment throughout my studies at MIT. Special thanks are due to Dr.Shin-Suk Park,
Mr. Peter U. Park, Mr. Andy S. Kim, and Mr. Hyun-Gyu Kim. I would like to express my
deep gratitude for my dearest old friends, Mr. Jae-Hong Kim and Gaette, Mr. Jin-Wook
Jung, Mr. Won-Shik Shinn, Mr. Jae-Won Song, and Mr. Jin-Young Jung.
Finally, and most importantly, I would like to express my warmest gratitude to my
family, my father and mother, my brother and sister-in-law, and my sister and brother-inlaw. I feel profoundly indebted to them for their continuous support, encouragement, and
prayer.
3
Contents
1. Introduction ........................................................................
20
1.1
Research Objective and Scope ..............................................
22
1.2
Thesis Structure ..............................................................
24
1.3
General Form ulation ............................................................
25
1.4
Dynamic Effect on the Crash Resistance ..................................
36
2. Biaxial Bending Collapse of Thin-walled Beams ...........................
39
2.1
Formulation of the Problem and Finite Element Modeling ............... 40
2.2
Results : Moment - Rotation Angle .......................................
46
2.3
Development of Moment Interaction Curve ..............................
49
2.3.1
Initial Failure Locus ...................................................
2.3.2
Shrinking of the Failure Locus and Normality Rule ................. 56
2.3.3
Formulation of Generalized Interaction Curve with Softening .... 60
3. Crush Behavior of Thin-walled
49
Prismatic Columns under Combined
Bending and Compression .......................................................
65
3.1
Formulation of the Problem and Finite Element Modeling ............... 65
3.2
Results : Axial Force and Bending Moment Response ..................
71
3.3
Construction of the First Failure Locus ....................................
76
3.4
Shrinking of Failure Locus ....................................................
83
3.4.1
Numerical Results .....................................................
4
83
3.4.2
Analytical R esults ........................................................
4. Analysis of Crushing Response of S shaped Frame ........................
86
93
4.1
One-hinge Pin-pin Supported Model .........................................
94
4.2
Planar S Shaped Frame .....................................................
99
4.3
4.2.1
Peak Force .................................................................
100
4.2.2
Load-Deflection Relation ................................................
106
Three-Dimensional S Shaped Frame .........................................
110
4.3.1
Finite Element Modeling ................................................
110
4.3.2
Determination of the Bending Axis ....................................
112
4.3.3
Calculation of Crushing Force of Three-Dimensional S Frame ...
117
5. Strengthening of Three-Dimensional S Shaped Frame .......................
119
5.1
Effect of the Cross-Sectional
Shape on Crash Behavior of a Three
Dimensional "S" Frame ........................................................
122
5.1.1
Formulation of the Problem and Finite Element Modeling ......... 122
5.1.2
R esults .....................................................................
125
5.1.2.1
Empty Column .....................................................
125
5.1.2.2
Analytical Prediction .............................................
127
5.1.2.3
Effect of Different Reinforcing Member .......................
128
5.1.2.4 Effect of Foam Strength ..........................................
132
5.1.2.5
Partial Foam-Filling ...............................................
135
5.1.2.6
New Design of Diaphragm Type 2 ..............................
136
5
5.1.2.7
5.1.3
5.2
Effect of Imperfection .............................................
D iscussion .................................................................
145
5.2.1
Formulation of the Problem and Finite Element Modeling ......... 145
5.2 .2
Results .....................................................................
149
5.2.2.1
Empty Model ......................................................
149
5.2.2.2
Model with Inner Stiffening member ...........................
152
5.2.2.3
Specific Energy Absorption ......................................
156
5.2.2.4
Aluminum Foam-filled Model ...................................
158
5.2.2.5
Model with varying orientation of the inner stiffener ......... 160
5.2.2.6
Effect of trigger and partial internal stiffener ..................
163
5.2.2.7
Specific Energy Absorption ......................................
165
5.2.3
D iscussion .................................................................
166
Optimization of the Aluminum Foam-filled Three Dimensional "S" Shaped
Frame .............................................................................
5.4
144
Effect of the Cross-sectional Shape of Hat-type Cross-sections on Crash
Resistance of an "S"-frame ....................................................
5.3
139
16 8
5.3.1
Optimization Formulation ..............................................
5.3.2
Crash Reponses and Energy Absorptions of S-Frames ............. 169
5.3.3
Solution Algorithm ......................................................
172
5.3.4
C ase Study ................................................................
173
168
Numerical Optimization of Aluminum Foam-filled Front Side Rail ... 176
5.4.1
Overview of the Model ..................................................
178
5.4.2
Redesigning of the Trigger .............................................
180
6
5.4.3
Description of Foam-Filled Model ....................................
181
5.4.4
Numerical Optimization ................................................
183
5.4.5
Optimization Results .....................................................
187
5.4.6
Weight Efficiency ........................................................
191
6. Conclusions and Recommendations ..............................................
194
6.1
C onclusions ......................................................................
194
6.2
Future R esearch .................................................................
196
B ib liograph y ...............................................................................
197
A p p en d ix ....................................................................................
205
7
List of Figures
1-1 Simplified crash pulse of a passenger car ..............................................
23
1-2 Crash pulse of 1998 Nissan Altima [32] ..............................................
23
1-3 Conceptual cut of the zone of plastic deformation ..................................
26
1-4 Cross-section deformation mode of the plastic hinge ...............................
28
1-5 Comparison of the components of moment .........................................
28
1-6 Energy equivalent flow stress ..........................................................
31
1-7 Bending about non-principal axis .....................................................
32
1-8 The interaction curve between two normalized bending moment components ... 34
1-9 Stress distribution for combined bending and compression (fully plastic state),where
Od is the position of neutral axis .......................................................
1-10 Interaction curve between axial force and bending moment .....................
34
35
2-1 Global coordinate system and the specimen subjected to biaxial bending ......... 41
2-2 Relation between the components of the bending moment vector in the global
coordinate system and local coordinate system .....................................
42
2-3 Constraint and loading conditions .....................................................
43
2-4 Finite elem ent m esh ........................................................................
44
2-5 Stress-strain curve used for BS 1775 ERWi 1 steel ...................................
45
2-6 Resultant moment vs. rotation of a rectangular section 1 .............................
47
2-7 Resultant moment vs. rotation of a rectangular section 2 .............................
47
2-8 Resultant moment vs. rotation of a square section ..................................
47
2-9 Function ()
vs Normalized fully plastic bending moment ......................
8
49
2-10 Initial failure loci of Rectangle 1 ...................................................
50
2-11 Initial failure loci of square section ................................................
50
2-12 Distribution of axial strain at the plastic hinge resulting from independently applied
increm ental rotation ...................................................................
51
2-13 Distribution of total strain increment and tensile/compressive stresses over the beam
51
circum ference ..........................................................................
52
2-14 Example of the first yield locus .....................................................
2-15 Plots of the normalized failure locus predicted by Eq.(2-1 1) and obtained from FE
. ... 53
resu lts ................................................................................
2-16 Two deformation modes in biaxial bending collapse .............................
54
2-17 Critical orientation angle W, of analytical prediction and obtained from FE
... 55
calculation ............................................................................
2-18 Direction of increment of rotation versus loading ....................................
58
2-19 Derivation of trajectory from the interaction curve ...............................
59
2-20 Trajectory of top end node predicted analytically and FE simulation results .... 59
2-21 Distribution of axial strain at the plastic hinge resulting from independently applied
incremental rotation in the post-buckling range ..................................
60
2-22 Distribution of total strain and stress over the circumference in the post-buckling
ran g e .......................................................................................
2-23 Determination of peak moment .....................................................
60
63
2-24 Analytical prediction of uniaxial bending collapse, square section, t=0.7mm ... 63
2-25 Analytical prediction of failure locus in biaxial bending square section, t=0.7mm
........................................................................................
9
. . .. 6 4
2-26 Analytical prediction of failure locus in biaxial bending, with exact moment-rotation
characteristics in uniaxial bending ..................................................
64
3-1 Configuration of the model ............................................................
66
3-2 Displacement vector and the dimensionless yield locus ...........................
69
3-3 Relation between rate of translation and rotation ....................................
69
3-4 AA 6063 T7 stress-strain curve .......................................................
71
3-5 Moment generated by the axial force ................................................
72
3-6 The effect of axial force to the bending moment ....................................
72
3-7 Translational velocity generated in pure bending (t=1.6) ............................
73
3-8 D eform ed shapes ........................................................................
74
3-9 Axial force (b/t=50) .....................................................................
75
3-10 Bending moment (b/t=33.3) ............................................................
75
3-11 Position of the neutral axis ............................................................
76
3-12 Stress distribution in Case (a) ...........................................................
77
3-13 Stress distribution of Case (b) .......................................................
79
3-14 Analytically constructed first failure locus .........................................
81
3-15 Initial failure locus (b/t = 50) .........................................................
82
3-16 Initial failure locus (b/t = 33.3) .......................................................
82
3-17 Norm of displacement vector ..........................................................
84
3-18 Shrinking of failure locus (b/t=50) ..................................................
85
3-19 Shrinking of failure locus (b/t=33.3).................................................
85
3-20 Failure locus for circular dented tube by Wierzbicki and Suh [28] ................ 86
3-21 Discretization of column ..............................................................
10
87
3-22 Force-displacement behavior of spring .............................................
87
3-23 Deformation of springs ..................................................................
88
3-24 Possible cases for Shanley spring model ...........................................
90
3-25 Comparison between analytical prediction and numerical results (b/t=50) ...... 92
3-26 Comparison between analytical prediction and numerical results (b/t=33.3) .... 92
4-1 Idealized front end of a car ............................................................
93
4-2 One-hinge pin-pin supported model ..................................................
94
4-3 Free body diagram of the column ........................................................
96
4-4 Division of a plastic hinge into two pairs of compression and bending ............ 98
4-5 Comparison of Eq.(4-6),(4-7), and (4-17) ...........................................
99
4-6 Planar S shaped frame with three plastic hinges .......................................
100
4-7 Elastic response of S shaped frame with the external force ..........................
100
4-8 Equilibrium of force and moment for each component of S frame .................. 101
4-9 Change of the tip angle ....................................................................
102
4-10 Deformation of springs in Shanley spring model....................................
104
4-11 Peak spring force ........................................................................
105
4-12 Pairs of the compression and bending in the planar S shaped frame .............. 107
4-13 Crushing force of S frame ...............................................................
109
4-14 Three dimensional S shaped frame .....................................................
110
4-15 Finite element mesh of a three-dimensional S frame ................................
111
4-16 Deformed shape of 3-D S frame (a/b = 1.0 : square cross-section) .............
111
4-17 Deformed shape of 3-D S frame (a/b = 1.3) ..........................................
111
4-18 Deformed shape of 3-D S frame (a/b = 1.4) ..........................................
111
11
4-19 Deformed shape of 3-D S frame (a/b = 2.0) ..........................................
112
4-20 Orientation angle of bending axis ......................................................
112
4-21 Normalized fully plastic bending moment ............................................
114
4-22 Front view of the deformation ..........................................................
115
4-23 Sectional collapse modes in three plastic hinges (a/b = 2.0) ........................
116
4-24 Contribution of the torsion to the total magnitude of the sectional moment
(alb = 2 .0 ) .................................................................................
116
4-25 Comparison of crushing force of square cross-section S frame : Analytical
prediction vs FE results .................................................................
118
4-26 Comparison of crushing force of rectangular cross-section S frame : Analytical
prediction vs FE results .................................................................
118
5-1 Cross-sectional collapse at the plastic hinge under uniaxial bending ............... 119
5-2 Cross-sectional collapse at the plastic hinge under biaxial (diagonal) bending ... 119
5-3 Comparison of crushing resistance - constant M0 and decaying function of M(0)
................................................................................................
1 19
5-4 Configuration of the model. All dimensions are in mm ..............................
123
5-5 Loading condition ..........................................................................
123
5-6 Various cross-sectional shapes considered .............................................
124
5-7 Finite elem ent model ......................................................................
124
5-8 AA 6063 T7 stress-strain curve ..........................................................
125
5-9 Force response of empty member ........................................................
126
5-10 Deformed shape of empty member ....................................................
126
5-11 A sequence of cross-sectional collapse in a central part of the frame ............. 126
12
5-12 A construction paper model of square S-frame. The member can be completely
flattened out with bending deformation only without membrane tension ........ 127
5-13 Various cross-sectional types ...........................................................
128
5-14 Sectional force response .................................................................
129
5-15 E nergy absorbed ..........................................................................
129
5-16 D eform ed shapes .........................................................................
130
5-17 Comparison of Specific Energy Absorption ..........................................
131
5-18 Force response of foam-filled member ................................................
133
5-19 Energy absorbed by foam-filled member .............................................
133
5-20 Comparison of Specific Energy Absorption ..........................................
134
5-21 Deformed shapes of foam-filled model ................................................
134
5-22 Partially foam-filled model ..............................................................
135
5-23 D eform ed shapes .........................................................................
135
5-24 Force response of partially foam-filled model .......................................
136
5-25 New design of diaphragm type 2 .......................................................
137
5-26 Deformed shapes of new diaphragm type 2 ..........................................
137
5-27 Comparison of force response for two types of diaphragm 2 ....................... 138
5-28 Comparison of energy absorbed for two types of diaphragm 2 ....................
138
5-29 Imperfection type 1 .......................................................................
139
5-30 Imperfection type 2 .......................................................................
139
5-31 Deformed shapes at displacement = 150mm ..........................................
140
5-32 Force response - Imperfection type 1 and imperfection type 2 ....................
140
5-33 Deformed shapes of model with new diaphragm type 2 and imperfection type 1
13
..............................................................................................
14 1
5-34 Deformed shapes of model with new diaphragm type 2 and imperfection type 2
..............................................................................................
14 1
5-35 Force response ............................................................................
142
5-36 Energy absorbed ..........................................................................
143
5-37 Comparison of Specific Energy Absorption ..........................................
143
5-38 Configuration of the model. ( all dimensions are in mm ) ........................... 146
5-39 Loading condition ........................................................................
146
5-40 Various cross-sectional shapes considered ............................................
147
5-41 Finite elem ent model .....................................................................
148
5-42 Stress-strain curve of the material used in this study ................................
148
5-43 Force response of Type 1-4 ............................................................
150
5-44 Force response of Type 5-8 and Type A, B ..........................................
150
5-45 Energy absorbed by various cross-sections (refer to Fig.5-40) .....................
151
5-46 Deformed shapes of the various empty models .......................................
152
5-47 Force response of Type 1-4 with inner stiffener .....................................
153
5-48 Force response of Type 5-8 and Type A and B with inner stiffener .............. 153
5-49 Energy absorbed by various cross-section model with inner stiffener ............ 154
5-50 Deformed shapes of various models with inner stiffener ...........................
155
5-51 Comparison of Specific Energy Absorption: Empty model ........................
157
5-52 Comparison of Specific Energy Absorption: Model with inner stiffener ........ 157
5-53 Force response of foam-filled model ...................................................
159
5-54 Deformed shapes of aluminum foam-filled model ...................................
159
14
5-55 Varying internal stiffener ................................................................
160
5-56 Change of orientation angle c ..........................................................
161
5-57 S-frame with varying inner stiffener ...................................................
161
5-58 Imperfection type 1 .......................................................................
161
5-59 Im perfection type 2 .......................................................................
161
5-60 Force response of the member with varying inner stiffener ........................
162
5-61 Deformed shapes of the model with varying inner stiffener ........................
162
5-62 Type 6 with new design of internal stiffener and trigger ...........................
163
5-63 Force response of the various cases of Type 6 model ...............................
164
5-64 Deformed shapes of various cases of Type 6 model .................................
165
5-65 Specific energy absorption ..............................................................
166
5-66 A foam-filled square cross-section .....................................................
171
5-67 The specific energy absorptions of optimized foam-filled and empty sections .. 175
5-68 General Procedure of the Crash Optimization ........................................
177
5-69 Front side rail and subassembly structure .............................................
178
5-70 Wall force response of original front rail structure ..................................
179
5-71 Deformed shape of original model with trigger ......................................
179
5-72 Elastic buckling mode shape of the cross-section ....................................
180
5-73 Front end parts for quasi-static crushing ..............................................
180
5-74 Force response of crushing of front end parts ........................................
181
5-75 Aluminum foam-filled front side rail .................................................
182
5-76 Mechanical properties of aluminum foam (compression) ..........................
182
5-77 The approximate optimization process ................................................
187
15
5-78 Wall force response of Optimized aluminum foam-filled front side rail ......... 191
5-79 Comparison of the specific energy absorption: original design vs foam-filled
optimum designs .........................................................................
192
5-80 Relation between weight and energy absorbed .......................................
192
A-1 Case 1 : Both Compression ...............................................................
205
A-2 Case 2 : fi
-
Compression, f2 - tension, subcase 1 ....................................
206
A-3 Case 2 : fi
-
compression , f2 - tension, subcase 2 ....................................
207
16
List of Tables
2-1 Geometry of the cross sections ...........................................................
43
5-1 Table.5-1 Properties of aluminum foam used ..........................................
132
5-2 Summary of the Specific energy absorption ............................................
156
5-3 Optimum solutions for empty and foam-filled S-frames ..............................
174
5-4 Optimal LHS design matrix and FE results summary .................................
185
5-5 Design variable and design space ........................................................
188
5-6 Optimum Designs .........................................................................
188
5-7 Result summary for Optimum Design 1 ................................................
189
5-8 Result summary for Optimum Design 2 ................................................
189
5-9 Result summary for Optimum Design 3 ................................................
190
17
Nomenclature
Rate of external work
W.
Rate of energy dissipation
f,
Velocity vector
e
Strain rate
H
half-length of the local plastic fold
@k
rate of rotation vector
F
external force
M
moment
c-,
yield stress
O
energy equivalent flow stress
K
curvature
Nap
membrane force
Map
bending moment of the shell
V
volume
A
area
No
fully plastic sectional force
Mo
fully plastic bending moment
N
normalized sectional force
M
normalized bending moment
W
orientation angle of the cross-section in biaxial bending
Wcr
critical orientation angle in biaxial bending
Mres
resultant bending moment
Ms
bending moment about strong principal axis
M,
bending moment about weak principal axis
ms
normalized bending moment about strong principal axis
mW
normalized bending moment about weak principal axis
normalized peak bending moment in biaxial bending
18
Q
first moment of inertia
0
rotation in bending
M
pre-buckling bending moment
M"'
post-buckling bending moment
k
norm of the displacement component
11
rate between rate of the rotation and the rate of transration
Mp
fully plastic bending moment per unit length
EI
bending rigidity
beff
effective width
SEA
specific energy absorption
19
Chapter 1. Introduction
Responding to an increasing public awareness about safety of transportation systems such
as cars, trucks, airplanes, trains and ships, the industry is putting crashworthiness as a
high priority feature [1-5]. The body of a vehicle is considered crashworthy if it deforms
under prescribed force and preserves sufficient survival space around the occupants to
limit bodily injury during an accident [6]. Controlling the deceleration within the distance
available requires controlling forces and moments, given the mass and other constraints
of the rest of the structure. Weight efficient and crashworthy car body is composed of
thin-walled prismatic members, such as spot-welded sheet metal beams or extruded
aluminum beams. The application of the thin-walled prismatic members into the structure
of vehicle has stimulated the study on plastic collapse behavior under large axial and
bending deformations.
The crushing response of axially compressed box beams has been extensively studied in
connection with the lumped mass-spring model which was a dominant tool to calculate
the crash response of the automotive structure in 1970's and early 1980's. Efforts were
made to model the major structural components which absorb most of crash energy with
nonlinear springs. For this purpose, Ohkubo [7] developed a simplified expression for the
mean crushing force of closed-hat section members and Masanori and Funahashi [8]
presented a simple equations to calculate peak and mean crushing force of spot-welded
box column with hat-type cross-section. Mahmood and Paluszny [9] investigated the
contribution of buckling of each component column wall to the crushing resistance of
whole box column. Meng et al [10] carried out an experimental study on the axial
crushing of square tubes made of PVC, aluminum and mild steel. Abramowicz and Jones
developed an approximate theoretical prediction for the axial progressive crushing of
square and circular columns in the series of their studies [11-15].
The first
comprehensive analytical solution for axial crushing of thin-walled prismatic box
columns was published by Wierzbicki and Abramowicz [16]. The subsequent publication
of Abramowicz and Wierzbicki [17] extended this theory to the multi-coiner prismatic
20
members. A more recent work by Wierzbicki et al [18] showed a way to calculate local
distribution of stresses at a given cross-section of the column, subject to axial crushing.
Abramowicz [19] and Kecman [20] addressed the problem of a deep bending collapse of
open and closed section, respectively. Both authors followed the so-called kinematic
approach to plasticity, and the method involved a determination of a suitable folding
mechanism with stationary and/or moving plastic hinges. Cimpoeru and Murray [21]
presented empirical equations of the moment-rotation relation of a square thin-walled
tube subject to bending collapse. Similar approach was taken by Kecman et al [22]
through the numerous bending crush tests of the simplified thin-walled single cell section
members. Mahmood et al [23] developed so-called "strip" method to solve the bending
strength and collapse mode of vehicle structural members. Wierzbicki et al [24]
developed a much simpler closed-form solution for bending collapse by generating first a
numerical solution using commercial finite element code ABAQUS, analyzing the stress
and displacement fields.
Based on the results published to date, the crushing response of thin-walled member
subject to either axial crushing or bending collapse can be well predicted analytically, but
these results cannot be applied directly to the structure under combined compression and
bending, or bending with two bending axes. In real world applications, many structural
members of the transportation systems are subject to a combined loading, and do not
collapse in a planar way. Several approaches were made to attack this subject by Civil
Engineering
literature
[25,26,27],
but they did not provide the comprehensive
understanding of complex crushing mechanisms. The only well documented work on the
interaction between bending and compression was published by Wierzbicki and Suh [28].
They carried out plastic analysis of tubular members subjected to local denting under
combined loading for offshore applications and developed a simplified model of a
circular tube and constructed a failure locus on a purely theoretical basis.
Development of simplified tools for crash calculations to predict the crash response
without use of the finite element tool or experiments is the ultimate goal in the field of
21
crash mechanics. Drazetic et al [29] analyzed a planar "S" shaped frame using nonlinear
springs by dealing with axial compressive collapse and bending collapse separately. A
similar approach was taken by Kim et al [30] to calculate the crash response of full scale
automotive structure as well as the occupant behavior. Park [31] applied this method in
the crash analysis of aluminum space frame car. However, the interacting behavior of the
compression and bending or bending between two different axes was neglected in their
studies. The present author was involved in these studies during his work at the Seoul
National University in Korea before coming to MIT. Clearly, these early analyses without
the interaction effects have overpredicted the resulting crash forces. However, because no
tests were made the inaccuracies have not been detected. To develop a simplified tool for
crashworthiness analysis of complex thin-walled structures, the analytical solution of the
crushing resistance under combined loads must be derived. The design of weight efficient
and crashworthy structure of vehicle will be far more difficult without understanding of
the complex crushing behaviors under combined loading.
1.1 Research Objective and Scope
The main objective of the present thesis is to develop a theoretical basis for predicting the
crushing resistance of three-dimensional thin-walled structures subjected to combined
loading. Based on this theory, several new design concepts of strengthening threedimensional S frame with regards to weight efficiency and energy absorption are
proposed.
The theory is developed in three steps. First, the limiting cases of axial compression and
planar bending are constructed analytically and numerically to serve as reference solution
for more complicated loading cases. Then the numerical analyses using nonlinear finite
element code PAM-CRASH are carried out to learn about a nature of the complex
interaction between the components of generalized cross-sectional forces and moments.
Next, simplified solutions for the interactions between biaxial bending, and axial force
and bending moment are developed and compared to the exact numerical solutions. The
22
analytical expression of crushing resistance of three-dimensional "S" frame is derived on
those theoretical bases. Methods of strengthening of the S frame are studied extensively.
Various types of cross-section and new idea of reinforcing "S" shaped frame with
diagonally positioned internal members as well as the ultralight metallic foam-filling are
investigated. As a systematic way of the design of thin-walled frame structures, the
application of numerical optimization process into the S frame and a front side rail
structure of mid size passenger car is carried out. The optimized crashworthy design of
each structure is determined with regards to weight efficiency and energy absorption.
The generalized crash pulse of a passenger car subject to frontal collision is shown in
Fig.1-1 and compared with the crash pulse of a real passenger car (Fig.1-2) [32].
,
Deceleration
G2 ------- ---------------
G 1 ..-----L2
Time or Deformation
Fig. 1-1 Simplified crash pulse of a passenger car
7ZEnrie apke, Igd
&umprEffec
.
Elfectim Psk
. cceetion &tkm
U.
DAD0.020
U.
UPU l e iJ
.QU 0.0 010
.
.J'
; ... ......
deio nir
40
0. 60
O.AK
Time (s)
M.OO
0.120
0.140
Fig. 1-2 Crash pulse of 1998 Nissan Altima [32]
23
Mahmood and Aouadi[32] and Motsumoto et al [33] studied the relation between the
injury level of occupants and the shape of the crash pulse. It was shown that to lower the
injury level, GI should be increased and G2 lowered. Also, increasing GI yields lowering
G2. The front side rail plays the most important role in L2 range. Considering the threedimensional "S" shaped frame can be regarded as an idealized front side rail structure,
the scope of this thesis will be focused on L2 region. However, as the G1 and G2 levels
are related to each other, the results of the current study cover the overall characteristics
of the crash pulse.
1.2 Thesis Structure
The mechanics of bi-axial bending collapse will be revealed in Chapter 2. The
computational model used in this study follows the test program conducted by Brown and
Tidbury [34]. Prismatic thin-walled beams with square and rectangular cross-section are
subjected to bending collapse about two different axes. Two different loading conditions
are used :
1) Prescribed axis of bending deformation
2) Prescribed axis of bending moment
The physical tests are replicated using nonlinear finite element method. Based on the
careful observations of the finite element analysis results, the analytical expressions for
the first and subsequent failure locus are developed. The collapse mode and the normality
rule in plasticity will also be analyzed.
In Chapter 3 the initial and subsequent failure locus in combined bending and
compression is generated numerically. This is followed by analytical derivation. A square
cross-section beam with two different aspect ratios is chosen as a representative structure.
The beam is subject to combined bending and compression by changing the ratio between
the rate of prescribed translational displacement and rotational displacement. The first
failure locus is derived analytically considering the stress distribution over the cross-
24
section, which depends on the position of the neutral axis. The subsequent shrinking of
failure locus, which is expected as the deformation progresses, is constructed analytically
using the concept of "Superbeam Element" and Shanley Spring model [35,36]. All the
analytically predicted failure loci are compared with the results of finite element analysis.
The analytical derivation of the crushing resistance of three-dimensional "S" shaped
frame is presented in Chapter 4 on the basis of the results developed in the previous two
chapters. The model used in this chapter can be considered as the simplified front side
rail commonly used in the automotive structure. The computational model is developed
in three stages with increasing levels of complexity: A simple one-hinge pin-pin
supported prismatic thin-walled column with square cross-section, planar "S" shaped
frame, and finally three dimensional "S" shaped frame. In each case the finite element
analysis is carried out to verify the analytical prediction of the crushing response.
The problem of weight efficient and crashworthy design of thin-walled structures under
complex loading situation is formulated and solved in Chapter 5.
The structure
considered is the same three-dimensional "S" shaped frame. An extensive study on
strengthening of the structure is performed. Two types of internal reinforcing member are
introduced : 1) diagonally positioned sheet metal stiffener and 2) ultralight metallic foamfiller. The optimization process involves varying the cross-sectional shape and the type of
reinforcing member for both aluminum extruded member and spot-welded hat-type crosssection member. Using the analytical expression of the crushing force of S frame, the
optimization process based on "Sequential Quadratic Programming" (SQP) is carried out.
As a more realistic application, a front side rail and subassembly structure of a mid size
passenger car is analyzed, and the combinational optimization process of "Design of
Experiment" (DOE) and "Response Surface Method" (RSM) is carried out with the
objective of weight minimization while maintaining the same or higher level of crash
energy absorption.
Finally, a summary of the new results is presented and discussed in Chapter 6.
Recommendations for future studies are given.
25
1.3 General Formulation
The general expression of the equilibrium of a single structural element can be given in
the form of the global balance between the rate of energy dissipation and the rate of
external work[37].
where Wex, is the rate of external work, and
*in
is the instantaneous rate of energy
dissipated at the plastically deforming region. u is the velocity vector and t is the strain
rate resulting from
n.
It has been observed experimentally [11,13,20] that the plastic
collapse or large shape distortion of prismatic members is localized over relatively
narrow zones while the remainder of the structure undergoes a rigid body motion or
elastic deformations. The assumption of a localized zone of plastic deformation enables
one to describe the mechanics of the crushing process at the local level, which is used
throughout in this thesis.
X2
X1
M2
P
Fd
Fig. 1-3 Conceptual cut of the zone of plastic deformation
An illustration of a conceptual cut which contains the zone of plastic deformation is
shown in Fig. 1-3. Consider a section of a prismatic member of length 2H, where H is the
half-length of the local plastic fold. Inside of the two cuts, the arbitrary large
displacement takes place, and the shell is assumed to be rigid outside of the deforming
26
zone. The relative rates of translation and rotation vectors of the rigid part are noted as
u 0 and yY 0 , respectively. Thus, the rate of work done by the external force F and moment
M is expressed as,
Wext
= F60 + M@ 0
(1-2)
Or in the expanded form
Wext = F1 1i + F2
2
+F3d 3 +Mif +M 2 Vf 2 +M 3 V 3
(1-3)
The coordinate system used is shown as attached to a centroid of the conceptual cut.
Shown in Fig.1-4 is the cross-sectional deformation mode of a center cut through the
"plastic hinge" of the 3-D "S" shaped member. At the same time the relative transverse
displacement and rotations of the end sections of the plastic hinge are vanishingly small,
so that the deformation producing shear is very small.
0 =0 = 0
(1-4)
3a=
u2
At the same time the reaction shear forces F1 and F2 are non-zero from equilibrium.
However, the shear work is small and will be neglected in the present study [38,39].
The plot of three components of the bending moment (M1 , M2 , and M 3) taken from the
clamped end of 3-D "S" shaped member with square cross-section is given in Fig.1-5.
These are the results of finite element analysis using PAM-CRASH. The magnitude of
twisting moment (M1 ) is negligible compared to other moment components, thus
MI =0
(1-5)
Also, there is a negligible relative rotation on both sides if the plastic hinge.
VI = 0
(1-6)
Therefore, the contribution to the work from the shear and twisting moment are
neglected.
27
Fig. 1-4 Cross-section deformation mode of the plastic hinge
2000
M (Twisiting moment)
-- --- M2 (Bending moment)
1500
.-----
1000
500
-
E
M 3 (Bending moment)
0
. . .. . . . .. . . .. . . .. . . .. .
-500
------------------
----
--
.
-
E0
-1000
...............
-1500
-2000
0
100
50
150
200
Displacement of moving end (mm)
Fig. 1-5 Comparison of the components of moment
F2 0 = F 3 0 = M Ifl = 0
(1-7)
Consequently, the components of the velocity vector considered in this study are
t0 = ao, 0, 0
0"= to,
o2",
# 1
(1-8)
(1-9)
Accordingly, vectors of generalized loading should have the following non-vanishing
components.
F ={F, F2 , F3 }
28
(1-10)
M = {0,M 2 ,M 3}
(1-11)
where F1 = F is the axial force and M2 and M 3 are magnitudes of bending moment about
two orthogonal cross-sections.
The rate of external work is expressed as follow.
Wet
= Fa0 + M 2 V2 + M
3V3'
+J
(1-12)
p vidS
where the last term on the right-hand side of Eq.(1-12) includes both distributed and
2
or
W=
3
-
concentrated transverse loads and v is the transverse velocity, i.e. ii = I
Assuming that the rate of displacement is constant over the cross-section '= 8, the last
term on the right-hand side of Eq.(1-12) becomes,
(1-13)
Js pwdS = P$
where P is the total lateral load acting on the transverse velocity 8.
With these assumptions, the final formula for the rate of external work is
S=Fa
+M
2 2/4
+MAO + PS
(1-14)
The rate of internal energy dissipation for a general shell of volume V is defined as Eq.(115).
Wint =afl
taIdV =
I
JioatagdAdL
(1-15)
The components of the stress tensor are related by the yield condition. The Von-Mises
yield condition in plane stress takes the form [40]
O(U"
wheret
2
aai
Ca/
1 e a
-2|= 0
2
is the uniaxial yield stress.
The strain rate tl. is obtained from the associated flow rule
29
(1-16)
tafl
=,ao
(1-17)
au-a
Using Love-Kirhihoff's hypothesis,
(1-18)
' # = '6" + zka)
where z is the local through-thickness coordinate.
The rate of energy dissipation per unit area is expressed as,
Wnt per unit area = ta
r
O'a-dz + ka8 f,
O-a zdz
(1-19)
=ta/Na +ka Ma
where membrane force NO and bending moment MOg are defined as,
Na =
j
MaI =
(1-20)
adz/
(1-21)
6Aaa/zdz
Considerable simplifications can be obtained for prismatic beams for which the only
stress component is the direct stress a-= ai with the Euler-Bernoulli assumption for a
beam
E = g 0 + K 2X 2 + K 3X 3
(1-22)
and
dV = Adx
(1-23)
Eq.(1-15) for the rate of internal work can be transformed to
2H
'it =
JF tdx
0
2H
2H
+ JM 2k 2 dx + JM ksdx1
0
0
where
30
(1-24)
(1-25)
F = Jho-d'
r
M2
(1-26)
= Jhox 2d'
r
M3
(1-27)
= Jh-x3 dF
Assuming that the generalized forces are approximately constant within the plastic hinge,
the integration with respect to xi can be easily performed to give
W*
= Fai +M 2
2
+ M3
3
(1-28)
which is a special case of Eq.(1-2).
In this thesis the material is considered to be rigid-perfectly plastic with the flow stress ao
defined as an average stress over a given strain range (0, Ef), following the theory by
Abramowicz and Wierzbicki [17].
o =
(1-29)
-(e)d
to the
This is illustrated in Fig.1-6, which shows the equivalence of the product GOFa
actual value of the dissipated energy. Consequently, ao given by Eq.(1-29) is referred to
as an energy equivalent flow stress and used as a nominal value of the stress in the plastic
range in rigid perfectly plastic shells.
ef
Fig. 1-6 Energy equivalent flow stress
31
Interaction curves for solid cross-sections
When a solid prismatic member with rectangular cross-section is subjected to the bending
about non-principal axis, the general configuration is given in Fig. 1-7. This is a textbook
problem both in the elastic and plastic range [41,42]. However, a simple derivation is
presented here as an illustration with more complex cases of thin-walled members.
Zero
stress axis
b
M
zb
-a
0X
70
b
a
a
Fig.1-7 Bending about non-principal axis
From the figure the axis of the applied bending moment M makes an angle 0 with O
,while the zero-stress axis for full plasticity is assumed to make an angle cc with the
principal axis O, where, from the figure,
tan a =
b
z
a
(1-30)
-
By taking moments about O and Oy, the components M and My of the bending moment
M are given by
MX = M cosO = 2ab2ao(1_1- z2
MY = M sin6 = 2a2b '(2z)
32
(1-31)
(1-32)
Note that these expressions hold for -1
z 5 1, that is, for the value of tan cc less than
b/a. Normalizing M, and My with the fully plastic bending moment about O (M,o, =
2ab2),
M
-
1 - z2
M
M
M
(1-33)
3
2~~
- MY
M
- az
3
(1-34)
M2
(1-35)
M =
34Y
-M2
(1-36)
mY =
3(-{j2M
mY =1
32
m2
4
By eliminating parameter z,
m, =1- 3b
Similarly, for Iz|>
,
For the square cross-section member,
(1-37)
For the square cross-section member,
(1-38)
The interaction curve between two bending moments components for solid square crosssection member of Eq.(1-36) and (1-38) are plotted in Fig.1-8. The normal at the point P
is shown. The slope of the normal at P is -dmx/dmy, and from Eq.(1-30), (1-33) and (134), this slope is tan a. Thus the normal at P forms an angle a with the direction of the
axis mx. The normality rule is clearly satisfied in this case. Also note that in the above
derivation the cross-section is constant and equal to the original cross-section. The case
of collapsing section is treated in the next chapter.
33
y
1
III
x
-1
iI
SI
-11
Fig. 1-8 The interaction curve between two normalized bending moment components
As a second example consider the solid prismatic member subjected to the combined
bending and compression. The fully plastic stress distribution is illustrated as in Fig.1-9.
b
GO
d
2a.
Pd
d
Fig. 1-9 Stress distribution for combined bending and compression (fully plastic state)
,where Pd is the position of neutral axis
Then the axial force N and bending moment M are calculated as,
(1-39)
M =bd 2
(1-40)
,
N = 2fibdo- = 3N0
0(1 _g2)= (I _g2)M
34
where N and M, are fully plastic axial force and bending moment, respectively, defined
by
No = 2bdo
(1-41)
MO =bd2 O
(1-42)
N
n= N
No
(1-43)
By normalizing N and M
M
m=1-fl
2
(1-44)
Eliminating the parameter 0 between the above equations, the normalized axial force and
bending moment are simply related by,
m+n 2 =1
(1-45)
Eq.(1-45) is plotted in Fig.1-10.
N/N
0
0
-1
j
1
Fig. 1-10 Interaction curve between axial force and bending moment
35
The initial yield locus in the form of the interaction curve between the loading
components for solid cross-section is derived using classical plastic theory. In this thesis,
this approach is generalized for the thin-walled structures. Furthermore, the analysis will
be carried out up to deep collapse range, and this will place a solid basis for the
derivation of complex crushing behavior of a three dimensional structure.
1.4 Dynamic Effect on the Crash Resistance
A dynamic collapsing process in the thin-walled structures involves many interacting
effects which are not present in static collapse [43,44,83]. There are two most important
factors identified in dynamic collapse under the impact velocity in the most of crash
accidents where the strain rates are below 102 sec-. The first is the strain rate effect which
is a material property whereby the yield or flow stress is raised. The second factor is the
inertia effect developed within the structure by the rapid accelerations during the collapse
[45].
While the inertia effects were shown to be responsible for peak magnitudes of the
instantaneous resisting forces and therefore do not contribute the crash energy
dissipation, many studies have been conducted on the strain rate effect empirically or
analytically yielding the simple equations relating the dynamic crash resistance and the
static crush resistance. Ohkubo et al [7] suggested an empirical formula for dynamic load
factor in closed-hat axially compressed columns. By fitting experimental data a ratio of
dynamic to static crushing force was approximated by a straight line,
=+0.0668Vo
1d
(1-46)
PS
where Pd is the dynamic crash force, P, is the static crushing force, and Vo is the initial
impact velocity. A different empirical formula was obtained by Wimmer [46] for square
mild steel columns,
36
L=
1+ 0.07 V 0082
(-47)
PS
The Cowper-Symons equation [47] is used widely to relate the dynamic flow stress (o)
to the static flow stress (a-).
(N1Iq
d
a-
"d =I+
0
-D)
(1-48)
where t is the strain rate and D and q are material constants to be determined from the
dynamic tensile tests on the material.
Based on the strain rate sensitivity on the yield stress of material, Masanori and
Funahashi [8] derived the following equation for mild steel structure by simply applying
the Cowper-Symonds equation with one-dimensional uniform deformation assumption.
+245
Pd
P,
(1-49)
04L0.2
2.475 x10~4L
where L is the crushing distance.
Wierzbicki and Akerstr6m [48] derived the following equation with the consideration of
complex folding mechanism of axially compressed mild steel box column.
d
=1+0.11V0
7 14
"
(1-50)
P,
Most recently Jones [15] suggested the following equations of mean dynamic crushing
forces for top-hat (Eq. 1-51) and double-hat (Eq. 1-52) section members.
Pd =32.89M,(p/t)1/3
37
.3
(1-51)
Pmd = 52.20M,(P It) 1
(
6+
jf")}(1-52)
where p is the total width of the cross-section, t is the thickness, D and q are the constants
in Cowper-Symonds equation, and Mp is the fully plastic bending moment per unit
length.
For impact velocities used in crash barrier tests the dynamic correction factor is in the
range of Pd/Ps = 1.2-1.4. This of course applies to unitized steel body structures.
Aluminum alloy have no or very little strain rate sensitivity. For all practical purposes no
dynamic correction factor has to be introduced in all aluminum car bodies.
It was observed from the comparison of static and dynamic tests of crushing of thinwalled structures that the deformation patterns of sheet metal components differ little
between static and dynamic loading conditions [48]. Thus the analytical derivations
mostly on the static loading conditions obtained in this thesis could be easily applied to
the actual crash calculation for practical purpose. However, the elevation of the crushing
resistance in the dynamic cases should be considered using the equations given above.
38
Chapter 2. Biaxial Bending Collapse of Thin-walled Beams
Biaxial bending is defined as the bending about the non-principal axis. For example, a
rectangular section beam bent along the diagonal line of the cross-section is considered to
be subject to biaxial bending. Biaxial bending of cold-formed profiles has been studied in
the Civil Engineering literature with regards to stiffness, buckling, and ultimate strength
(Razzaq and William [25], Zhou and Chen [26], Liew and et al [27]). TodorovskaAzievska and Kecman [49] carried out the theoretical and experimental analysis of the
multi-axial collapse modes in rectangular section tubes, and showed that multi-axial
hinge collapse characteristics can be derived from the uniaxial (bending and/or torsion)
moment - rotation curves. The only well documented experimental study on this topic
known to the author was reported by Brown and Tidbury [34].
The most important result of this chapter is a construction of the interaction curve
between two principal bending moments. Brown and Tidbury distinguished between two
cases of loading : 1) prescribed plane of bending and 2) prescribed direction of moment
applications. Tests were run on mild steel prismatic beams with a square and rectangular
cross-section. The empirically determined failure locus will be used to validate a part of
our own numerical and analytical solutions. Brown and Tidbury tests were arranged to
measure only the resistance of the beam near the peak moment.
The "uniaxial" bending collapse of thin-walled structures was studied by several authors
[19-24]. A comprehensive analytical study of a deep bending collapse of thin-walled
beam was reported by Wierzbicki et al [24]. The approach taken was to generate first a
numerical solution (ABAQUS code was used), then analyze the stress and displacement
fields and on that basis develop a much simpler closed-form solution.
A similar strategy was adopted in the present paper. Here the nonlinear computer
program PAM-CRASH is used as a convenient tool to study a complex response of thinwalled square and rectangular beams subjected to a biaxial cantilever bending. A number
of important questions were asked in the study.
39
*
What is the peak load under the action of two mutually orthogonal bending
moments?
*
How the yield locus shrinks with hinge rotation?
*
What is the effect of the aspect ratio and loading conditions on the evolution of
the yield surface?
* How well the 'normality' rule is preserved in this problem with structural
softening?
*
How the folding mode in the hinge depends on the orientation of the bending
moment application with respect to the principal inertia axes?
The performed numerical simulation (94 computer runs 14 hours each) gave an
exhaustive answer to all of the above questions. Based on the general formulations
described in the previous chapter, simple formulas were derived and closely followed
numerical results.
2.1 Formulation of the Problem and Finite Element Modeling
Consider a thin-walled beam with a rectangular cross-section axbxt where a and b are
widths of two sides and t is the wall thickness. The length of the beam is denoted by 1,
Fig.2-1. The bottom end of the beam is fully clamped, (for example encastred in a
concrete foundation, Kecman [20]). The top end is laterally displaced and the vertical
displacement is unrestricted. In this formulation the axial force is zero. Of interest in this
study is the resisting force P and the resulting bending moments described by the
components of the bending moment vector Mi, i=1,2,3.
Two rectangular Cartesian coordinate systems were introduced in the analysis. One is the
global, laboratory coordinate system (x, y, z) in which the load is always applied parallel
to the x-axis. The other is a local coordinate system (s, w, z) aligned with the principal
moment of inertia axes of the cross-section. Referring to the moments of inertia I, and I,
40
the subscripts 's' means the 'strong' direction and 'w' the 'weak' direction. In general the
above coordinate systems are rotated with respect to one another by the angle W about the
column z axis, Fig.2-2. In the present paper V was a parameter of the orientation which
was constant in each experiment and was varied in the range of 0 <W 90 in increments
of 150. The components of the bending moments in the two coordinate systems are
related by the linear transformation
cosY
--siny
[M,]
MW
sin1f] M 1
cosqfMy
I
(2-1)
where the twisting moment M 3 = M, is the same in both systems.
z
a
y
Load
t
1
III
SM
- X
y
Fig.2-1 Global coordinate system and the specimen subjected to biaxial bending
In order to compare the present numerical calculation with the available experimental
results, the beam was loaded in the same way as in the Brown and Tidbury [34] tests.
Thus, two types of loading conditions were considered:
Loading case 1 : Prescribed plane of bending
Loading case 2 : Prescribed direction of loading, no constraints on the beam
deformations
41
In both cases the velocity boundary conditions were applied. The center of the top crosssection was pulled in the global x-direction causing the member to collapse with a
generalized plastic hinge formed at the root of the cantilever beam.
Y
Y
MX
-My
Mres
--
WMes
(a) Loading case 1
Y
Y
X
x
-
Mres
Mres
CC=J
~'
~44
(b) Loading case 2
Fig.2-2 Relation between the components of the bending moment vector in the global coordinate
system and local coordinate system
Loading case 1
In the Brown and Tidbury [34] experiments the test member HB, oriented at the desired
axis angle V was constrained by two side stays (DB and BG) connected between the
cantilever tip B, and spherical anchor-joints at D and G (refer to Fig.2-3 (a)). The anchors
were arranged to the line on the horizontal straight line DHG passing through the
encastered base H of the test piece. Hence, the axis of hinge rotation was constrained
parallel to the global y direction. In the finite element modeling, the displacement u, of
the top cross section was constrained to represent the experimental condition explained
above. The bottom of the beam was clamped. In the above loading configuration the
42
reaction force develops in the side stays and the resulting bending moment vector Mres
forms an angle o to the x axis which is different from W. (see Fig.2-2 (a))
Loading case 2
The constraint imposed on the top cross-section was removed. The load was applied in
the global x direction. Hence, the axis of moment exerted on the member was constrained
as global y direction. Consequently cc=i. This situation is explained in Fig.2-2(b).
Three different cross-sections were selected for the analysis. The specifications are given
in Table 2-1. The rectangular section 1 and a square section are chosen to compare the
results with those by Brown and Tidbury. In addition, the rectangular section 2 with
45x30 (mm) was also considered. The aspect ratio (a/b) of rectangular section ranging
from 1:1 to 2:1 covers most of profiles encountered in practical applications. Two
different thickness were considered for each case.
Table 2-1. Geometry of the cross sections
I
a
b
thickness
[mm]
[mm]
[mm]
[mm]
Rectangular section 1
306
51
25.4
1.2, 0.7
2
Rectangular section 2
270
45
30
1.2, 0.7
1.5
Square section
228
38
38
1.2, 0.7
1
alb
z
B
Y
Load
y
B
W
(a) Loading case 1
(b) Loading case 2
Fig.2-3 Constraint and loading conditions
43
Load
The geometrical model was established using the mesh generator program HYPERMESH.
The finite element model was then completed with the pre-processor PAM-GENERIS.
Actual calculations were performed on a SILICON GRAPHICS 02 workstation with R10000 processor using the explicit finite element code PAM-CRASH. The post-processor
PAM-VIEW was used for visualization.
Following Wierzbicki et al [24], the width of the expected bending hinge (the folding
wavelength 2H) can be calculated from
(2-2)
2H =.3V
where c = (a+b)/2 for a rectangular section. Using the values from Table 1, the width of
the hinge was predicted to be equal to 2H = 30mm. The length of the beam segment with
a finer mesh l should be larger than 2H. In the present analysis the localized length was
taken as 1, = 40mm. Outside the localized plastic hinge the beam is deforming elastically.
A coarse mesh with 5-8 elements per side is sufficient. The mesh density is doubled in
the localized area.
(a) Rectangle 1
(b) Rectangle 2
Fig.2-4 Finite element mesh
44
(c) square
The stress-strain curve for the mild steel beam used in the calculation is shown in Fig.2-5.
An effort was made to ensure that the above curve characterizes the material used in
Brown and Tidbury tests. However, these authors did not provide the material data other
than giving a coded name BS 1775 ERW.1I for the steel used. A material handbook was
used to recreate the stress-strain curve for this particular steel with an understanding that
considerable error could be generated when correlating numerical results with
experimental data.
300
-
EE200
-
400
100
0.00
005
0.10
0.15
020
strain
Fig.2-5 Stress-strain curve used for BS 1775 ERW11 steel
Both ends of the column were connected to a rigid body mechanism. The rigid body was
allowed to move in six degrees of freedom (6-DOF) so that it could perform rigid
rotations and translations. With the above structural configurations, warping of the end
section was not allowed. The rate of bending rotation was applied at the center of gravity
of the rigid bodies. At the clamped end all six degrees of freedom were restrained. The
treatment of the top degrees of freedom was different for the two loading cases.
Loading Case 1 : To ensure 'planar' response, the displacement in the y-direction and the
rotation in z-direction were set equal to zero, uy=O, Oz=O at the top end. The component ux
was monotonically increasing by applying a ramped velocity in x-direction.
Loading Case 2 : The constraints present in Loading case 1 were removed. The beam was
free to displace in the y-direction, which it did. As in Loading case 1, the deformation
was controlled by prescribing time variation of the component ux.
45
Repeated calculations were run by incrementing the orientation angle ig by 150 in the
intervals between 00 and 900. Seven cases were considered for rectangular sections
(W=0 0 , 150, 300, 450, 600, 750, 900). For the square section, because of symmetry,
modeling was done at 150 intervals between 00 and 450 (W=00 , 150, 30', 450), and the
results for the remaining cases were obtained by reflection about the 450 axis.
2.2 Results : Moment - Rotation Angle
Considering three cross-sectional geometries, two different loading schemes and seven
orientation of the cross-section, altogether 72 computer runs were made. Additionally, 22
computer runs were made to determine lfcr which will be discussed later in this paper. In
each case the components of the bending moment vector in the global coordinate system
were acquired at the bottom end. In each case of the geometry and loading the following
quantities were determined and plotted:
*
Components of the bending moment vector(Ms, M., Mz) versus hinge rotation 0.
They are calculated from the global components using Eq.(2-1).
*
Sequence of collapsing cross-sections.
*
View of the deformed plastic hinges.
*
Trajectory of the loaded top end of the beam (components ux, uy).
Based on the above results interaction curves were constructed between Ms and Mw
components of the bending moment.
The bending rotation angle of the beam 0 and the resultant moment Mres are defined as
follows.
sin 0 =
Mes = }M
2
+
ux+uy
U2
'
+M2= M+M2
(23
(2-3)
(2-4)
The resultant moments are plotted in Fig.2-6-8 as a function of beam rotation for each
cross-section and constraint condition.
46
1000
500
-A
. ..
=-v600
-
-
.-.
--....
y=600
z
.. .. ...-
w--w-=90o
300
200
0
..
.... ..-
10
-
20
30
=75o
......-..
........
...
100
0
=45-
.....
.........
--.
--..
...
..
. .....
...
...-.-.-..
200
=5
..........-.............
E
- - ~~ ~~~~ -.-.-
---
---- .. =300 - - -
-750
400
aD
_-9~=00
............ ....--.. ....--..
400
1=90'
-
800
-
4
W= 50
-300
-
. -..
800
W=0-
0
40
0
10
20
rotation (deg)
30
40
rotation (deg)
(a) loading case 1, t=1.2mm
(b) loading case 2, t=0.7mm
Fig.2-6 Resultant moment vs. rotation of a rectangular section 1
400
800
-W=00
300 .-
-
-..................
.............
z
-
----
- -.....................
%V=00 ----- V=150
=90-
..........-.
................ ...
.................... .....
10
20
=900
0
40
30
-
-%--300 -- -W=450
- 600 -=750
..--.....
..
-...
.
-.
..-..
..
...-...-.
----...-
200
0
0
--
E 400
.
......
-
f
..
................... ....................
0
a100
--.-...-..
-
600
-
E 200
y1o
--
=....-- - --. =4...
---w=600 ---- w=750
0
10
rotation (deg)
20
30
40
rotation (deg)
(a) loading case 1, t=0.7mm
(b) loading case 2, t=1.2mm
Fig.2-7 Resultant moment vs. rotation of a rectangular section 2
800
--W=00
----- 50
-.......
....
. ..--...
---- -- 300
---
- ------------.
400
--... -..
........
.. ..--....--
600
.
-
=450
-...-.--.--
400
z
30
40
-00
-
-..........
-----.
..
.........- -----
_
.-.---.
-.
........----.
E
....
----- V=150
-
...-....... ..... -.-
600
2
200
200
0
0
10
20
30
40
0
rotation (deg)
10
20
rotation (deg)
(a) loading case 1, t=1.2mm
(b) loading case 2, t=1.2mm
Fig.2-8 Resultant moment vs. rotation of a square section
47
The moment-rotation characteristics are in most part smooth and regular. The M-0 curve
increases up to the point of plastic buckling and then gradually decays (refer to Kecman
[20], and Wierzbicki et al [24]). The case ig = 0' corresponds to bending about the
weaker axis,'w', while xj = 900 describes bending about the stronger axis, 's'. The ratio
of Mimax/Mwmax depends on the aspect ratio of the cross-section a/b. In most cases the
Mres-O curves corresponding to the intermediate values of W are shifted in the vertical
direction. However, the rate of decay is somewhat different at W=6 0 '-75*. This range
corresponds to the so called 'critical' angle Wcr, discussed later in this chapter.
The fully plastic bending moment of any cross-section is defined by
M, =
-x.dA=-,Q
(2-5)
A
where c-. is the flow stress and A is the cross-sectional area, and
Q is the
first moment of
inertia.
We can define
as a normalized peak bending moment.
f)
The plots of the function ()
= (M)
(Me ),=
Q(
(2-6)
QV=O
as determined from FE results for the rectangular cross
section are shown in Fig.2-9. The result for the square cross section members are not
plotted as in this case the peak bending moment is observed not to vary much with
respect to the orientation angle W.
From the plots, it is clearly seen that the analytically derived normalized peak bending
moment (Eq.2-6) of each orientation angle y has closer relations to the relative peak
moment obtained in the course of numerical calculations. So the largest bending
bending moment of the initial orientation and multiplying it by the function
48
.
resistance of any orientation angle W can be estimated by calculating the fully plastic
_________________________
.
.
.
....
3 .0 ..
.
Mpeak/MpeakO
2.5
.......
-- .
:
peak
.......
................
............
2.5 -
. --
..
........
.
3 .0
peakO
--
..
.-----
.
. .... .. ..................
....
....
..
5
.
.....5
-..
...
..
....
.....
....
-
2 .5 - .
..
-....
-....
...
.
.... ................. ......... ...
1 .0
0
15
30
45
60
1 .0
75
............ ........ ..... .....
0
v (deg)
30
45
60
w (deg)
(a) rectangular section 1, t=1.2mm
Fig.2-9 Function ()
15
(b) rectangular section 2, t=0.7mm
vs Normalized fully plastic bending moment
2.3 Development of Moment Interaction Curves
2.3.1 Initial Failure Locus
Calculated pairs of maximum bending moments (Ms, M,) for each orientation angle 1f
were plotted in Fig.2-10 and 2-11 to form initial failure loci. On the same diagram shown
are experimental results due to Brown and Tidbury [34]. Note that tests were available
only for two types of cross-section a/b = 2, and a/b = 1. The agreement appears to be
good considering the fact that exact stress-strain curve of the material used in the above
tests was not available. Actually, the spread of the test results was of the same order as
the difference between the theory and experiments. The largest discrepancies are
observed for the square section where the numerical results provide consistently a lower
bound. It is also apparent that the type of constraint does not have any appreciable effect
on the initial interaction curve.
49
75
1000
--
Test - loading case 1
-
- Test - loading case 2
Numerical - loading case 1
Numerical - loading case 2
800
600
z
400
200
0
200
400
600
800
1000
Ms (Nm)
Fig.2-10 Initial failure loci of Rectangle 1
900
800
----
700
600
D
500
400
300
-
zE
-'--Test - loading case 1
--'--Test - loading case 2
--Numerical - loading case 1
Numerical - loading case 2
200
100
0
0
100
200
300
400
500
600
\
1
700
800
900
Mstrong (Nm)
Fig.2-11 Initial failure loci of square section
An interesting insight into the initial failure interaction curve can be gained by assuming
material to be rigid perfectly plastic with no sectional collapse. Consider a local
coordinate system and assume that incremental rotation (dOs, dOw) are applied to the
cross-sectional axes of the beam. A typical distribution of axial strain resulting from the
50
rotation around s and w axes respectively is shown in Fig.2-12. The total strain increment
is
dE = de, + de,
(2-7)
Distribution of total strain increment (tensile +, compressive
) is shown in Fig.2-13
-
along with the corresponding stress distribution.
SdE
T2
bL+
----------..-..-.-.---..
.
.....
k 11111111
....- ..
d
w
- 1 11111E ]
L -1
dEg= adO
Fig.2-12 Distribution of axial strain at the plastic hinge resulting from independently
applied incremental rotation
..........
2
--------- ..................
Fig.2-13 Distribution of total strain increment and tensile/compressive stresses over the
beam circumference
The strain increment (positive or negative) can be mapped into the stress space using
the uniaxial constitutive equation for the material
- = o-sign(de).
Define the bending moments
MS =
J-wdA
A
MW=
fa s dA
A
51
(2-8)
one gets
M= (oaibt
=
0
abt+
2
2
(1-_)
2
)
M
for
dO
a
dOS
b
O>-
(Yv<
cr)
(2-9)
e,.)
(2-10)
and
M =coabt+ 2a(1-_)72)
2
M==o-0biyat
for dO<
(
dOs b
>
Eliminating the parameter il in Eq.(2-9) and (2-10), one can obtain normalized branches
of the interaction curve. The normalizations are done with respect to Mwo, and MSO, which
are fully plastic bending moment along w axis and s axis, respectively.
(2+a)2
m
=1-
b
a
(8- +4)
b
for dO > - ( < Vf,.)
dOs b
2
(2-11)
(2a+1)2
m, =1S
b
+8)
a(4-a+8)
b b
W
for dO <- (
d8s b
> f,.)
The interaction curve in the first quadrant is described by two parabolas with coefficients
depending on the ratio a/b. In Fig.2-14 shown is an example of construction of first
failure locus.
W
W
------ for W<W.r
--.-------- for
for W>M cr
. .---... .----.
---------
for
-
"
-1
-1
.1
Fig.2-14 Example of the first yield locus
52
cr
First failure locus
IM
I
~-1
I<NI
>
The comparison of predicted failure loci by Eq.(2-1 1) and FE calculations is shown in
Fig.2-15. The plots show good agreement. Therefore, qualitative information on the fully
plastic interaction curve can be extended to elastic-plastic response with local plate
buckling and sectional collapse.
1.0
..
loading case 2
Eq.(2- 11)
-
0.8
N
0.6
E
0.4
1.0
loading case 1
..
0.8
~
-
0.4
E
0.2
n I
0.0
0.2 -
.
0.2
i
0.4
0.6
'
loading
case 1
i
Eq.(2-11)
...--
0.0
0.0
'
0.8
1.0
0.2
m. (normalized)
(a) rectangular section 1, t=0.7mm
N
0.4
0.6
0.8
1.0
m (normalized)
(b) rectangular section 2, t=1.2mm
1.0
1.0
0.8
0.8
0.6
N
0
0.6
0
0.4
0.2 -
C
-- -
loading case 1
-
loading case 2
Eq.(2-11)
-
0.0
0.0
0.2
0.4
0.6
0.4
E
loading case 1
loading case 2
Eq.(2-11)
...
0.2
- -.
-
0.8
0.0
0. 0
1.0
ms (normalized)
-
0.2
I
0.4
-
I
.
0.6
.
.loading case 2
0.8
1.0
m (normalized)
(c) rectangular section 2, t=0.7mm
(d) square section, t=0.7mm
Fig.2-15 Plots of the normalized failure locus predicted by Eq.(2-1 1) and obtained from FE
results
53
(b) high W
(a) low W
Fig.2-16 Two deformation modes in biaxial bending collapse
From the careful observation of each plot of FE results, it can be noticed that the curves
are actually composed of two parts intersecting at some orientation angle W, and this also
can be verified in the construction of the first failure locus, each corresponding to
formation of the fold on different sides of the section. In the case of lower value of W, the
main fold is formed on the wider side of the panel (see Fig.2-16(a)), and in the higher
value of ig, on the narrower side (Fig.2-16(b)). The orientation angle at the bifurcation
point is defined as critical orientation angle
icr.
The critical orientation angle in the
loading case 1 is observed to follow the relation
(2-12)
,cr= tan 1
In the loading case 2, Wcr can be calculated analytically from the intersection point of two
branches of Eq.(2-11). Let the coordinate of intersection point be (msm, mwm). By
multiplying msm and mwm with Mso and Mwo, (Msm, Mwm) can be calculated. Considering
M. = Mressinl and Mw = MrescOSI for the loading case 2, Icr can be computed as
Ycr = tan -1 (M
h
For example, for the rectangular section 1, a/b = 2. So the Eq.(2-1 1) becomes
54
(2-13)
=1--m
=1- (2+2)2
m
(8x2+4)
(2x2+1)
m =1
4M2
2
2
5
25
3
2
2(4x 2+ 8) W =1
S
(2-14)
m2
32
2
The coordinates of the intersection point are
mW =0. 9 1=mwm
(2-15)
m, = 0. 3 4 = mM
Therefore Wcr is calculated as
tan-' 0.91x 986.14
0.91M ,0
_tan
0.34M,
tan 4=80.12
0.34 x 459.53)
0
)
cr
(2-16)
80.120 is very close to the "exact" value 80' obtained from FE calculation.
The comparison between analytically predicted critical orientation angle and cr obtained
from FE calculation is shown in Fig.2-17. The results show very close agreement
between the values in both loading conditions.
90
80
- ------ ----. ..................I
--
----------
70
60
-.....
0)
~0
------- ........------------....
50
40
30
-
-
loading condition 1
.
--u--loading condition 2
20
--A-- predicted, loading condition 2
p ei
0
1.00
1.25
1.50
ct
d
lo
1.75
d
n
o
2.00
d
toI
----------
2.25
2.50
2.75
-
10
3.00
aspect ratio (a/b)
Fig.2-17 Critical orientation angle Wcr of analytical prediction and obtained from FE calculation
55
2.3.2 Shrinking of the Failure Locus and Normality Rule
The first failure locus defined above shrinks as the bending process progresses, and the
set of these curves fully defines the moment interaction curve in bi-axial bending.
Selected numerical results are shown in Fig.2-18. The locus drawn as solid line is the first
failure locus. Each subsequent locus corresponds to the increased but constant hinge
rotation angle 0. Some irregular behaviors were observed for the square section W=45'.
This abnormal weakening was caused by the instability of the folding mode in
geometrically prefect sections with symmetry. This effect is not so significant in practical
application.
From the observation of these graphs the following partial conclusions can be made.
* Each subsequent yield locus generally lies within the previous ones. This corresponds
to the so called 'isotropic' softening. In the thinner thickness cases some of the plots
does not obey this behavior, especially for the larger values of W's. This is due to the
internal contacts of collapsing walls.
*
The shape of the curves changes from a smooth interaction curves to the flattened
piece-wise linear interaction curve similar to the Tresca yield condition in the first
quadrant. As discussed earlier this behavior supports the concept of the critical
orientation angle Wr. In O<Ag<Icr, the beam collapses in the weak direction, and
vr<I<90* in the strong direction.
*
Loading condition 1 exhibits more irregularities near M, axis.
The understanding of the dependence of the interaction curve on the hinge rotation angle
is very important for practical application. Not only the moment resistance of each
orientation but also increments of generalized displacement can be derived from yield
locus using the concept of the so called normality rule.
The increment of plastic work is defined by a scalar product of the moment vector and
the increment of rotation vector.
56
dW, = Mid
(2-17)
1
where the components of the vector dO are [dOx, dOy, dOz].
In the loading case 1, as the axis of hinge rotation is fixed, the hinge rotation angle can be
resolved into component (0,, 0,).
d6,=dOsinV
d8~
V
dOW = dO cos
=~cos~(2-18)
In the case loading 2, the components of the hinge rotation angle are defined by
sin O= -
u
(2-19)
sin O, =
(2-20)
where the pair (ux, uy) defines the displacement of the top end of the beam.
The ratio of rotation increments is determined from Eq.(2-18), (2-19) and (2-20).
dO--=dO
(2-21)
du
x
2
The components of the rotation vector, transformed to the local coordinate system are,
FdO, ]
LdO,_
cos f
- sin V
sin V ldOx 12
cos Vi dOj
Now, at any point of the loading trajectories the direction of the rotation vector increment
can be found from Eq.(2-21) and (2-22). Those directions are shown in Fig.2-18 as
arrows attached to the corresponding points on the yield loci.
57
500
-- Initial
0=5o
Initial
-
- 0=50
-
-.-.-
400
0=100
- ...
0=100
-0=150
600
-- 6-=1 5 o
300
-....0=20 o
=250
z
z
..0=250
400
200
200
100
.
...
1*~~*,
0
0
100
200
300
400
0
500
200
M (Nm)
400
600
800
M (Nm)
(a) rectangle 1, t=0.7mm, loading case 2
(b) rectangle 2, t=1.2mm, loading case 1
400
400
-. - Initial
--. -- 0=50
.--e 0=100
[
300
300
....
200
0=50
-. ,.,
0=100
e-.- 0=200
E
0=250
------
200
4-..
711...
.-
100
0
0
100
200
300
0
400
100
200
300
400
M (Nm)
M (Nm)
(c) rectangle 2, t=0.7mm, loading case 2
(d) square, t=0.7mm, loading case 1
Fig.2-18 Direction of increment of rotation versus loading
It is clearly seen that the rotation increment vector in finite element calculations is
approximately normal to the current yield loci. Thus, for all practical purposes the
normality hypothesis holds in the case of biaxial bending of thin-walled rectangular
beams.
58
-.-
-+ -0=25o
J..........
100:
0
Initial
---..
0=150
0=1 50
0=200
z
-.-
2530
15
20
() MM 05 10
As an application, the trajectory of the top end node can be predicted analytically using
the normality rule. As the portion of the beam above of the plastic hinge undergoes small
elastic deformation, the displacement of the top end of the beam represents the rotation
characteristics of the plastic hinge part. In the previous section the initial failure locus
was predicted analytically and verified with the results obtained from FE calculations. On
the graph of failure loci we can indicate the direction of rotation vector using normality
rule. This direction determines the trajectory of the top end center node.
An example of calculation of i=30' rectangular section 1 is presented in Fig.2-19. The
calculated direction in this case is 19.3' counterclockwise from the global x axis (10.70
clockwise from s axis in local coordinate system), and the FE results show 210
counterclockwise from the global x axis (90 clockwise from s axis). The comparison of
predicted trajectory from the interaction curve obtained through FE calculations results is
shown in Fig.2-20. In general good agreement was achieved.
30
__
__
w--
_
W--1
=50,
sim
Liaton
25 -
20
-i-i=150, analyfical
-----.W=300, simuLafion
-. ......
-- =30, analytical
- -
=45o, simulafion
--=450, anytical
-
-
-- =
-------
-
dO15
- - - - - - ---M-d-
d0
6
00,
sim Lafion
W-=600, analytical
--
=750, simdaton
-
....................................
0
__
-
M
y
---- 750, analytical,nyba
10
10.70 for rectangle 1, t=1.2mm
M
U
x
Fig.2-20 Trajectory of top end node predicted
Fig.2-19 Derivation of trajectory from
analytically and FE simulation results
the interaction curve
59
2.3.3 Formulation of Generalized Moment Interaction Curve with Softening
Earlier in this chapter the initial failure locus was derived from the analysis of stressstrain distribution over the cross section. After buckling occurs, the neutral axis is
observed to shift dramatically toward to the tensile flange [24]. To predict analytically
correct stress and strain distribution, the shift of neutral axis has to be considered (see
Fig.2-2 1). Only one neutral axis is shifted ; opposite to the inward collapsing flange.
--- - - - -- - ---------- --
b
-
a
-
neutral axis
Fig.2-21 Distribution of axial strain at the plastic hinge resulting from independently applied
incremental rotation in the post-buckling range
.
.........
..
k.......
---------J ----------F
ia
Fig.2-22 Distribution of total strain and stress over the circumference in the post-buckling range
In Fig.2-22 shown is the corresponding distribution of the stress is shown. From this plot
the bending moments are calculated as follows.
MS= iabto
MW =abta'0 +(1-
2
)b2 t
60
for V
0
< Vr
(2-23)
M = abto +(1 -,
2
)a 2 to-
for
V >
(2-24)
cr
M = 77abto
Normalizing Eq.(2-23) and Eq.(2-24) by abtoo, and eliminating the parameter
m =1+-(I-m_
a
a
for
M
S
one gets
icr
(2-25)
for
m, = 1+-(1-M2)
S
VI<
TI,
b
Y >
cr
By comparing Eq.(2-25) with Eq.(2-1 1), change of the shape of moment interaction curve
from the first failure locus and the subsequent loci in the post-buckling range is well
explained. From Figs.2-6 ~ 2-8, it is clear that the bending resistance of the section is
diminishing with hinge rotation. Equation (2-25) can be generalized by considering the
declining of moment after buckling as Eq.(2-26).
b
a
a
for
In = f(O)+ - (I-- M2)
In =S g(O)~b+-a
y< i,
(2-26)
for
(-M)
V/> l/,
where
and
f(0) =
M~(6) |
g(O) = M
M,ni| x
MS,na
I4r
(2-27)
|V,=90"
Heref(O) and g(O) denote the moment-hinge rotation characteristics in uniaxial bending
collapse.
In order to use Eq.(2-27) in simplified calculations, the dependence of the bending
moments Ms(O)kw=9o and Mw(O)Iw=oo on the hinge rotation will be established analytically.
The solution is composed of two phases. Assuming that bit is relatively small so that all
side plates are fully effective, the pre-buckling bending moment M is related to the
I2
geometry of the rectangular section (a, b, t) and the average stress c
u-bt a+b
a
o-at b +-2
61
for y,=O0
(2-28)
for
=990'
Denoting by cy = cy(E) the material characteristics in uniaxial tension the average (or flow)
stress is defined as the stress corresponding to the average value of the strain in the
generalized plastic hinge
0
=
(2-29)
-(Eav)
where
Ob
4H
Oa
14H
0
(2-30)
for V =9 0
*
av
for f =
using Eq.(2-2), the final expression for the average strain is
02b)1/3
for Vf= 0
(2-31)
ay=I0.2O6
for w= 90'
0.2O(
In the post-buckling or folding region the moment rotation characteristic of the
rectangular section beam M'() was derived by Wierzbicki et al [24] in the form
M (0) = 22.06bM
a+b
30.576+
2
I
(2-32)
where the fully plastic bending moment per unit length Mp is
M P=
a
ot2
4
(2-33)
and the flow stress at large strain can be approximately by the equation [50]
O-o =
-oy-,
where cy and au are respectively the initial yield and ultimate (UTS) stress.
62
(2-34)
500
400
M
'.....
-
M"
momen
analytical prediction
-
E
I.
....
-
peak
E
00
0
rotation
Fig.2-23 Determination of peak moment
5
10
rotation (deg)
20
15
Fig.2-24 Analytical prediction of uniaxial bending
collapse, square section, t=0.7mm
An intersection of curves Eq.(2-28) and (2-32) defines the theoretical peak moment,
Fig.2-23.
It was shown in [51] that a transition from the pre-buckling to the post buckling and
folding stage is not instantaneous, as shown in Fig.2-23, but requires a considerable
additional rotation. The 'shift' in the moment rotation curve is governed by the aspect
ratio bit and i/b, according to Ref.[5 1], and is expressed as follow, Eq.(2-35).
2
5
2H 1.92
3.65
(2-35)
+1
(
sh
The theoretical curves predicted by means of Eq.(2-28),
(2-32), and (2-35) were
compared with the results of calculations showing very good agreement, Fig.2-24.
Now, we are in the position to determine the change in shape and size of the first failure
locus defined by Eq.(2-11) with increasing hinge rotation. A comparison of analytically
predicted and numerically calculated failure loci is shown in Fig.2-25. The agreement is
satisfactory considering the complexity of the problem. The present theory predicts
63
correctly the shrinking of the interaction curve as well as the development of a "kink".
Small discrepancies are noticed on the axes due to the approximate character of Eq.(232). A comparison of the shape between the analytically predicted curves with the FE
results, assuming that one has the exact moment-rotation response in uniaxial bending, is
shown in Fig.2-26. For practical application, the agreement is quite satisfactory. It is
interesting to note that the theoretical solution capture the "flattening" effect of the
interaction curve from the initial circular arch to a square shape for large rotation.
-
1 2.
Initi al
-.-
-e-=5o
1.0
*
0.8
0.6
7-
from
FE results
-- -9=200
------ e=250
-andytical
predictioin
......
--------
-
E
0=100
e=-=150
.
0.4
0.2
I
0.0
0. 0
0.2
0.4
0.6
0.8
1.0
1.2
m
Fig.2-25 Analytical prediction of failure locus in biaxial bending square section, t=O.7mm
1.2
---
0.81
Initial
E)
1.0
50
....-=100 from
----E=I5P FE results
-i
-
e=20>
..-
analytical
predictioin
0.6
0.4
'
0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
m
Fig.2-26 Analytical prediction of failure locus in biaxial bending,
with exact moment-rotation characteristics in uniaxial bending
64
Chapter 3. Crush Behavior of Thin-walled Prismatic Columns
under Combined Bending and Compression
The objective of the present chapter is to analyze the response of thin-walled prismatic
member subject to combined compression-bending collapse loading. Of interest is the
construction of failure locus of prismatic member in the bending - compression plane
(M,N). First, the numerical calculations using the nonlinear explicit code PAM-CRASH
are carried out and a pair of (M,N) is plotted for various loading histories and deformation
stages. Then, analytical solution of the crushing response is derived using a simplified
Shanley [36] model with softening. Both methods are shown to compare well in terms of
general trend and specific shapes of the interaction curves. The results of this study can
be applied to the development of simplified crash-oriented design tools.
3.1 Formulation of the Problem and Finite Element Modeling
The structure considered in this study is the thin-walled prismatic column with square
cross-section. Two aspect ratios (bit) (80/1.6=50 and 80/2.4=33.33) are considered. Here
b is the flange width and t is the thickness. The bottom end is fully clamped, and the load
is applied on the top end as the velocity boundary conditions. The rate of axial
displacement t and the rate of rotation 0 are applied resulting in the combined
compression/bending response. The subscript z and x denote the direction in the local
coordinate system. Two coordinate systems are used in this study. One is the global
coordinate system (X, Y, Z) having its origin at the center of the bottom of the
undeformed beam, and the other is the local coordinate system (x, y, z) attached at the
center of top cross-section. The local coordinate system is translating and rotating with
top cross-section as the deformation progresses.
65
b
x
dA
b
E-
--
t
- -
-
i
z
Fig.3-1 Configuration of the model
Following the general procedure developed in Chapter 1, the rate of strain energy
dissipation in the beam is defined as follows.
E= fr-dV = dz fI dA
I
V
where V is the volume and
(3-1)
A
A is the cross-sectional area. Using Euler-Bernoulli's
hypothesis,
t= to +xk
(3-2)
and the definition of axial force N and bending moment M,
N= fodA
(3-3)
M =f ox dA
(3-4)
The integration of Eq.(3-1) with respect to the area can be easily performed.
Sper unit length
=
, + xk)dA
= tN+kM
66
(3-5)
Noting that
* dur
dz
(3-6)
dO
dz
(3-7)
the second integration with respect to z can also be easily performed.
k=
fkper unit lengthdz
(3-8)
= dN +8 M
It is convenient to introduce dimensionless axial force n and dimensionless bending
moment m.
N
(3-9)
No
M
(3-10)
Here Mo is the fully plastic bending moment and No is fully plastic sectional force defined
as follows.
No = PodA =cr-A
(3-11)
MO = fuoxdA =OQ
(3-12)
where ac is the flow stress of the material and
Q is the first moment of inertia of a cross-
section.
With the above definition, the rate of plastic energy can be transformed to
E = (N
a,)n + (Mod)m
67
(3-13)
or in the normalized form
M0
No
A
"Un+Om=--d n+On
Q'
Mz
(3-14)
For a square cross-section,
A = 4bt
(3-15)
Q = -b2t
(3-16)
2
and Eq.(3-14) reduces to,
-M0
8
-abn+Om (3-17)
3b
Assuming that the plastic normality rule (the associated flow rule) holds, one can see
from Eq.(3-17) that for a square section the components of the generalized strain rate
vector are ( 8t / 3b,0 ), Fig.3-2. For an arbitrary section, these components are
(d A IQ, ).
Denote by p the angle between the strain rate vector and the n - axis of the dimensionless
yield locus. The ratio between the two components of the displacement vector is then rj
=
tan(p). The parameter T determines the relative contribution of compression and bending.
77
OQ
Adi,
3b
8di,
(3-18)
7 computer runs were made for each constant value p (or TI) over the range from 0' to
90' with the increment of 15', Fig.3-3. Correspondingly, I is changing from 0 to
00.
The
case when T = 0 is simply a pure compression, while I = oo corresponds to pure bending.
68
85/I3b
n
Fig.3-2 Displacement vector and the dimensionless yield locus
6
(P=600
=
(P=45 0
(P =30
(P =150
p
m-
U8/3b
Fig.3-3 Relation between rate of translation and rotation
Numerical simulations are made for the column made of the aluminum extrusion AA
6063 T7 with mechanical properties of Young's modulus E = 6.9x104 N/mm 2 , initial
yield stress acy = 86.94 N/mm 2, and Poisson's ratio v = 0.3. The detailed stress-strain
relation for this material is shown in Fig.3-4.
The geometrical model is established using the mesh generator program HYPERMESH.
The finite element model is then completed with the preprocessor PAM-GENERIS.
Actual calculations are performed on a SGI Octane machine with dual R-10000 processor
using explicit finite element code PAM-CRASH. The post-processor PAM-VIEW is used
for visualization and data acquisition.
69
The whole structure is modeled using 4-node Belytschko-Tsay shell element. Material
Type 103 provided by PAM-CRASH is used for the shell element, and it corresponds to
elastic-plastic isotropic thin shell material models. Also this material type uses an
enhanced plasticity algorithm that includes transverse shear effects, precisely updating
the element thickness during plastic deformation [52]. The width of each side flange is
composed of 16 elements, which corresponds 5x5 mm of element size. Altogether 2304
elements are used to construct the model, and more than 600,000 time steps are needed to
complete a simulation. The overall effect of the mass scaling is controlled by ensuring
that the calculated kinetic energy is insignificant (less than 0.15%) when compared with
the strain energy absorbed by the model.
The nodes in the top cross-section are connected with rigid body constraint. In other
words, these nodes can translate and rotate as a rigid body. This configuration can be
interpreted as a virtual rigid massless plate attached to the top end cross-section of the
beam. The load in the form of ramped velocity (translational and rotational) boundary
condition is applied at the center of gravity of this rigid body. The ramping time is
0.05sec, and the constant rotational velocity is 1.513rad/sec. The translational velocity
and rotational velocity applied are related by the Eq.(3-18) by the parameter 71 which is
held constant for each run. For the bottom end, all six degrees of freedom of the nodes
are fixed.
As the internal self contact between the elements near the plastic hinge is expected, the
highly improved self-contact provided PAM-CRASH (contact algorithm #36 using "3D
bucket" global search algorithm) is applied.
70
200
160
..... ...-.
.....
....
-..- - ---
....
-..
--
- -
--
-
-
1 8 0 -.
120 -
--
100 -
--
-
-.-
-
80-1'
-
E
E
-
C7
-
140 --
6040-
20
0.00
0.02
0.04
0.08
0.06
0.10
0.12
0.14
strain
Fig.3-4 AA 6063 T7 stress-strain curve
3.2 Results
-
Axial Force and Bending Moment Response
In Fig.3-9 ~Fig.3-1O shown is the evolution of the sectional force and bending moment.
Pictures of the deformed mesh are presented in Fig.3-8. The force and moment are taken
not from the mostly deformed section but from the bottom section. However, of interest
is the bending moment at the location of the plastic hinge. Consequently, a correction
must be introduced to account for the additional bending moment caused by the force
eccentricity e.
M =Moom
- Ne
(3-19)
where M is the bending moment at the deformed cross-section (which is of interest in this
study) and Mbott0 0 2 is the calculated bending moment at the bottom. As shown in Fig.3-6,
the magnitude of the correction is quite large. In the plots shown in Fig.3-9 ~-Fig.3-10, the
corrected, true moment is used.
In the cases of low rj, it is observed that the axial crushing is dominant, while for high rn,
the bending collapse is dominant. A change in the shape of the moment-rotation curve
71
and thus the change in the failure mode occurs between i = 0.5774 (p = 300) and q = 1
(p
450).
Moment arm e
Fig.3-5 Moment generated by the axial force
An interesting feature of the responses is that the sign of the force or moment is changing
as the crushing progresses. For low
TI,
the bending moment increases up to the peak
point, and decays dramatically to negative values. This negative bending moment is due
to the eccentricity effect discussed above.
600 - ----------- ---
------- Ti=tan 300, raw Moment
-- i=tan 300. Axial compression considered
400
-. --- --...- ----.-..
200
z
0
.-
..
-...........-..--............
---- ................
--..
E
0
-200
-400
....---I ........---- ---- -..--I -----------600
0
5
10
15
20
25
Rotation (deg)
Fig.3-6 The effect of axial force to the bending moment
72
30
While the sign of moment is changing in the cases of low ri, the sign of force is also
changing in high il regime. This is caused by the traslational displacement occurring in
pure bending to ensure zero axial force. When a pure rotation on the top tip of prismatic
member is applied, the top cross-section rotates with the applied rotational velocity, and
at the same time must move by a certain amount toward the bottom end with a unique
translational velocity. This is because the neutral axial is shifted dramatically to the
tensile flange after the sectional buckling. Originally, the axis of rotation is located at the
centerline of the cross-section, but in the postbuckling range it is positioned near the
tensile flange, which causes the translation of top cross-section. When the rotational
velocity 0=1.5 (rad/sec) is applied on the top cross-section, the translational velocity in
the local coordinate system is as large as 57mm/sec (Fig.3-7). Therefore, when the
translational velocity ti is lower than the velocity profile in Fig.3-7, the tension is
generated in the column in the post-buckling range. The prescribed velocities in the case
i=45', 600 and 75' are 45.4, 26.2, and 12.16 (mm/sec) respectively, which are below
velocity profile in Fig.3-7. Thus, somewhat counterintuitively, tension is generated in
those cases.
N
4 0 --------
-
--- -- -
-
------ -------------
---- - ---
---- - - - - -- -
---
.
..-.
- --- ------
-
I -- ---5 0 ------ - ---- - ---- - - ---
-
-.---.
..-.
.-...---------------------. . . .. .. . .
2 0 -------- -.
10 ....... .- . .
0
0.00
Velocity generated in Pure bending
.......
0.05
0.10
0.15
0.20
0.25
0.30
Time (sec)
Fig.3-7 Translational velocity generated in pure bending (t=1.6)
73
(b) Tj = tan(15*)
(a) il = tan (0*)
(d) il = tan (90*)
(c) i = tan (60')
Fig.3-8 Deformed shapes
74
0
LL
50 000
---------------- ------ ------------------------------------------------ -----------------------
400 00
---------------------- ------------------------ --------------------
30 0 00
---------------------- ----------------------- :------------------------
2 00 00
-- ------------------------------------------------------------------ -- - ---------------
10000
.................
..............
.
................. ......
-ij=tanOO
0
- -- - ii=tanl5o
'Pol-
...... ii=W300
-10000
.............................................
......... - ------ Ti=tan450
Ti=W600
-20000 . .....................
0
Tl=tan750
...............................................
20
40
60
80
Displacement (mm)
Fig-3-9 Axial force (b/t=50)
3000
- -- - 11=150
------ I ............... I---------------- ............... ............ ...... T1=300
250 0
2000
........ ..... ............
............... ............... -------------
Ti=450
------- Ti=600
71=750
A
TI=900
E 1500
..........
--------
--------------------
...............
z
E 1000
0
--------------- ------------- ---
...........................................
500
-
1 ............. ............... .............. --------------- ---------------- --------------
0
---------............... ............... ........
-500
0
5
10
15
20
Rotation (deg)
Fig.3-10 Bending moment (b/t=33.3)
75
25
30
3.3 Construction of the First Failure Locus
Considering the stress distribution over the cross-section, the interaction curve between
axial force and bending moment can be derived analytically. In the derivation, the
material behavior was assumed to be rigid-perfectly plastic with the energy equivalent
flow stress cY.
From the Euler-Bernouilli hypothesis the position of neutral axis is calculated
(3-20)
0
Depending on the value of , the following three cases can be distinguished.
" Case (a) - neutral axis is positioned between centerline of the cross-section and
tensile flange.
" Case (b) - neutral axis is within the tensile flange.
*
Case (c) - neutral axis is outside of the cross-section.
x
t
bi
Bending
Axis
Case (a)
Case (b)
Case (c)
Fig.3-11 Position of the neutral axis
76
b2
(b2
Case (a)
~2)
2
The sectional force with the stress distribution shown in Fig.3-12 is calculated as
(3-21)
N = a dA = 4tJ470
A
The maximum axial force or the squash load is,
x
b2/2
b2/2
LI----
a,,
--a
.3-12 Stress distribution in Case (a)
=
(3-22)
2aot(b + b 2
)
No
The normalized axial force is,
max
4t 0
No
2 0 ot(b +b 2 )
_
2_
b, +b
(3-23)
2
= b 212
2 b2
n-
= n (-) =
"a =
1
(3-24)
-
For Case (a)
N( )
-
77
b +b
2
1+bi lb2
2
Thus, for a square cross-section, nmax= 0.5.
M
L2
= fax dA = 2cotb
b2
/2
+ 4c-t fx dx
) 2
oob2
-
= o-tbib2+2-ot
2(3-25)
The maximum or fully plastic bending moment is,
MO =o-tb2 b +
2
(3-26)
and the normalized bending moment is,
1 _- 2j
b +b
(3-27)
m
M(f)
2
b+ b2
M
b21~J
2
The parametric representation of the yield condition for Case (a) is summarized as
follows
2
(
i+
M
b2)
-b
bi +b21-2
(3-28)
2
m =F
M
b
2
b2
b2
b
b2
78
2
(3-29)
As a special case, when b, = b 2 (square column)
4
m =1
_2_ t <
Case (b)
2
<
3
n2 (0 < n < 0.5)
(3-30)
2
2
The stress distribution in the lower flange changes rapidly from compression into tension.
The stress distribution in the upper flange is the same (tensile). The stress distribution in
the web is approximately (to the second order) constant and equal to the stress at the end
of Case (a),
=
b 212
-=-YO
-
-=(T
Fig.3-13 Stress distribution of Case (b)
M =C-
Ltb
2
+ 0b
L2(2A -
2
t) = Obb 2A
N =4tLo-+ob t+(t - A)- A]
(3-31)
(3-32)
2
Normalizing,
M M
MO
M"
b1A
aO-bIb2A
cr~tbj(bi
+ 2j
79
t~b b1 2
1
+>2
(-3
b
n =
N
No
4
0-+
4t- 2
b
t
2-ot(b + b 2 )
-)
b
__b2b,[ _
-tab
(3-34)
b +b 2
Eliminating the parameter X,
b
m = (1- n)
1
2
b2
b
(3-35)
b2
For a special case, b 1 = b 2 (square cross-section),
4
m = -(1- n)
3
Case (c)
2
b2
2
2
(3-36)
The neutral axis is outside of the cross-sectional profile and the section is either is pure
compression or tension.
m=0
(3-37)
n = +1
(3-38)
Thus, the yield condition is composed of a section of the parabola and a straight line.
Assuming that the yield stress is the same in tension and compression, o- = a-, a closed
yield condition become a close, convex curve with two vertices, Fig.3-14.
80
1.0
Case (a)
Case (b)
Case (c)
-1.0
1.0
II
-1.0
Fig.3-14 Analytically constructed first failure locus
The above derivation gives also some clues as to the shape of subsequent yield loci for
larger deformation. Soon after peak load, the neutral axis is shifted towards the tension
flange. Thus, the parabolic portion of the interaction curve will shrink and subsequent
yield loci should be represented by straight lines.
In Fig.3-15 and Fig.3-16, the comparison between analytical prediction of the first yield
locus and the results from numerical analysis is given. Note that additional two runs
(ri=tan(360 ) and ij=tan(38.20 )) were made between rl=tan(3 0 ') and ri=tan(450 ). It is
observed the change of failure mode occurs in that range. In both cases the correlation is
quite good. In the cases of lower value of TI, some discrepancies are observed in the
results of thick column. This is caused by the fact that the hardening of material is not
considered in the derivation of the analytical solution when highly unstable sectional
collapse occurred for low il. In high ri cases, the correlation is almost prefect. Thus, using
the stress analysis over the cross-section, the resistance of the column can be calculated
analytically in combined compression/bending cases.
81
1.0
Analytical
Numerical
-+
0.8
0.6
0.4
0.2
u.u
0.2
0.6
0.4
1.0
0.8
N/N
Fig.3-15 Initial failure locus (b/t = 50)
1.0
-----
Analytical
Numerical
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.6
0.4
N/N
0.8
c
Fig.3-16 Initial failure locus (bit = 33.3)
82
1.0
3.4 Shrinking of Failure Locus
3.4.1 Numerical Results
The initial failure locus will shrink as the deformation progresses. In picking the data
points for the plots, an effort was made to assure the same "degree" of deformation in
translational and rotational displacement. For this purpose, a parameter k is introduced in
this study representing the norm of the displacement component, defined by Eq.(3-39).
k=
U
2
+02
=1
T3
2 +1
(3-39)
where uz is the displacement of top center node in local coordinate system, and 0 is the
amount of bending rotation. Both uz and 0 can be calculated from the prescribed ramped
velocity profile.
u =I
t 2 (0<t
uz = Li (t - 0.025)
(3-40)
0.05)
( t > 0.05)
where t is the prescribed top velocity. The rate of rotation 6 is related to
(3-41)
ai
by the
parameter il according to Eq.(3-18) in the case of a proportional loading.
Using the above equation, the strain norm k can be uniquely related to the time parameter
in PAM-CRASH calculation, see Fig.3-17.
83
2.4
-
.
=00
---
2 .0
.. . . . .. . . . . .. .. . . . .. . . - - - -- - - --
- -- - -
-
2.2
- 11 -T=150
1.6
----
1.4
0.6
0.0
- --
0.00
- -
-
-
-
-
-
-=00
--- -- --
----
--- -
-
-
11=360
=38.20
--
0 .4
0.2
- --
-
.....-- 1=300
--
-
--
-
-
0.05
------
-
1.8
0.10
0.15
0.20
0.25
0.30
Time (sec)
Fig.3-17 Norm of displacement vector
The plots of shrinking of failure locus are shown in Fig.3-18 and Fig.3-19 for the two FE
models considered in this study. Both results show some consistent behavior. After the
first failure, the shape of the failure locus changes to be linear, and shifts as the
deformation progresses. The locus in lower ri cases and higher il cases show different
characteristics, which implies the transition from compression dominant range to the
bending dominant range. As in the first failure locus, the transition is observed to occur
between when Tj = 300 and when i = 45*.
In Fig.3-20, the failure locus for dented circular tube under axial force - bending moment
combined loading is shown, as derived analytically by Wierzbicki and Suh [28]. The plot
shows similar characteristics to those acquired by numerical calculation in this study. The
failure locus shrinks as the deformation progresses, and changes shape from almost
parabolic to almost linear. Both negative moments and negative forces are observed.
84
M/M
1.0
----=-
Initial
k=0.1124
k=0.2164
-v-k=0.3204
-
-+-k=0.4244
N/N
-0.5
1.0
0
-- 0.5
Fig.3-18 Shrinking of failure locus (b/t=50)
M/M
1.0
-A-
Initial
k=0.1644
k=0.2164
k=0.3204
-v-+-k=0.4244
-N/N
-0.5
1.0
- -0.5
Fig.3-19 Shrinking of failure locus (b/t=33.3)
85
0
M/MQ
1.0
-1.0
/N/N,
1.0
Fig.3-20 Failure locus for circular dented tube by Wierzbicki and Suh [28]
3.4.2 Analytical Results
The shrinking of failure locus can be constructed analytically using the concept of the
'Superbeam Element' and the Shanley spring model [36]. The Superbeam Element
concept is basically an extension of the concept of the generalized plastic hinge. Such
"hinge" is characterized by its reference length 2H, which corresponds to the typical
length of a folding wave in the process of progressive collapse. A Superbeam element
can be considered as a single 'spring', which has the property of collapse characteristics
of structure. In the present study, the cross-section is discretized into four equal springs
(Fig.3-21), which has the force - displacement characteristics shown in Fig.3-22.
86
2H
Fig.3-21 Discretization of column
t
Compression
FEcrEA ----
-2H Ey
2HF-cr
2H
Tension
--- - oA
Fig.3-22 Force-displacement behavior of spring
The spring which represents the flange of the tensile or compressive part will extend or
be compressed depending on the deformation mode of whole column. The following two
cases can be considered in Fig.3-23.
87
lu
622
Uf__
f2
f: compression
Both compression
tension
f2
Fig.3-23 Deformation of springs
Kinematics of Spring
The relations among uz, 61, 62, and 0 are summarized as follows, where uz is the
displacement of the center of the cross-section,
8 1 is
the displacement of the spring in the
compressive part, 82 is the displacement of the spring in the tensile part, and 0 is the
rotation of cross-section by bending.
Case 1. Both Compression
be
3
3
u+Z 22 sin8~-r -+2
4q/
beg
bO
3
3
2
2
4i
4
4)(-2
Case 2. fj - compression , f2 - tension
beff
g2
= -U+
+ -sin
2
b
~
2 4
4(
bO
3
34
Isin
--~- 2( 4q 4
+
s,=
bO( 3 +3)
-
2
88
(-2
(342
where beff= 3b14 so that the ratio of the strain rates in the real structure and the Shanley
model is the same.
Determinationof the Spring Characteristic
The typical force-displacement relation of the spring used in this study is given in Fig.322. The equation of the force-displacement in each range was derived in Ref.[18]. The
derivation was based on the concept of the Superfolding Element and rigid-plastic
material idealization described extensively in Ref[16,17].
i)
8 >2Her (Compression)
C
(3-44)
where, according to Ref.[18,24]
C=1.17(3rMP)t
ii)
-2Hey
I1/3J
(3-45)
8 2HEc (small axial displacement) - The response is assumed to be
elastic
AE
2H
iii)
(3-46)
6 5 -2Hey (Tension)
f = -Ao 0
(3-47)
where co is the energy equivalent flow stress, Ey is the yield strain, Ec is the critical strain,
C is the coefficient, and A is the cross-sectional area represented by the spring.
89
The strain corresponding to the buckling of the section, Ec is defined as,
(,)
4)T 2
cr12(1-
2
3.615
b
2v)
(3-48)
where Mp is fully plastic bending moment per unit length of the plate of thickness t
defined as Eq.(3-49)
2
=
(3-49)
4
Altogether, there are six different cases which are schematically defined in Fig.3-24.
I
62
6
6
62
6
6
-j
j
(a)
(b)
(c)
8-
6
6
6
62
J2
(d)
62
(e)
Fig.3-24 Possible cases for Shanley spring model
90
(f)
In practical crash calculations, of interest is the post-buckling range. The solution of the
case 1 (Fig.3-24(c)) is given in Eq.(3-50) and Eq.(3-51) is for case 2 (Fig.3-24(e) and (f)).
For the detailed analytical derivation, the reader is referred to the Appendix.
t_3)2111
1.9493
1
2
4/r/3
493t
3/4i/+3/4
Case1
+
4
1.9493 t
/
3
47
1
3/4/3/
-
1
3/417+3/4
(3-50)
1
3/41 -3/41
5
Case 2
4
t6
m =--n + 4.084 3
b
1
VT
(3-51)
Eq.(3-51) shows that the failure locus in Case 2 is linear line with the slope of -4/3.
Therefore one can see that the transition from compression dominant range (Fig.3-8(b))
.
to bending dominant range (Fig.3-8(c)) occurs when 1l = 3/4, which is equal to (p=36.90
This corresponds to the numerical result. Plot of Eq.(3-50) and (3-51) for varying values
of k as superposed on the numerical results is shown in Fig.3-25 and 3-26. Note that the
analytical initial failure locus is derived in the preceding chapter. It is seen that the
theoretically predicted failure loci behave similarly to those obtained via numerical
calculation. The yield condition could be closed by determining other combination of
spring responses, not shown in Fig.3-24. However, the tension quadrant is of little
interest in crashworthiness research. Therefore this problem is not pursued any further.
91
1.0
k
M/M
Nume rical
1.0
Initial
-A-
k=0.1124
k=0.2164
k=0.3204
k=0.4244
Analyt ical
Initial
k=0.1124
k=0.2164
k=0.3204
k=0.4244
- ,N/N
-0.5
.
-- 0.5
Fig.3-25 Comparison between analytical prediction and numerical results (b/t=50)
M/M
Numerical
Initial
-s-k=0.1644
-A- k=0.2164
-v- k=0.3204
1.0
-+-
V
Analytical
-Initial
- - - - -k=0.1644
-.......
k=0.2164
----- k=0.3204
-------- k=0.4244
N
A.
II
*1I.
.4
*
A.
4.
..
'I
.
*.
I
" *
-0.5
k=0.4244
I
.v
\~'A
N/N
'\
1.0
- -0.5
Fig.3-26 Comparison between analytical prediction and numerical results (b/t=33.3)
92
Chapter 4. Analysis of Crushing Response of S shaped Frame
Design, calculation and testing of lightweight and energy absorbing front rails of
passenger cars has been the subject of extensive studies over the past two decades
[53,54,55]. A highly idealized model of the front end of a car body is shown in Fig.4-1
e2
Fig.4-1 Idealized front end of a car
An inherent difficulty in dealing with this type of structure is the presence of a vertical
and horizontal eccentricities el and e2. In the event of a frontal collision, the member is
subjected to a bi-axial bending and torsion, in addition to an axial compression.
In the previous two chapters, the interaction curves for the sectional collapse under
complex loading conditions were developed. The large rotation bending response about
non-principal axis was studied in Chapter 2. The analytical initial and subsequent failure
loci were derived and the corresponding deformation modes were identified. The force
and moment response of the thin-walled prismatic member under combined bending and
compression was calculated theoretically using the concept of Superbeam element and
Shanley spring model in Chapter 3.
93
The application of the theories developed in the previous two chapters to a practical
problem is presented in the current chapter. The considered structural model is the threedimensional S shaped frame with rectangular or square cross-section. Upon crash loading
the structures undergo biaxial bending and combined compression and bending. Also this
model can be considered as the simplified front side rail commonly used in the
automotive structure. Because the geometry of the model is three dimensional, the axial
compression, bending, combined compression and bending, and bi-axial bending collapse
are all developed in the structure. All the analytical predictions are verified with the FE
analysis using PAM-CRASH.
4.1 One-hinge pin-pin supported model
A simple model with pin-pin support prismatic column is analyzed first. The initial
configuration and the deformation mode are shown in Fig.4-2. A single plastic hinge is
assumed to be placed in the middle of the column.
U
21l
1
Fig.4-2 One-hinge pin-pin supported model
From the geometry and moderately large rotation angles,
62
u=21-2lcos6~ 21 --
0~5_
94
2
=l02
(4-1)
(4-2)
Case 1) Constant Bending Moment
As the simplest case, the resisting moment in the plastic hinge is assumed to be constant
and equal to the fully plastic bending moment M0 of the cross-section. The balance
equation of the rate of energy dissipation is expressed as,
Pa = 2M0 0
31
MO =-b 2 tc- 0
and
2
(4-3)
(4-4)
where b is the width of the cross-section, t is the thickness and o-, is the energy equivalent
flow stress.
From Eq.(4-2),
U
"
8=
(4-5)
P=PM
(4-6)
,
Thus the force response P is,
The above derivation neglects the interaction between the compression and bending.
Case 2) Decaying Function of Bending Moment
The post-buckling moment rotation characteristic of the square section beam derived by
Wierzbicki et al [24]. The constant bending moment in Eq.(4-6) is replaced with the
actual decaying function of bending moment response,
11.030- b4/3t5/3 0.576+ 11/4
P=
1.3r2u14(4-7)
95
Again, no interaction is considered.
Case 3) Combined Loading of Bending and Compression
From Fig.4-3, the axial force N is expressed as
N=
(4-8)
cos 0
P
H
M
N
Fig.4-3 Free body diagram of the column
In Chapter 3, the normalized interaction curve of bending and compression in deep
plastic deformation range was derived in the form,
4
3
m=--n +C(k)
(4-9)
t
C(k) = 4.084j
(4-10)
where
k was defined as the norm of the displacement components in the previous chapter.
96
k
4I
S
~ F
2
b=+02
I
3
2
+(
+
(4-11)
where 8 is the axial displacement.
As Eq.(4-9) is the straight line with a slope rj = 0.75, from the normality rule one can
infer that
(4-12)
k ~4 0
and from the relation of the rate of axial displacement and bending rotation,
3b9
(4-13)
be9
=2
(4-14)
Then Eq.(4-10) becomes,
5/6
C(k) = 4.084 t
11
V
C0
0"F
(4-15)
where
C5=3
(4-16)
It is assumed that the generalized plastic hinge is divided into two pairs of the bendingcompression (Fig.4-4), then the balance equation of the rate of energy dissipation is
expressed as,
97
.C
------------------*---Fig.4-4 Division of a plastic hinge into two pairs of compression and bending
(4-17)
N = 26 N + 26 M
=2
M
M
S
cos6
+20 M
P
8M
cosO
3 No
----
3N0
6
P
0-+
cos0
+
5
20M
(4-18)
Co
"
=2-
+ 20M
"
P4 NC
Cos 0
+
Pd = 2$
=2GM CO
Using Eq.(4-2) and (4-5),
M 0 C0
1
ifU3 /4
(4-19)
or in a non-dimensional form,
Pi
C0
MA (u /1)
3
/4
(4-20)
The peak load can be calculated from the equation of Euler buckling load [56].
Pmax =412
or the squash load,
98
(4-21)
Pq =Z
A
(4-22)
whichever is smaller.
Plots of Eq.(4-6), (4-7) and (4-19) for b = 80mm, t = 1.6mm, 1 = 300mm, and c
=
10OMPa are compared in Fig.4-5. The cut-off value of Pmax given by Eq.(4-21) or (4-22)
falls beyond the scale of the graph.
.
,4
---- Constant M (Case 1 - Eq.(4-6))
---- With decaying function of M(0) (Case 2 - Eq.(4-7))
Considering combined loading (Case 3 - Eq.(4-19))
.
t=1.6mm
30
C/)
C
0
*
40
-
20
.--- .--- .-------- - -------- -------- ---- --- --- --------- ------- --- --------
CL,
C/)
a)
CD,
0
U-
10 --
.............................. ......................
----------------
-----------
-----------......
0
0
50
100
150
200
Displacement (mm)
Fig.4-5 Comparison of Eq.(4-6),(4-7), and (4-19)
4.2 Planar S shaped frame
Initial and deformed configurations of the planar S shaped frame are shown in Fig.4-6.
The analysis is divided into two parts. First, the peak force is determined from the first
yield criterion. Then, the post-collapse analysis is presented
99
L
0
12
0
13
U
12
Y
13
Fig.4-6 Planar S shaped frame with three plastic hinges
4.2.1 Peak Force
The reaction forces and moment at both ends necessary for the determination of the peak
crushing force responses are calculated through the elastic analysis. When the external
force P is applied on the moving end, the S beam deforms elastically until yield is
reached at the most stressed section. The beam can be divided into three segments and the
corresponding free body diagram is defined in Fig.4-8.
Deformed
Undeformed
Fig.4-7 Elastic response of S shaped frame with the external force
100
v
Ma
Vb
M Na
1
Nbe
Va
2N
Mb
Vd
M2
i P
MYF d
V
Fig.4-8 Equilibrium of force and moment for each component of S frame
The equation of the equilibrium of force and moment for each segment is expressed as
follows.
For segment 1,
P=Na
(4-23)
VI = Va
MI =-Ma +V l,
Continuity between segment 1 and segment 2,
Nb =Na cosO+Va sinO
Vb
=-Na sin8 +Va cos0
A b =Ma
(4-24)
J
For segment 2,
NC
= Nb
(4-25)
VC = Vb
MC +Mb +Vbl
2
=0
Continuity between segment 2 and segment 3,
Nd =
N Ccos0+Vc sin
Vd =-N Csin 0 +Vc cos 0
/Id =MC
For segment 3,
101
1
(4-26)
Nd =P2
=V2
Md =-M 2 + V 2 13J
Vd
(4-27)
The global equilibrium of reaction moment on both ends is expressed as,
M1 + M
2
+ P1 2 sin
=V(11 +l 2 cosO+ 13 )
(4-28)
The 17 unknown force and moment components must be expressed in terms of the
external force P, but only 15 equations are available. Thus the kinematic continuity
should be used to solve this indeterminate system.
Considering the continuity of the rotational displacement, the change of the rear end tip
angle of component 1 must be equal to that of the component 2 front end tip angle (see
Fig.4-9). This gives,
Val1
2EI
Mali Mi2
EI
3EI
Mcl 2
(4-29)
6EI
which reduces to,
M1 (61, + 212) = M 2 12 +v (3
+ 21112 -1213)
(4-30)
Fig.4-9 Change of the tip angle
Likewise, the change of the rear tip angle of component 2 must be equal to that of the
component 3 front end tip angle.
102
----
-
2EI
EI
1-2
3EI
Mbl 2
(4-31)
6EI
which reduces to,
(4-32)
M 2 (61 3 +2 2 )= M1 1 2 + V 2(3l+2213 -112)
From Eq.(4-23)
(4-32), the reaction shear force and moments are expressed in terms of
-
external axial force P.
= V2 =
12 sin 0(611+ 1)
2si 061+2)+1CO
(6 cos 0 - 3)1 11 2 +6 1 3 +31213 +l2cos
M, =M 2 =
12
sin 0(3l +21112 -1213)
+6113 + 31213 +l 2 cos
(6 cos0-3) 11 2
(4-33)
P
(
V
p
(434)
Using the equations (4-23)-(4-27), (4-33) and (4-34), the pair of sectional axial force (Ni,
i = 1,2,a,b,c,d) and bending moment (Mi) at any critical section can be calculated.
Normalizing Ni and Mi with No and M0 , respectively,
.
N
ni =
'
No
M -=
Mo
(4-35)
(4-36)
ni and mi are related as,
m
NoM
ni
MoNi
(4-37)
The relation between ni and mi is calculated from the Shanley spring model [36] at the
point of peak crushing resistance. The relations among the displacement components are
shown in Fig.4-10.
103
0
f2 7f
Fig.4-10 Deformation of springs in Shanley Spring model [36]
In the initial stage, the displacement of each spring is expressed as,
5f=u3+
32 =
(4-39)
U
2
bbeff sin
sn
2
427
b O( 3
k
2 4r7
4
3)
4
The spring forces are expressed as,
AE beff (1
A--+1
2H
2
(4-40)
y
(
~
3sin 0~- + (4-38)
2
fAEbeff
2H
1
2
(4-41)
i]
The point of peak crushing resistance is when the spring force f1 in Fig.4-10 reaches the
point P in Fig.4-11.
104
f
P
qA
AE"
8C
8
Fig.4-11 Peak spring force
At this stage, 0 is calculated from
1
fyA - AE b(ef
2H 2
7
+ijj
(4-42)
,which gives
0
4o-,AH
AEbeff
-+1
a 4H 1
E (bef 1+77
C
(4-43)
where cr is the ultimate stress.
Then the sectional force and bending moment are calculated as,
N = f + f 2 = 2u, A 1
1+ 7
(4-44)
M
(4-45)
-b
beff (2auA) I
2
1+17
2
Two extreme cases are easily checked when il = 0 (pure compression) and rj = oo (pure
bending).
No = 2a- A (pure compression)
M=
(4-46)
b
ff (2a,, A) (pure bending)
2
Normalizing N and M with No and Mo, respectively,
105
(4-47)
1
N
(4-48)
22u=
-A
M nbeff (2-, A)
(4-49)
1+7
2
Eliminating parameter 11,
m =1-n
(4-50)
Combining Eq.(4-37) with (4-50),
n =-
M0 N1
M
1N
MONi + NOMi
(4-5 1)
The axial force on each section is generally expressed as,
N1 = a1 P
(4-52)
where ci is constant value and can be calculated from Eq.(4-23)-(4-27), (4-33) and (434).
Thus, from Eq.(4-5 1), the peak crushing force is calculated as,
min
i=1,2,a,b,c,d
N MN
a,(M0 N + NOM ,
(4-53)
)
Peak =
The peak force response obtained in Eq.(4-53) is used as the cut-off value for the entire
crushing resistance of the S shaped frame.
4.2.2 Load-Deflection Relation
From the geometry of the deformed configuration (Fig.4-6),
106
L-u=l1 cosa+l2 cosy+13
(4-54)
8 =a+ y
(4-55)
l 2 siny=l1 sina+l 2 sin 0
(4-56)
There are five pairs of the compression and bending in three plastic hinges. They are
shown in Fig.4-12.
P -P-
E3N
M,
M2
N,
N2
N3
M4
N4
N5
Fig.4-12 Pairs of the compression and bending in the planar S shaped frame
The effect of the shear forces is ignored in this study as the rate of shear energy is
negligible compared to other rate of energy terms (see Chapter 1.2 General
Formulations). The axial forces are expressed in terms of the external force P as follows.
N1 =Pcosa
N 2 =Pcosa
N3 = Pcosy
(4-57)
N4 =Pcosy
N5 = P
The equilibrium of the rate of energy dissipation is expressed as,
Pd=(
Pd =6 1 N1 +&M + 6 2 N 2 +-M
2
2
+SN3
Ni+A8Mi)
(4-58)
+1M3 +S4N4 +$M4 +$ 5 N5 +-M5
2
107
2
2
(4-59)
Using Eq.(4-14),
P
d
2
NI+dMl+ b/N2 +-M2 +
4
22
4
+
N
2 M4
2N+AM
2
(4-60)
2N+-M5
which becomes,
2
+
-N
+
-
C, (k) +
3
3
+
M
N2 +
- 4N + C3(k) + bkN4
N 5 + M(- 3N
+
m
2+
3N
+C 2 (k)
+
3No
+r~r
0 ~ 4N
3 No4
+2
4(
N
(4-61)
C5 (k)J
This reduces to,
Pa=M
C(k)+f C 2 (k) +AC3 (k)+ -C4 (k)+kC5 (k)
2
2
2
2
(4-62)
=M C
+
+
O
From Eq.(4-54),(4-55), and (4-56), and with the assumption of moderately large angle of
P, and, y,
the force response is expressed as a function of u as follows.
- Au+ B (-1112sinO+l 2Au
11(11+12)
l2{}Au+ B -2
+Bf
(4-63)
where
108
1 1+2
+
U,
12(-11
sin0+
Au
+B)12
2
A
211(11 +12)
(4-64)
12
B=l sin 2 8- 21(l1 +l2)
12
L-l-
-12
(4-65)
sin2 0-l
+
2
Two examples of the crushing force of planar S frame is given in Fig.4-13. In the
calculation of Eq.(4-53) and (4-63), l1
13
=
= 333mm, 12= 463mm, and cO = 10OMpa are
used.
25
Eq.(4-53) and (4-63), t=2.4mm
----- Eq.(4-53) and (4-63), t=1.6mm
------ ----------------: ------------ -------
---- ---
-20 ---
Peak force from Eq.(4-53)
-7
-- - - ----- -- -- --
. ...
. . .
. . . . . . .
q (
- - - - --
. . .-
- - - - - - - - - . . . . .
-
- - -
5
0
- -- -- -- ----- - -- - -- ----------
. . . . ..
--
. .
. . ..
- -
-
- -
. . . .
200 - - --
-
L0
U-
4. .. . . . .
Eq (4 6 3
.
0
CL
Ci,
CD 10
.
15
-
(D,
0
50
100
150
Displacement (mm)
Fig.4-13 Crushing force of S frame
109
200
250
4.3 Three Dimensional S shaped Frame
L
00
e2
Fig.4-14 Three dimensional S shaped frame
The configuration of the three dimensional S shaped frame is shown in Fig.4-14. Because
of the vertical and horizontal eccentricities, the structure is be subjected to the complex
bending well as the combined loading of bending and compression when the external
crushing force is applied. In this section, the finite element analysis is carried out to
investigate the complex crushing mode, and the analytical explanation of the deformation
mode and the analytical derivation of the crushing resistance is followed.
4.3.1 Finite Element Modeling
The FE analysis of the three-dimensional S frame is carried out using non-linear finite
element code PAM-CRASH. Altogether four FE models of three-dimensional S frames
are made. Dimensions of 11, 12, 13, and 0 are fixed, and four different aspect ratios a/b (1.0,
1.3, 1.4, and 2.0) of the rectangular cross-sections are considered, where a is the width of
the wider column wall, and b of narrower width. The range considered in this study
covers the most cases of practical significance. Shown in Fig.4-15 is the S frame with the
square cross-section. The finite element mesh is constructed using HYPERMESH and
110
PAM-GENERIS. The rear end of the S frame is fully clamped by fixing all six degrees of
freedom, and only translational degree of freedom along the length is set free for the front
end. The velocity boundary condition is imposed for the moving front end of each
member. For the definition of the material property, Young's modulus E and energy
equivalence flow stress a, are chosen as 69000MPa and 10OMPa, respectively. Two
thickness, 1.6mm and 2.4mm are used for each case.
33049
Fig.4-15 Finite element mesh of a three-dimensional S frame (dimensions of length are in mm)
The deformed shapes of the 3-D S frames are shown in Fig.4-16-19. The deformed
shapes are taken at approximately 180mm of the end displacement. The actual finite
element calculation is done using PAM-CRASH on the SGI Octane workstation with dual
R10000 processors. Three localized plastic hinges are observed in all deformed shapes.
(b) Deformed
(a) Initial
Fig.4-16 Deformed shape of 3-D S frame (a/b = 1.0 : square cross-section)
(b) Deformed
(a) Initial
Fig.4-17 Deformed shape of 3-D S frame (a/b = 1.3)
111
(b) Deformed
(a) Initial
Fig.4-18 Deformed shape of 3-D S frame (a/b = 1.4)
(b) Deformed
(a) Initial
Fig.4-19 Deformed shape of 3-D S frame (a/b = 2.0)
4.3.2 Determination of the Bending Axis
The analysis of the orientation of bending axis on the fully plastic bending moment is
conducted to determine the bending axis of the plastic hinges in 3-D S frame. Shown in
Fig.4-20 is the relation between the cross-section and the orientation angle of bending
axis W.
Weak
Principal Bending Axis
a
Bending Axis
Strong
Principal Bending Axis
Fig.4-20 Orientation angle of bending axis
112
The fully plastic bending moment Mf is expressed as Eq.(4-66) and (4-67) depending on
the orientation angle W.
M f=
a 0-t
cos
/(
Mf = cot sin V/
+
72
tan)2 V+ab
_ t2
+ b+abi
2)
2
(05 V <900 - tan-1
(90 -tan-
b
V/
(4-66)
90)
(4-67)
Normalizing with the fully plastic bending moment about the weak bending axis
Mf ,weak =
obta +j
(4-6 8)
One gets the following expressions of normalized fully plastic bending moment about
any orientation angle W.
V
m =
(0< V<90' -tan
2(
sin
cot 2
+1
V+
2
~a)
(4-69)
+1
mf =
(90' -tan-'
2
b)
)
tan 2
+
+ 2CaI
cos
b <
< 90.)
(4-70)
+1
Plots of Eq.(4-69) and (4-70) with various aspect ratios are shown in Fig.4-21. It is shown
from Fig.4-21 that the point of the minimum normalized fully plastic bending moment mf
shifts from W = 450 to V = 90' as the aspect ratio changes from a/b = 1 to higher value.
This implies that the square and rectangular cross-section member with the smaller aspect
ratio than a certain critical aspect ratio will bend along non-principal axis while the
113
rectangular cross-section member with higher value of aspect ratio bends uniaxially along
weak principal bending axis.
1.8
E
C
E
0
a/b
1.6
=
2
a/b = 1.5
a/b = 1.4a/b = 1.3
CM
C:
1.4
.
(D
M
C.o
CZ,
1.2
a:
.
LL
_0
a/b =1.1
1.0
a/b =1 (sqaure)
0
Z
0.8
0
10
20
30
40
50
60
70
80
90
Orientation angle W (deg)
Fig.4-21 Normalized fully plastic bending moment
The orientation angle of minimum mf can be calculated by Eq.(4-7 1).
dmf
-0
(4-71)
d yf
This yields,
(
sin qi =
2
_
)2
(4-72)
-
1+2
(a
By inserting Eq.(4-72) into Eq.(4-70) and equating Eq.(4-70) with unity, the critical
aspect ratio is calculated.
114
2
2
1+2C
________
-~2
>=b1
1+2Qa -
(4-73)
2Qa+1
J
The calculated critical aspect ratio from Eq.(4-73) is,
bI~cr
= 1+12
F =1.366
(4-74)
Thus, the rectangular cross-section 3-D S frame member with higher aspect ratio than
1.366 shows uniaxial bending deformation on the plastic hinges, and square or
rectangular cross-section member with aspect ratio less than 1.366 bends diagonally or
biaxially. Shown in Fig.4-22 is the front view of each 3-D member. Clearly, a change of
deformation mode is observed between a/b=1.3 and alb=1.4. But the deformation in
a/b=1.4 and a/b=2.0 does not show "perfect" uniaxial bending mode. This is caused by
the fact that the bending deformation of each plastic hinge has to "accommodate" the
eccentricity along the strong principal axis of the cross-section. The perfectly uniaxial
bending deformations on all three hinges cannot be compatible with the deformation
mode of the entire structure.
(a) square (a/b = 1)
(b) a/b = 1.3
(c) a/b = 1.4
Fig.4-22 Front view of the deformation modes
115
(d) a/b = 2.0
Because of the eccentricity mentioned above, the torsion is developed on the plastic
hinges. Shown in Fig.4-23 is the sectional collapse mode for each plastic hinge (a/b=2.0).
The sectional deformations seem to be the superposition of the sectional collapse mode of
pure torsion [84] and pure bending. However, the collapse mode is dominated by pure
bending.
Fig.4-23 Sectional collapse modes in three plastic hinges (a/b = 2.0)
400
-Magnitude of sectional moment
-------- Torsional moment
-------------- ----------------------------
200
E
0
- --
-- - -
-
-- -
--- ---
E
-
z
-
300
100
-................
.
--.......
-.--.-.-..-..-..-.-.---.-
0
0
I
I
50
100
150
I
I
200
250
300
Crushing displacement (mm)
Fig.4-24 Contribution of the torsion to the total magnitude of the sectional moment (a/b = 2.0)
Also, the contribution of the torsion to the entire magnitude of the moment is shown in
Fig.4-24 from PAM-CRASH result. Clearly torsion is quite small compared to the other
moment components, which are bending moment. According to the experimental study
by White et al [57], a small magnitude of the torque seen in Fig.4-24 (about 10% of the
116
total magnitude of moment) from the torsion affects very little the bending moment. This
observation further supports the assumptions made in Chapter 1 regarding neglect of the
torsional action in crashworthiness calculations. Thus, in the calculation of the crushing
force of S-frame, torsion is neglected, and the calculated results show quite good
agreement with the finite element analysis results shown in the following section.
4.3.3 Calculation of Crushing Force of Three-Dimensional S Frame
Using the analytical expression derived in the previous chapters, the crushing response of
3-D S frame can be calculated. For the S frame with smaller aspect ratio than 1.366, MO
in Eq.(4-63) must be replaced by min(M W)) which is the minimum of actual resultant
fully plastic bending moment in bi-axial bending(Eq.(4-67)). This can be calculated by
inserting Eq.(4-72) into Eq.(4-67).
For example, a/b of the square cross-section member is unity. From Eq.(4-72),
sin f =1/ -5
(4-75)
Inserting this into Eq.(4-67),
Mf =
where Mf,o
olb 2t = 0. 9 4 3 M f~.
(4-76)
is the fully plastic bending moment of square cross-section when W = 0'.
The analytically calculated crushing resistance of three-dimensional S frame with square
and rectangular (a/b = 2.0) cross-section is compared with the FE analysis results in
Fig.4-25 and 4-26 for two different column wall thickness. The correlation is almost
perfect in both cases.
117
0
a
CO)
20
...
----...
---- - ----
15
oS 10
0
---- PAM-CRASH (t=2.4mm)
--------- -Analytical Prediction (t=2.4mm)
--------
---- PAM-CRASH (t=1.6mm)
----- Analytical Prediction (t=1.6mm)
.
Z
*
25
--~
-------...---.----..--.
..
-.-..............-~
-
LL
0
50
0
...........
-
-.
5~ --.-
250
200
150
100
.-
300
Crushing Displacement (mm)
Fig.4-25 Comparison of crushing force of Square Cross-section S frame
Analytical prediction vs FE result
25
20 ............
-....... ---------
----- PAM-CRASH (t=2.4mm)
Analytical Prediction (t=2.4mm)
..-.--. PAM-CRASH (t=1.6mm)
----- Analytical Prediction (t=1.6mm)
15
-- ----------- ------+
*
0
U-
01
C1)
0
. ..% - -
.. . . . . . . . . . .
-- -
50
100
150
200
...
.....I ..
.I ..
.
0
250
300
350
Crushing Displacement (mm)
Fig.4-26 Comparison of crushing force of Rectangular Cross-section S frame
Analytical prediction vs FE result
118
Chapter 5. Strengthening of Three Dimensional "S" Shaped
Frame
In the previous chapter, a detailed analysis of crushing of a three dimensional S shaped
frame was presented. It was observed that most of deformation occurs at the highly
localized plastic hinges leading to a global structural collapse. This is accompanied by a
very pronounced deformation of cross-section at the location of the plastic hinges.
Fig.5-2 Cross-sectional collapse under
Fig.5-1 Cross-sectional collapse under
biaxial (diagonal) bending
uniaxial bending
35
1
I
.
Constant sedicanal moment M, (Eq. (4-6))
. . . Decaying fLnction of M(E)(Eq(4-7))
30
I
I
25
Potential for increasing of energy absorption
20
15
10
5
0
I
0
50
100
150
200
Displacement (mm)
Fig.5-3 Comparison of crushing resistance - constant M. and decaying function of M(O)
119
Examples of change of cross-section at the plastic hinge for uniaxial and biaxial bending
are given in Fig.5-1 and Fig.5-2. In both cases, severe collapsing of cross-section is
observed. This sectional collapse causes the sectional bending moment to drop, which
results in the decrease of load resistance of the entire structure. A qualitative illustration
of the comparison of crushing energy absorption between constant sectional moment M
and decaying function of M(0) is shown in Fig.5-3. The structure is one-hinge model
considered in the previous chapter. Note that the shaded area between two curves denotes
the loss in the energy absorption by the sectional collapse.
Thus, in the design of a crashworthy S frame a key factor of reinforcing the S frame is to
increase the resistance to the sectional collapse. Two types of internal stiffeners are
considered in this chapter :
A diagonally positioned internal reinforcing member - This method is based on the
in-depth understanding of crushing response of 3-D S frame. The empty frame develops a
general flattening deformation mode in addition to localized plastic hinges. The presence
of internal stiffeners effectively prevents both failure modes. The diagonally positioned
internal stiffening member is introduced for aluminum extruded S frame (Chapter 5.1)
and spot-welded S frame with the hat-type cross-section (Chapter 5.2). The reinforcing
effect of the stiffening member on the progressive axial collapse is investigated for
different cross-sectional shapes as well as methods of triggering.
An aluminum foam filler - A lightweight foam core acts like a foundation and prevents
or retards cross-sectional collapse. The moment elevation due to the presence of
aluminum foam filler is quantified. The optimization technique based on the Sequential
Quadratic Programming (SQP) is introduced to maximize the crush energy absorption
and weight efficiency of the structure (Chapter 5.3). As a more realistic application, an
aluminum foam-filled front side rail of a mid-size passenger car Ford Taurus is taken for
the analysis. The combined optimization technique based on the numerical simulation
called "Design of Experiment (DOE)" and "Response Surface Method (RSM)" is used to
get the optimum design in terms of energy absorption and weight efficiency without any
120
prior analytical expression of the crushing resistance of the structural member (Chapter
5.4).
All of above methods result in efficient ways of strengthening a three dimensional S
shaped frame. The methodologies covered in this chapter can be used in the design of
front or real side rails of automotive structures.
121
5.1 Effect of the Cross-Sectional Shape on Crash Behavior of a Three
Dimensional "S" Frame
Improved crash performance of automotive components can be achieved by introducing
suitably designed internal stiffeners into crash energy absorbing member such as front
side rail [58]. The complex nested sections have been used traditionally by industry, but
the present research has proved that these stiffeners should be in the form of a flat
diaphragm or flanges.
In this chapter, a comprehensive study of strengthening the front rail members made of
extruded aluminum profile by an introduction of diagonally positioned reinforcing
member is made. Also, internal reinforcement by the aluminum foam-filling is studied at
length and shown to achieve highest specific energy absorption. A beneficial effect of
triggering on the progressive axial folding on the overall crushing response is also
investigated. Because of a complexity of the problem, the impact response of the
reinforced column cannot be solved analytically.
5.1.1 Formulation of the Problem and Finite Element Modeling
Consider the thin-walled S-shaped frame with square cross-section with or without an
inner stiffening member. The model can represent an idealized front rail of a car. The
aspect ratio of the cross-section (b/t) is 50 (80/1.6 = 50), where b is the flange width and t
is the thickness. The S-shape is composed of two circular arches in both Z plane and Y
plane. The detailed dimensions are given in Fig.5-4.
One end of the model is fully clamped, and the load is applied on the other end as the
velocity boundary condition in -X direction. All the degrees of freedom except for X
translation are fixed on the moving end to model the actual deformation of the front side
rail of a car under frontal collision. Because the geometry of the model is S-shaped, the
122
member will be subject to a combination of axial compression, bending, and torsional
moment.
380
80
80
380
Fig.5-4 Configuration of the model. All dimensions are in mm.
M
Fig.5-5 Loading condition
In this formulation the front end of the frame hits a rigid wall while the rear end is
continuously pushed by the remaining mass of the car, Fig.5-5.
The various methods of internal strengthening are investigated using an inner stiffening
member or filling sections with ultralight metallic core. Six cross-sectional shapes,
investigated in this study are shown in Fig.5-6.
123
LJNZ
tUltialight
nxal~Hic core
Fig.5-6 Various cross-sectional shapes considered
Fig.5-7 Finite element model
The finite element model, consisting of 8064 shell elements is shown in Fig.5-7. The
nodes in the moving end are connected with a rigid body constraint. In other words, these
nodes can translate and rotate as a rigid body. This configuration can be interpreted as a
virtual rigid massless plate attached to the moving end of the beam. The load in the form
of ramped velocity boundary condition is applied at the center of gravity of this rigid
body. The ramping time is 0.05sec, and a constant velocity is 2000mm/sec. The dynamic
effect is avoided by reducing the mass density of the model by the factor of 1000.
The geometrical model is established using the mesh generator program HYPERMESH.
The finite element model is then completed with the preprocessor PAM-GENERIS.
Actual calculations are performed on a SGI Octane machine with dual R-10000
processors using explicit finite element code PAM-CRASH. The post-processor PAMVIEW is used for visualization and data acquisition.
The column is made of the aluminum extrusion AA 6063 T7 with mechanical properties
of Young's modulus E = 6.9x10 4 N/mm 2 , initial yield stress ay = 86.94 N/mm 2, and
124
Poisson's ratio v = 0.3. The detailed stress-strain relation for this material is shown in
Fig.5-8.
200
-- ---..--..-.
180
160
-
.--... . . . -- . ..-- ..- . ..- . . . .. .
140
.
.--.
--.-..
-..
..
.-..
-.--..
120
100
Mn
80
- --.-.--.--.- .....
-...
--.-...
.
z
-.
...
-...
--.
.
---.
...
60
40
20
0
)0
0.02
0.04
0.06
0.08
0.10
0.12
0.14
strain
Fig.5-8 AA 6063 T7 stress-strain curve
As the internal self contact between the elements and segment-to-segment contact in the
severely deformed part are expected, the highly improved self-contact (contact algorithm
#36 using "3D bucket" global search algorithm) and segment-to-segment contact (#33
using "3D bucket" global search algorithm with edge treatment) provided by PAM-
CRASH are applied.
5.1.2. Results
5.1.2.1 Empty column
The evolution of sectional force response and the deformed shapes of the empty model
are shown in Fig.5-9 and Fig.5-10, respectively. The force response is taken from the
clamped end, and the computer run is made up to deformation of 60% of the initial
length. The force increases up to the peak value, decays dramatically, and stays constant
around 2kN. Global bending collapse is dominating the response, as observed in the
125
previous chapter. An important feature of the deformation mode is the lateral collapse in
the middle part of the model. The originally squared cross-section is entirely flattened
over the central part of the member, as shown in Fig.5-10 and Fig.5-11. The bending
resistance of the collapsed section is very low, which explains a small magnitude of the
barrier force.
20000
C.
.ry,
-I
15000
Th.
E..(
Th o y
Eq (
...
10000
0
0-
5000
...
0
0
100
300
200
400
)
. . . .. . . . . .
.
.. . . . . ... . . .
.. ..
500
600
Displacement (mm)
Fig.5-9 Force response of empty member
Ik
Fig.5-10 Deformed shape of empty member
Fig.5-11 A sequence of cross-sectional
collapse in a central part of the frame
126
5.1.2.2 Analytical prediction
The above deformation mode can be nicely reproduced on a simple construction paper
model of the S-frame, see Fig.5-12. It can be easily shown that the original square profile
can be entirely flattened out over its length by bending deformation of four corners. The
plastic work done to flatten the column is
Eb
= 4 )TMP11
2
(5-1)
where Mp = ayt2/4 is the fully plastic bending moment of the tube wall and 11 = 1.41 is the
total arc length of the S-frame. The plastic work Eb should be equal to the external work
done. Equating the above expression to the external work Eext = Pm 0.6 1, the mean
crushing force becomes
P, =3.65c0 t 2
(5-2)
The expression for Pm should be understood as an asymptotic value of the crush
resistance. In our example, the energy equivalent flow stress O= 160Mpa and the wall
thickness t = 1.6mm. The calculated mean crushing force is Pm 1.5kN, which agrees
well with the asymptotic value of the numerically obtained crushing force, see Fig.5-9.
Fig.5-12 A construction paper model of square S-frame. The member can be completely
flattened out with bending deformation only without membrane tension.
127
5.1.2.3 Effect of different reinforcingmember
It is expected that double cell extruded profiles will have a higher level of energy
absorption than the original square profile. Four different types of double cell profiles
were considered with diagonal or parallel position of diaphragms, see Fig.5-13. For
comparison, a foam-filled section was also included.
Empty
with inner wall 1I
with diaphragm I
with inner wall 2
with diaphragm 2
foam-filled
Fig.5-13 Various cross-sectional types
The thickness of diaphragm and inner wall is same as the column wall. For the aluminum
foam-filling model, cyf = IMPa of HYDRO aluminum foam is chosen, and the calculation
of the mechanical properties of foam is based on Ref.[59]. The force response, the energy
absorbed, and the deformed shapes of all models considered are shown in Fig.5-14, 5-15,
and 5-16, respectively.
All the models have similar characteristics in force response. After reaching a peak value,
the force decays and stays at a certain value. The initial peak force increases in all five
cases compared with the empty model. Among five types of reinforced members, foamfilling and diaphragm type 2 show the highest reinforcing effects. The diagonal
diaphragms are much more efficient in the way of reinforcing than parallel diaphragms.
Diaphragm type 2 shows more strengthening effect than diaphragm type 1. The
orientation of the diagonal flange to the plane of the S-frame is also very important. A
128
lateral collapse is observed on two bending hinges in the model with diaphragm 1, while
this behavior is not found in diaphragm type 2. Considering the direction of global
bending of the model, the diaphragm type 2 is positioned to have the strongest bending
resistance. Two types of sections with parallel inner wall show similar force responses
and deformation modes. The complete flattening mode by lateral deformation in global
bending zone occurs in both inner wall models.
30000
empty
- - ---
w/ diaphragm 1
------- w/ diaphragm 2
w/ inner wall 1
---- w/ inner wall 2
foam-filled, o =lMPa
20000
. - -...
1 5 00 0
- - -- - - - -
-- - ----- - --- --.. . . .. . . . .. . . .. . . . .... .... ...
0
15000 --10000. .
1000020
0
0
........................................................................
0
100
200
300
40
50
400
500
600
Displacement (mm)
Fig.5-14 Sectional force response
.
5000
-
empty
w/ diaphragm 1
4000
w/ diaphragm 2
--- w/ inner wall 1
--- w/ inner wall 2
foam-filled, ,=1MPa
-o 3000
-__-__-_-_-__-_-_-
-
-2
---
...
---- ----
- --- --..... ....
.
> 2000 -----
1000 - -
0
0
-.
100
200
300
400
Displacement (mm)
Fig.5-15 Energy absorbed
129
500
- ----
600
In terms of energy absorbed, the aluminum foam-filled model shows about 300% more
energy absorption than the empty model, and the model with diaphragm type 2 absorbs
200% more energy than the empty model.
(b) with diaphragm 2
(a) with diaphragm 1
(d) with inner wall 2
(c) with inner wall 1
(e) foam-filled
Fig.5-16 Deformed shapes
The weight efficiency of the structure is assessed through the ratio of the total energy
absorbed to the total structure weight. This measure is often called Specific Energy
Absorption (S.E.A.), according to;
130
S.E.A.
(53)
Energy absorbed
Structure weight
The calculated S.E.A. is compared in Fig.5-17 for six models. In the cases of diaphragm
type 2 the specific energy absorption is doubled. The relative benefit increases to 2.83 for
a foam-filled section.
2.2_ - -.
-
-_- - - - - -- _
. . --
-- - -
-
2 .0
C
.21
10.44
28
1.3
.3
.3
0.4U)
0.0
empty
w/ diaphragm 1 w/ diaphragm 2 w/
inner wall
1
w/ inner wall 2
foam-filled
cr,=1 MPa
Aluminum Extrusion Models
Fig.5-17 Comparison of Specific Energy Absorption
All the above double wall sections can be easily manufactured by the extrusion
technique. Triangular double-wall profiles with diagonal flange have so far not been
considered in the design of the aluminum space frame car bodies. The main reason is that
section with triangular geometry can never develop a stable progressive folding under
axial load. However, deformation modes of the S-frame must include bending at
generalized plastic hinges. It has been shown that superior bending resistance is achieved
for sections with diagonal internal flanges. A combination of single and double wall
131
member, optimum with respect to both bending and compression folding mode, will be
considered in the last section of the paper.
5.1.2.4 Effect offoam strength
The reinforcing effects by three different foam densities are compared for the S-frame
with a fixed wall thickness of 1.6mm. The specification of aluminum foam is given in
Table.5-1.
Table.5-1 Properties of aluminum foam* used
Density (relative
Foam strength a7
Weight fraction
density) (g/cm 3 )
(MPa)
of the foam
Foam 1
0.0486 (1.8%)
0.3519
18.9%
Foam 2
0.0945 (3.5%)
1.0000
28.0%
Foam 3
0.2025 (7.5%)
3.0799
44.8%
*
HYDRO aluminum foam with 150.4Mpa of plastic flow stress CJf is used.
The plots of force response, energy absorbed and specific energy absorption are shown in
Fig.5-18, 5-19, and 5-20 respectively. The plot of empty model is shown for comparison.
The deformed shapes are given in Fig.5-21.
The characteristics of force response are similar in all cases, but the transition from prepeak to post-peak stage is delayed more in the case of higher foam strength. More than
600% increase of energy absorbed is achieved by foam-filling, and 200% increase of
S.E.A. is observed. Note that the stronger the foam is , the higher is S.E.A.
The ultralight metallic foam-filling method appears to be the most efficient way to
reinforce the member under crush loading. This finding confirms the result of previous
work [60,61,62] on filling metal tubes with lightweight metallic core such as aluminum
foam or honeycomb under axial, bending, and torsional crushing. This study shows that
higher strength foam is preferable for the higher S.E.A., but it will increase the weight
132
contribution of foam at the same time. An optimization process can be adopted to get the
highest S.E.A. by varying the column wall thickness and foam density. This will be a
shown later in this chapter.
..................
. . . .
.
35000
.
S...
..... ...
30000
---------
25000
=Mpa
.......
-Y -=3Mpa.
_
__..
_
. .I. . . .. . . .. .
.
.
empty
af=0.352Mpa. ......
Z 20000
- ..
-....-.-----..
...
0
.
u_ 15000
5000
...
. .
.
...
.
-
.-.-. ...
--
---5
.
------....
. . . . . ..-
-
.
.
. ..
10000
0
0
100
200
400
300
600
500
Displacement (mm)
Fig.5-18 Force response of foam-filled member
8000
empty
-- --a=0.352Mpa
-
6000
-..-- =1Mpa
-- - ---- a=3Mpa
5000
-- --- ---------- --------
o 4000 --- ----- ........
------------
7000
......-------------- --
---
-----.
......---.3000 -.... ......e..... ........-
LLI
2000 - --- ---/ ....
............ ...-.-.- .-.-.-
-.
1000
0
-..
--------......... ... . .
.
a>)
)
100
200
300
400
500
Displacement (mm)
Fig.5-19 Energy absorbed by foam-filled member
133
--
600
2.82.62.42.2-
2.0
...
-........-.
....
.
.. ..
--.
-.
.
-.
..
.-ll
-..-oam
-t.
... .
-......
-.. . . ...
.
o 1.8-
.
1.6-
-1<-
1.4>1.2-
(D 1.0-
0.80.6CL0.4-
1.00
1.78
2.84
empty
ToaM-Tfmea
toam-tiiieci
o,=0.352Mpa
o,=1MPa
3.67
0.2
-
--
0.0
toam-tilled
af=3Mpa
Aluminum Extrusion Models
Fig.5-20 Comparison of Specific Energy Absorption
(b) Foam 2, aT = LMPa
(a) Foam 1, aY = 0.352MPa
(c) Foam 3, ay = 3MPa
Fig.5-21 Deformed shapes of foam-filled model
134
5.1.2.5 Partialfoam-filling
In this section we shall see if partial foam-filling a more weight efficient way of
strengthening than full foam filling. It is observed that most severe plastic deformations
are concentrated on one or two bending hinges and the remaining portion of the model
behaves elastically. Thus, a partially foam-filled model shown in Fig.5-22 is investigated.
For this model the same aluminum foam as in the foam model 2 is used. Two areas with
potential hinge location are reinforced by aluminum foam and the middle part of the
model is also filled with foam to prevent lateral sectional collapse.
Fig.5-22 Partially foam-filled model
As seen from the deformation mode given in Fig.5-23, the bending hinge moves to the
non-filling part, and the strengthening effect of foam is not utilized. Therefore, the partial
foam-filling method does not appear to be efficient way of reinforcing when the position
of plastic hinges is not known.
(b) displacement = 350mm
(a) displacement = 150mm
Fig.5-23 Deformed shapes
135
The force response of partially foam-filled model is shown in Fig.5-24 as compared with
the fully foam-filled model. The SEA of partially foam-filled model is 0.998kJ/kg, while
that of fully foam-filled model is 2.115kJ/kg. It is still higher than the SEA for the empty
structure (0.745kJ/kg).
30000
--- EEmpty
25000
Partially foam-filled
....
.-Fully foam-filled
.
20000
15000
0
0_
10000
-
............
5000
- --
A
0
100
300
200
400
..---
500
600
Displacement (mm)
Fig.5-24 Force response of partially foam-filled model
5.1.2.6 New design of diaphragmtype 2
The diaphragm type 2 shows the best strengthening effect among all models with an
internal flange stiffener. It prevents the lateral collapse of the member in the global
bending collapse zone, and yields 100% increase of S.E.A. The deformation mode of this
model shows the development of two generalized bending hinges. The new design of
diaphragm type 2, which is shown in Fig.5-25, is proposed for still higher weight
efficiency. The front and rear end portions of the diaphragm are removed because the
main plastic hinge is not formed in this part. The removal of those parts would induce a
progressive axial deformation in front and rear ends of the member. The middle part
cannot be removed to avoid the lateral sectional collapse of the structure.
136
(b) new diaphragm
(a) original diaphragm
Fig.5-25 New design of diaphragm type 2
The deformation mode of S-frame is shown in Fig.5-26. Two bending hinges are formed
in the reinforced part.
Fig.5-26 Deformed shapes of new diaphragm type 2
The plots of force response and absorbed energy are given in Fig.5-27 and Fig.5-28,
respectively. The overall characteristics and values are similar to those of the original
diaphragm 2. The new diaphragm 2 reduces the total weight by 10%, but the S.E.A. is
increased only 7% as the absorbed energy is slightly lowered due to the new diaphragm
2.
137
30000
25000 I
-----
---
20000
Empty model
Old diaphragm 2
New diaphragm 2
-
--..-..
-......--------------
z
C
0
L-
15000
1
10000
" ....
- -- --..-.
0
0
100
.-.
..-.-.
.
5000 1------- ----
200
300
400
500
600
Displacement (mm)
Fig.5-27 Comparison of force response for two types of diaphragm 2
3500
Empty model
iaham2------ Old d
iaphragm 2
-------- New
2
-
3000
------------ --------
--------.
.diaphragm
2500
~0
2000
-0
.0
500
0
- --------
0
-- - ---
-- --- - - --- - -.
. .. ..
.
1500
c,)
-a)
w: 1000 - ------ -- --
---
--
100
-----
200
- - - .........
300
400
500
600
Displacement (mm)
Fig.5-28 Comparison of energy absorbed for two types of diaphragm 2
138
5.1.2.7 Effect of imperfection
The models investigated so far have a "perfect" geometry and do not show much
deformations in the front clamped end. The geometrical imperfections are imposed in the
front end of the model to induce more axial deformation in this part. Two types of
imperfections are considered in this study. The first one is based on the elastic buckling
mode of a square cross-section. Concave and convex buckles are applied on each pair of
facing column wall (See Fig.5-29). The other one is based on the formation of plastic
fold. A pair of deep dents is applied on the opposite column walls(See Fig.5-30).
Fig.5-29 Imperfection type 1
Fig.5-30 Imperfection type 2
First, these imperfections are applied to the original empty model. The deformed shapes
of both models are shown in Fig.5-3 1. The severe lateral collapse occurs in the central
part, and the member cannot maintain high bending resistance in this part for the front
end to collapse axially. Thus the imperfections do not seem to work well in this case. The
plots of force response are given in Fig.5-32. The strengthening effect is not apparent in
both models.
139
(a) Imperfection type 1
(b) Imperfection type 2
Fig.5-31 Deformed shapes at displacement = 150mm
17500
-..... - - w/o imperfection -.
imperfection type 1
w/ imperfection type 2
-
15000
12500
-
10000
-
-
-
.-- .
-- -.
-.-.
-w/
-
........
zi
- .-- ....
5000
-.
2500
.
-..
.....-..
-..
.........
. ..-.
-.
.
7500
.-.-.-.-..
..-.
..-.
....-.
.............-.
-
0
0L
.-.
.--.
...
-.-.
-...-.-.
..
-.......
- ....
A
00
100
200
300
400
500
600
Displacement (mm)
Fig.5-32 Force response - Imperfection type 1 and imperfection type 2
Now, the imperfections are applied to the model with new diaphragm type 2. Because
this model has a high resistance to the lateral sectional collapse, the likelihood of the
development of axial folds is greater. Therefore the two types of imperfections discussed
previously are imposed on the model. As seen from the deformed shapes shown in Fig.533 and Fig.5-34, considerable plastic deformations are induced in clamped and moving
ends by imperfections. The lateral sectional collapse is not observed.
140
(a) Displacement = 150mm
(b) Displacement = 350mm
Fig.5-33 Deformed shapes of model with new diaphragm type 2 and imperfection type 1
(a) Displacement = 150mm
(b) Displacement = 350mm
Fig.5-34 Deformed shapes of model with new diaphragm type 2 and imperfection type 2
141
The plots of force response and absorbed energy are given in Fig.5-35 and Fig.5-36,
respectively. The formation of multiple plastic folds is apparently shown in force
response, and this gives rise to a high energy absorption. The comparison of S.E.A. is
given in Fig.5-37. Compared with the empty original model about 300% increase is
achieved. Thus, the triggering dents must be accompanied by the sufficient bending
resistance of the model. The S-frame with the partial diaphragm type 2 is sufficiently
strong in bending to induce the axial crushing efficiently.
30000
25000
-
--
----
20000
Empty model
- - - w/ full diaphragm 2, w/o imperfection
--...... w/ partial diaphragm, w/o imperfection
--w/ partial diaphragm, w/ imperfection type 1
w/ partial diaphragm, w/ imperfection type 2
1500le
--
--
-
4
10000
0
0
100
200
300
Displacement (mm)
Fig.5-35 Force response
142
400
500
600
5000
-
4500
----
-.--.. w/ partial diaphragm, w/o imperfection
---. w partial diaphragm, w/ imperfection type 1
w/ partial diafragm, w/imperfection type 2
3500
o 2500
-
-.-00-
-
-.
1000
500
0
0
0
100
300
200
400
6( 0
500
Displacement (mm)
Fig.5-36 Energy absorbed
9,
----------.....
-.
... .
2.6
-
wL
-
-r
-
- -
1500
--
2.4
2.2-
........
.
..............
0) 2.0
1.87
0
2-
1.6-
.0
1.4-
0)
CL
C-
- ---- - ---
-
c
.-..
-.s-..
-...-I.
-....
-...
-..
..
-..... .
-
2000
1.21.00.8-
..
.
0.6
1.00
U)
a. 0.4-
2.03
2,18
3.39
2.(
-
0)
-
i
/
---
0 3000
-
-
-
4000
-----
Empty model
w/ full diaphragm, w/o imperfection
0.20.0-
empty
full diaphgram 2
w/o imperfection
partial diaphragm partial diaphragm partial diaphragm
w/ impe rfection
w/o imperfection w/ imperfection
type 2
type 1
Aluminum Extrusion Models
Fig.5-37 Comparison of Specific Energy Absorption
143
5.1.3 Discussion
Several design aspects of front longitudinal members of a three dimensional aluminum
spaceframe are discussed and quantified. These are:
*
Position of internal diaphragm in the cross-section
*
Full and partial foam-filling
" Effect of foam density
" Double cell profiles with cut-out end portion of the internal member
*
Triggering dents
Over two dozen of computer runs were performed by a judicious choice of a combination
of all of the above parameters. Two types of design of the S-frame were found to be
superior over the remaining cases.
Optimum Design I consists of an internal diaphragm positioned diagonally in a square
cross-section and dividing the section into two triangular cells. Such a cross-section
offers a high resistance to plastic bending but behaves poorly in axial compression. This
deficiency is corrected by removing the end portion of the diagonal flanges and
introducing triggering dents. The total energy absorbed by such a modified member
increased 4.5 times compared to a baseline design without the reinforcing flange. At the
same time there was 20% increase in structural weight. Still the Specific Energy
Absorption increased by a factor of 3.39 compared to empty rails.
Optimum Design II uses an aluminum foam as a reinforcing agent. It was shown that for
best results the entire length of the member should be filled with the foam. The benefit of
the reinforcement increases with the foam density. For example, the foam with pf = 0.2
g/cm 3 density develops approximately 3MPa crushing stress. The S-frame filled with this
foam absorbs 600% more energy than empty member. Structural weight is higher by 63%
giving a spectacular 3.67 fold increase in the Specific Energy Absorption.
144
5.2 Effect of the Cross-sectional Shape of Hat-type Cross-sections on
Crash Resistance of an "S"-frame
In section 5.1 the design aspects of a front side rail structure of an automobile body,
which is an aluminum extrusion S-shaped member, from the point of view of weight
efficiency and energy absorption were addressed. Various ways of reinforcing the crosssection was investigated, and the result showed that the diagonally positioned diaphragm
with suitably designed triggering dents achieved 300% increase in the crush energy
absorption and 3.4 fold increase in the specific energy absorption as compared to
unstiffened member of the same cross-section.
In this section extruded members are replaced by spot-welded sheet metal structure. Thus
more practical aspects of designing of front or real side rails will be investigated [63,64].
Various methods of internal strengthening are investigated using inner stiffening member
or filling the spot-welded sheet metal S-frame with an ultralight metallic core. The
specific energy absorption which is the measure of weight efficiency and crashworthiness
is used to access the structural performance.
5.2.1 Formulation of the Problem and Finite Element Modeling
Consider again the thin-walled S-shaped frame with hat-type cross-section with or
without an inner stiffening member. The S-frame is an assembly of two or three main
sheet metal parts. Spot-welding is used to join the components. The model can represent
a highly idealized front side rail of a car. The reference cross-section without side flange
is square with the aspect ratio (bit) is 50 (80/1.6=50), where b is the web width and t is
the thickness. The width of side flange is 30mm. The S-shape is composed of two
circular arches in both Z plane and Y plane. The detailed dimensions are given in Fig.538.
145
1000
380
8030
80
Y
380
X
Fig.5-38 Configuration of the model. (all dimensions are in mm)
Fig.5-39 Loading condition
One end of the model is fully clamped, and the load is applied on the other end as the
velocity boundary condition in X-direction. All the degrees of freedoms except for X
translation are fixed on the moving end to model the actual deformation of the front side
rail of a car under frontal collision. Because the geometry of the model is S-shaped, the
member will be subjected to a combination of axial compression, bending, and torsional
moment. Therefore, from the point of view of loading, the above model corresponds well
to a real front rail of a passenger car.
Various orientation of cross-section and methods of internal strengthening are
investigated using the inner stiffening member or filling sections with an ultralight
146
metallic core. The cross-sectional shapes investigated in this study are shown in Fig.5-40.
The ten different types of the reinforcing member exhaust all possible cases encountered
in practice. The position of flanges in the front rail of a unitized steel body structure is an
important design parameter. Therefore, the present study will give a much needed
physical insight into the crash behavior of actual design. The dashed lines in Fig.5-40
denote that two cases for one type of cross-section are considered, i.e. with or without the
inner stiffening member. Altogether 21 cross-sectional shapes are investigated as a base
study.
Z
Y
Type I
Type 2
Type 3
Type 4
Type 5
Type 6
Type 7
Type 8
Type A
Type B
Foam-filled
Fig.5-40 Various cross-sectional shapes considered
The finite element model of the reference rail without internal stiffener consists of 8288
shell elements and is shown in Fig.5-41. For the model with internal stiffener, the number
of shell elements increases to 11248. The nodes in the moving end are connected with a
rigid body. This configuration can be interpreted as a virtual rigid massless plate attached
to the moving end of the beam. The load in the form of ramped velocity boundary
147
condition is applied at the center of gravity of this rigid body. The ramping time is
0.05sec, and a constant velocity is 2000mm/sec. The dynamic effect is avoided by
reducing the mass density of the model by the factor of 1000. The detailed methods for
finite element modeling such as the contact interface and tools for modeling are same as
in the previous chapter.
Fig.5-41 Finite element model
- --6 00 -------- ---- -- - ---
... .. .. .
---- --- -- -... .. . ..
-..
700
----- -- - --- -- -- --5 00 --------- --- --
-.
..-...
-
-
z
..
.. ... .. .... . .. . .. . .. ..-...-...-C/ 30 0 - ----- ---- ----.. ..-..
1 00 ------0
0.0
-..-.-.-.--.-.-.-
------ -- --- -
0.2
0.6
0.4
0.8
1.0
Strain
Fig.5-42 Stress-strain curve of the material used in this study
The model is made of the sheet metal steel with mechanical properties of Young's
2
modulus E = 2.07x 105 N/mm 2, initial yield stress ay = 335.47 N/mm , and Poisson's ratio
v = 0.3. The detailed stress-strain relation for this material is shown in Fig.5-42.
148
5.2.2 Results
5.2.2.1 Empty Model
In the first round of calculations ten types of cross-sections shown in Fig.5-40 without
any inner stiffening member are analyzed. The evolution of sectional force response,
energy absorption, and the deformed shapes of the model are shown in Fig.5-43, Fig.544, Fig.5-45, and Fig.5-46, respectively. The force response is taken from the clamped
end, and the computer run is made up to deformation of 50% of the initial length. In all
models the force increases up to the peak value, decays dramatically, and then stays
constant around 5-lOkN. Global bending collapse is dominating the response, and a
single local axial fold is observed near the clamped end or the moving end. The original
square hat-type cross-section is entirely flattened out over the central part of the member
in all empty models, see Fig.5-46. This type of deformation pattern was also observed in
the empty column of the previous section on the aluminum extrusion S-frame member.
The bending resistance of the flattened out section is very low, which explains a small
magnitude of the force resistance.
The types of cross-section considered in this study can be categorized into three original
hat-type profiles. In Type 1-4 the side web flanges are located parallel to each other
while in Type 5-8 they are positioned at a right angle. Type A and B are typical doublehat profile. Difference in the crash response can be best seen on a graph showing
evolution of the absorbed energy with displacement, see Fig.5-45. In the case without any
internal stiffeners, the classical double-hat profile shows the highest force response from
all other profiles.
149
*
70
- -- -4 ------- ----- - --
02 0 --10
---..
- -.. --..
--- ----
---- -- --- --
- - - - - - ----
- - - -
--.
-----. . .---------. . . . . .-..
------- ...
.. .. . . . .
-
.
------ ----..
---- -- -- -- - ------------------ - -- - - - - - --- - - --
-
--
-
---- ------ --------.--..-.- - ---- -- ---
-
--..--
Type
--.-- Type2
----~-Type32
A-- Type 3
-- Type 4
all without inner stiffener
50
50
04300
--
-------------
--------------
-----------------
.
60
0
200
100
0
500
400
300
Displacement (mm)
Fig.5-43 Force response of Type 1-4
70
60
y
-
-
--
-
-m-Type 5
----
40
-Type
A
all without inner stiffener
30
------
Type7
- -Type 8.
-- -
-
...............- .......Type
B
Type 6
50
0
---
0
100
---
- - - --
-- --
200
300
-----
-
10
400
Displacement (mm)
Fig.5-44 Force response of Type 5-8 and Type A, B
150
500
--- Type 1
Type 2
---A, --Type 3
6000
-
Type 4
-0- Type 5
-- I--Type 6
-
--
-
-
7000
---
. --"aM-
-
--
-7A
Type7
Type 8
.-
-L"
--.
-A
Type A
. ------- TypeB
4000
-
5000
---
UO
<
--.- -.
--- ---- - --
--
-..
-...--...
-
---. ..
3 0 00 --------------
2)
(D
- - ---------- -------- ---- - - - ----- - - - --------------------
20 00 ------- - -- -
1000 --- ------------
0
0
---------
100
- - ----------
All without inner stiffener
300
200
400
----
--
500
Displacement (mm)
Fig.5-45 Energy absorbed by various cross-sections (refer to Fig.5-40)
Type 1
Type 2
Type 3
Type 4
151
Type 5
Type 6
Type 7
Type 8
Type A
Type B
Fig.5-46 Deformed shapes of the various empty models
5.2.2.2 Model with an inner stiffening member
A various types of double cell profiles are investigated with an inner stiffening member
positioned diagonally or parallel to column wall. Dashed lines in Fig.5-40 show the
position of the inner stiffener. Type 1
-
8 cross-sections are with diagonally positioned
inner stiffener and Type A and B are typical double hat cross-section with double cell
profile. The thickness of the inner stiffener is same as the column wall, 1.6mm. The force
152
response, the energy absorbed, and the deformed shapes of all models considered are
shown in Fig.5-47, Fig.5-48, Fig.5-49, and Fig.5-50.
------------- --
100
. -.----------
80
-
- ----
-
Type 1 with
-- *--Type 2 with
--A--Type 3 with
-. --. Type 4 with
-------------------
inner stiffener
inner stiffener
inner stiffener
inner stiffener
-
120
z
60
0
40
20
- - -
-------
-
-
--------
-
---- ----
I
0
100
- -
I
.
200
----------
-- - - - --
I
.
300
-
*
U-
I
400
500
Displacement (mm)
Fig.5-47 Force response of Type 1-4 with inner stiffener
120
-Type5withinnerstiffener
-- -- Type 6 with inner stiffener
-- A--Type 7 with inner stiffener
-Type 8 with inner
stiffener
- - - - T ype 8 w ith in
ner stiffe
n er
-Type A with inner stiffener
-*-Type B with inner stiffener
100
80
- --
- ---
60
\ - ----------
-----
----
- ---- -- ------------ --..
........--------------
z
0
0
L-
--.........
-.
----
- - - ---
-- --
40
-----
20
.
0
I
100
*
~
-. ..
I
I
200
300
400
500
Displacement (mm)
Fig.5-48 Force response of Type 5-8 and Type A and B with inner stiffener
153
12000
All with inner stiffener
-U--Type 1
10000
-- A-- Type 3
-r-- Type 4
--+--Type 5
Type 6
---X Type 7
)K Type 8
-B- Type A
-,-- Type B
6000
)
U
-
-
- ---T
-W-
-
8000
'a
A
-
-
-
-- *--Type2
nn
4 0 0 0 ----- - -----
----..-.--.-.---.--
C
20 0 0 -- -
0
--------
- - ---------- - ----- - ---
100
------- - - - - ----
300
200
--
400
-
w
500
Displacement (mm)
Fig.5-49 Energy absorbed by various cross-section model with inner stiffener
Type 1 with inner stiffener
Type 2 with inner stiffener
Type 3 with inner stiffener
Type 4 with inner stiffener
154
Type 5 with inner stiffener
Type 6 with inner stiffener
Type 7 with inner stiffener
Type 8 with inner stiffener
Type A with inner stiffener
Type B with inner stiffener
Fig.5-50 Deformed shapes of various models with inner stiffener
As in the empty section all the models displayed similar characteristics in force response.
After reaching a peak value, the force decays and stays at a certain value. The reinforcing
by inner stiffener is clearly observed in all models, especially in the models with
diagonally positioned inner stiffener. The constant force level 5-10kN in empty models
the
increases up to 10-20kN when the inner flange was added. One can see from
deformed shapes in Fig.5-50 that the flattening over the central part of the member is
avoided by incorporating the diagonal inner stiffeners. But in case of Type A and B, it is
observed that the central part of the beam still tends to be flattened as the member
155
deforms further. This explains low reinforcing effect in the force response of these
models. A careful inspection of the force responses and energy absorption curves reveals
that Type 1, 3, 5, and 6 with inner stiffener show highest force level and energy
absorption. According to Kim and Wierzbicki [65], the bending axes can be positioned at
450 in diagonally loaded square profiles. Thus diagonal internal stiffener can be
positioned to have a weak or strong bending resistance. In Type 1,3,5, and 6 the diagonal
inner stiffener is positioned to have stronger bending resistance. This result shows that
the double cell profile with diagonally positioned inner stiffener is superior to the typical
double-cell/double-hat profile in terms of energy absorption. Also the orientation of
diagonal inner stiffening member is shown to be very important.
5.2.2.3 Specific Energy Absorption
The calculated specific energy absorption, which is defined in Eq.(5-3) is compared in
Fig.5-51 and Fig.5-52 for all empty and reinforced models. The specific energy
absorption increases in the models with diagonal inner stiffener, but decreases in doublecell/double-hat profile models. The average numerical values are summarized in Table.52.
Table 5-2. Summary of the Specific energy absorption
Average S.E.A.(kJ/kg)
Hat type profile I
Hat type profile 2
Hat type profile 3
(Type 1 - 4)
(Type 5 - 8)
(Type A and B)
0.8283
0.8860
1.1360
1.1233
1.1853
0.9500
135.6%
T 33.8%
1 16.4%
- Empty
Average S.E.A. (kJ/kg)
-
With inner stiffener
Change
156
'
-~-w I ---I
1.4-
.......
1.21.0-
..... .................. .............
.
........
.
0)
0.80 0.6-
0.4U)
0.20.01
1
2
4
3
5
6
7
8
A
B
Cross Section Type
Fig.5-51 Comparison of Specific Energy Absorption: Empty model
--------- -0)
0
1 .2 -
. .... . .. . . ..
1.0 -
...
-..-.-----
-
~0 0.8
-
2) 0.6
0
U.
LU
0.4 -CL
0.2
-
C)
1
2
3
4
5
6
7
8
A
B
Cross Section Type
Fig.5-52 Comparison of Specific Energy Absorption : Model with inner stiffener
157
All the above double cell profile hat-type cross-section members can be easily
manufactured with classical hat type cross-section member manufacturing process. It has
been shown that superior bending resistance and higher weight efficiency is achieved
with the diagonally positioned inner stiffening member. Various extensions of this idea
will be considered in the remaining sections of this paper.
5.2.2.4 Aluminum Foam-filled Model
The reinforcing effects by aluminum foam-filling is investigated in this section. The 7.5%
HYDRO aluminum foam is used in this study. The mass density, foam strength, and
weight fraction are 0.2025g/cm 3 , 3.08Mpa, and 17.4%, respectively. An empty crosssection Type 6 is chosen for foam-filling, as this model shows highest energy absorption.
The force response and the deformed shape are given in Fig.5-53 and Fig.5-54,
respectively. The reinforcing effect is apparent in the force response both in the peak
force and the mean force level. The PAM-CRASH model used in calculation has a
provision for spot-weld failure. Two nodes combined as spot-weld behaves as a single
node. At failure, spot-weld are allowed to separate after violation of the following failure
criterion.
(
Shear Force
AFAIL
<1
(5-4)
(
AFAIL
I+1
"
Normal Force
where
AFAIL = Spot-weld strength x Area of single spot-weld
(5-5)
It is observed that the spot-welded joint in central part of the rail failed and opened up
widely. This clearly lowers the energy absorption. Failure of spot-welds in single and
double hat foam-filled profile was also observed in the experimental study by Chen et al
[66]. Two modifications are considered to prevent rupture of welds. First the welding
pitch is reduced from 30mm to 15mm, and secondly the strength of the spot-welds is
doubled. In both modification the splitting failure in the spot-welded joint is not
158
observed. But reducing spot-welding pitch will increase the manufacturing cost, and
doubling the spot-welding strength may not be realistic. Another modification can be
considered, for example applying dense spot-welding pitch near the global bending hinge
parts and less dense pitch in other part, maintaining same number of spot-weld joints.
Further modification of the weldment strength is left for a possible continuation of this
work.
100
Original foam-filled model
- - -- - Foam-filled model with dense welding pitch
------- Foam-filled model with higher welding strength
Empty model (Type 6)
-
-
-----.--.
80
60
---
--- ------------------------- --------------
Onset of splitting failure
0
IL-
40
4 ----
-
--
--
--
-
-
-
--
---
-
-
--
---- --------
20
0
0
100
200
300
400
500
Displacement (mm)
Fig.5-53 Force response of foam-filled model
Model with dense welding pitch
Original model
Fig.5-54 Deformed shapes of aluminum foam-filled model
159
5.2.2.5 Model with varying orientationof the inner stiffener
It is well known that rectangular or square double-cell profiles develop a stable
progressive folding under axial load. Such a folding cannot be activated for triangular
profiles. At the same time the diagonally positioned internal stiffener shows stronger
bending resistance than rectangular sections. Thus the idea of "inner stiffener with
varying orientation along the length of member" could combine advantages of an
optimally designed member for axial and bending response. This idea is investigated in
this section. The concept is explained in Fig.5-55 and Fig.5-56. Beginning from c-00 , the
orientation angle x changes sinusoidally into c=45 0 . The case a=00 matches Type B
cross-section, and x=45' Type 1 cross-section in Fig.5-40. The schematic view of the
internally reinforced member is shown in Fig.5-57. In the axially collapsed parts, which
are clamped front and moving ends, double-hat/double-cell profile (Type B with internal
stiffener) is introduced. Along the length of the member, orientation of the inner stiffener
gradually changes to be positioned diagonally. In the central part of S-frame the crosssection is Type 1 with inner stiffener in Fig.5-40.
Fig.5-55 Varying internal stiffener
50
CD
30 --------0)
-------- -------- *------Ty e
ss-s--t-n------
Type 1 cross-section
0
-----
---
Rear end
Front end
Length of the member
Fig.5-56 Change of orientation angle X
160
Fig.5-57 S-frame with varying inner stiffener
To encourage a more axial folding, geometrical imperfections are imposed in the front
end part. Two designs of trigger are considered in this study. The first one is based on the
formation of plastic fold and shown in Fig.5-58. The other one is based on the elastic
buckling mode of the double-hat/double-cell cross-section. Concave and convex buckles
are applied on each pair of facing column wall (see Fig.5-59).
Fig.5-58 Imperfection type 1
Fig.5-59 Imperfection type 2
The force response and the deformed shapes of this quite sophisticated design are shown
in Fig.5-60 and Fig.5-61 respectively. The major characteristics in the force response
remain unchanged, but the force level is considerably raised compared to Type 1 or Type
B with inner stiffener. The model with the trigger based on the buckling mode shape
shows higher force response than with the imperfection type 1. This new model absorbed
42% more energy compared to the Type 1 with inner stiffener, and 3.5% of weight
161
reduction is obtained. As a result, 47% increase in the specific energy absorption is
achieved.
100
I
I
I
I
120
..----.imperfection type 1
Model with varying inner flange with
imperfection type 2
with
flange
inner
varying
Model with
- - - - Type 1 with internal stiffener
--.--.
Type B with internal stiffener
80
60
- --------- -----
---. - - -- - -- .---.-.
----.
...
-.
.
0
0
U-
40
--
- -
---
- - -- -
-
a)
20
0
0
100
300
200
400
500
Displacement (mm)
Fig.5-60 Force response of the member with varying inner stiffener
Model with imperfection type 2
Model with imperfection type 1
Fig.5-61 Deformed shapes of the model with varying inner stiffener
More plastic deformation in the axially collapsed part can be induced by reducing the
the
thickness of inner stiffener to lower the strength in this part. But it will decrease
global bending resistance at the same time. An optimization process can be adopted to get
the highest specific energy absorption by varying the wall and inner stiffener thickness.
This will be the possible subject of a future study.
162
5.2.2.6 Effect of triggerandpartialinternal stiffener
The cross-section Type 6 shows the best energy absorbing capacity among the ten crosssectional types considered with or without inner stiffener. The deformation mode of this
model shows the development of two generalized bending hinges in the central part of the
S-frame. The new design of internal stiffener, which is shown in Fig.5-62, is proposed for
still higher weight efficiency. The front and rear end portions of the stiffener are removed
because the main plastic bending hinge is not formed in this part. The removal of those
parts would induce a progressive axial deformation in front and rear ends of the member.
The middle part of the stiffener is left to prevent lateral sectional collapse of the model.
The geometrical imperfection based on the buckling mode shape is also introduced in the
front end part.
Fig.5-62 Type 6 with new design of internal stiffener and trigger
To investigate the partial internal stiffener and the trigger, six different models are
compared.
" Without internal stiffener and without trigger
" With full internal stiffener and without trigger
" With full internal stiffener and with trigger
" Without internal stiffener and with trigger
" With partial internal stiffener and without trigger
*
With partial internal stiffener and with trigger
(All models are Type 6 cross-section)
163
The plots of force response and the deformed shapes are given in Fig.5-63 and Fig.5-64.
With trigger only, the progressive axial collapse is not induced and the severe lateral
collapse occurs in the central part. In all the models without trigger or the model with full
internal stiffener and trigger, the deformation in axially collapsed part is not efficiently
developed. The model with partial internal stiffener and trigger shows outstanding energy
absorbing capacity among the all models considered. The formation of multiple plastic
folds is apparently shown in force response, and this gives rise to a high energy
absorption. Thus, the triggering dents must be accompanied by the sufficient bending
resistance of the model. The S-frame with the partial internal stiffener is sufficiently
strong in bending to induce axial crushing efficiently.
'
I
120
-
------------
100
80
-------------------
-------------------------------
-
-
w/o inner stiffener and w/o trigger
inner stiffener and w/o trigger
- - with
-- A- with inner stiffener and w/o trigger
--v-- w/o inner stiffener and with trigger
--- - with partial inner stiffener and w/o trigger
with partial inner stiffener and with trigger
-
0
0
100
300
200
400
500
Displacement (mm)
Fig.5-63 Force response of the various cases of Type 6 model
with inner stiffener and w/o trigger
w/o inner stiffener and w/o trigger
164
w/o inner stiffener and with trigger
with full inner stiffener and with trigger
with partial inner stiffener and w/o trigger
with partial inner stiffener and with trigger
Fig.5-64 Deformed shapes of various cases of Type 6 model
5.2.2.7 Specific Energy Absorption
The specific energy absorptions of Type 1, 6, A, and B with inner stiffener, aluminum
foam-filled model, model with varying inner stiffener, and Type 6 with partial inner
stiffener and trigger are compared in Fig.5-65. The Type A model with internal stiffener
is taken as the reference value. Aluminum foam-filled model and Type 6 with partial
internal stiffener and trigger show highest values in all models considered. Compared
with the reference model about 200% higher specific energy absorption is achieved. In
other words S.E.A. increased by a factor of 3. This result confirms what was obtained in
the previous study on aluminum extrusion S-frame. Two optimum designs were obtained
165
in the previous study, which are the S-frame with partial diaphragm and trigger, and
aluminum foam-filled model. In these designs S.E.A. also increased by a factor of 3.
3.0
.
I
I
I
..... ........
2.5---C
0
4.0
2.0--1.5 1 -------
----------- -----------
C
1.0
C.,
0.5CL,
o.o-4 '
1'0 "
1
Type A
Type B
Type 1
Type 6
with inner with inner with inner with inner
stiffener
stiffener stiffener stiffener
Foamfilled
(7.5%)
' '1'I"
1"
Foamfilled
with dense
welding
pitch
Model
with varying
inner
stiffener
and
trigger
Type 6
with partial
inner stiffener
and trigger
Fig.5-65 Specific energy absorption
5.2.3 Discussion
Several design aspects of closed hat-type S-frame were discussed and quantified. They
are:
*
Type of hat-type cross-section
*
Orientation of the cross-section
*
Position of the internal stiffening member
*
Aluminum foam-filling
*
Hat type double cell profile with cut-out portion of the internal member
*
Triggering dent
166
Over 30 computer runs were performed by a judicious choice of a combination of all of
the above parameters. As in the previous study on aluminum extrusion S-frame, two
types of design of the S-frame were found to be superior over the remaining cases.
Optimum Design 1 - This design consists of internal stiffener positioned diagonally to
have strong bending resistance. Such a cross-section offers a high resistance to plastic
bending but behaves poorly in axial compression. This deficiency is corrected by
removing the two end portions of the inner stiffener and introducing triggering dents. The
total energy absorbed by this model and the specific energy absorption increased by
190% and 203%, respectively, compared to the typical double-hat/double-cell profile
member.
Optimum Design 2 - This model used an aluminum foam as a reinforcing agent. When
3MPa foam was used, about 160% increase in energy absorption and 184% in the
specific energy absorption were achieved. To utilize this design concept, further study on
the strength of spot welds must be carried out.
The above results can be applied to the weight efficient and crashworthy design of front
or rear side rail of an automobile structure. Especially Optimum Design 1 proposed in
this study can be directly applied to the S-frames without changing current technology of
assembling a unitized car body. Future research will focus on the application of this idea
to the full car model.
167
5.3 Optimization of the Aluminum Foam-filled Three Dimensional "S"
Shaped Frame
The strengthening of S frame by applying aluminum foam-filler is investigated in this
chapter. Using the analytical closed-form expression of the crushing force of S frame
developed in Chapter 4, the optimization technique based on the Sequential Quadratic
Programming is introduced to maximize the crush energy absorption and weight
efficiency of the structure.
5.3.1 Optimization Formulation
A general problem of structural optimization for minimum weight can be stated as
follows: find the set of design variables, X, that will
Minimize
m(X)
Subject to
S=L(5-6)
X
_
X
i =1,n
X,
The function m(X), which is referred to as the objective or merit function, is the mass of
the structural member. The gj (X) are referred to as constraints, and they provide bounds
on various response quantities. The region of search for the optimum is limited by the
side constraints X L < X
X '.
For a proper crash energy management, a crash member is usually required to absorb
certain amount of kinetic energy (target energy absorption), that is,
j= 1
gI =En -
En
>0
168
(5-7)
where En is the actual energy absorption of the crash member during the crash
deformation; En is the target value.
On the other hand, the cross-section of the structural member must provide enough elastic
bending stiffness for normal loading condition. Thus, a lower bound of the sectional
bending stiffness must be set
j=2
g2
(5-8)
EI
-0
where E is the Young's modulus; I denotes the second moment of inertia of the crosssection. bl is a target value.
Therefore, the optimization problem of a structural component for minimum weight
under crashworthiness and bending stiffness constraints can be formulated in the
following way
Minimize
m(X)
Subject to
g 1 (X)=-E +
g 2 (X
XL
EI+EI
< 0
<n
0
(5-9)
X <x,
We shall in the following solve such optimization problem of an S-frame under crash
loading condition.
5.3.2 Crash Reponses and Energy Absorptions of S-Frames
In the Chapter 4, a closed-form solution of the force-deflection response of a thin-walled
S-frame undergoing frontal impact was derived by using a three-plastic hinge model. (see
169
Eq.(4-63)). The energy absorption during the structural collapse of the S-frame can then
be calculated by integrating the force-deflection curve
(5-10)
En = f Pdu
where P is the force response during collapse, and was given in Eq.(4-63); uj is the final
displacement of the tip of the S-frame.
Implementing the integration in Eq.(5-10) results in the expression of the energy
absorption of an empty thin-walled square S-frame
/fO+ J
+af+
En=MC0 (
(5-11)
-)
where Ee is the energy absorption of the empty S-frame; M0 and Co were given in Eq.(4and yf are the final angles in the three-plastic-
4) and Eq.(4-16), respectively; af ,i
hinge model, and are given respectively by
12
af
sin 2 0+ 4 Auf -4A[l 2 (cos0+0.5 sin2 6-1)-uf -1112
=
sin6
1 1 (l 1 +12)
l2 sin 2 0+ 4 Au, -4A1l 2 (cos 0+0.5 sin 2 0-1)-uf
(5-12)
1
Yf
I2 sin 2 0+ 4 Au, -4A1l 2 (cos0+0.5sin2 O-1)-uf
=
+1 2
sin0
(l1 +12)
where 1 1 ,l 2 and 6 are geometrical parameters defined in Fig.4-6; The expression for A is
given by
+12)
212
A =1(11
170
(5-13)
In order to improve the weight-efficiency of the crash members in energy absorption, the
idea of introducing lightweight cellular material, such as aluminum foam, into the hollow
space of the thin-walled beams has been investigated by a number of researchers. Chen et
al [68,69] carried out theoretical, numerical and experimental studies on the
strengthening effect of aluminum foam filler on the plastic resistance of thin-walled
square beams undergoing bending collapse. They found that the bending moment at the
plastic hinge was elevated due to the foam filling. The moment elevation was related to
the sectional geometry and the aluminum foam properties
AM =0.95c-f0 b3
j
(5-14)
where AM is the bending moment elevation due to foam filling; b is the sectional width;
denotes
-Of is the plastic flow stress of the base material of the aluminum foam; pf / p,
the relative density of the aluminum foam relative to the solid aluminum.
Fig.5-66 illustrates a aluminum foam-filled square cross-section with width b, wall
.
thickness t, and foam density pf
b............
............... ............
Fig.5-66 A foam-filled square cross-section
Due to the bending moment elevation, additional energy will be dissipated at the foamfilled plastic hinges. The energy absorption of a foam-filled thin-walled prismatic
member during a crash event can therefore be calculated
171
3
Ef=
Ee
n
(5-15)
+ AMXAOL
n1
where Ef is the energy absorption of the foam-filled S-frame; En is the energy
absorption of the non-filled S-frame, and is given in Eq.(5-1 1); AOi denotes the bending
angle at each plastic hinge of the S-frame,
A0 1 = af
A0 2 =/if
-0
A0 3
-0
-y'
(5-16)
The elastic bending stiffness of a foam-filled square cross-section can be calculated
EI = EO(-b
3
3t+-
6
bt3)+- (b-t)4 Ef
12
(5-17)
where E is the Young's modulus of the wall material; Ef is the elastic modulus of the
foam material, and is related to the foam density via [69]
2
Ef
E, S
A
KPs)
(5-18)
where ES is the Young's modulus of the base material of the aluminum foam.
5.3.3 Solution Algorithm
The problem formulated above is a constrained optimization problem with both objective
function and constraints being nonlinear. Closed form analytical and/or graphical
172
solutions for practical optimization problems are difficult to obtain if the number of
design variables is more than two and/or the constraint expressions are complex.
Therefore numerical methods must be used to solve most optimization problems. In these
methods, an initial design for the system is selected which is iteratively improved until no
further improvements are possible without violating any of the constraints.
In the present study, a Sequential Quadratic Programming (SQP) method was used to
solve the optimization problem. Given the general nonlinear problem, the principle idea
of SQP is the sequential linearization and formulation of a quadratic programming subproblem based on a quadratic approximation of the Lagrangian function. This quadratic
subproblem is then solved to find a search direction so that a sufficient decrease in a
merit function is obtained. For details readers are referred to reference [67].
5.3.4 Case Study
As a case study, let us consider a square foam-filled S-frame, with lengths
11 = 13 = 300mm, 13 = 490mm, initial angle 0 = 300, wall material being aluminum alloy
AA6060 T4 (po = 2.7g / cm 3 , EO = 69GPa, o- = 106. 1MPa ) and the foam material being
Hydro
aluminum
foam
(p, = 2.7g / cm 3 , ES = 69GPa, 0 -of = 150.4MPa ).
The
design
variables in the problem are the width b, wall thickness t, and the foam density pf, that
is, X={b, t, pj }.
Several values of target energy absorption are specified, while the target bending stiffness
remains constant EI = 25kNm 2 . The feasible range of the wall thickness is [0.7
+
3],
which is practical for aluminum automotive structures. The foam density is with the
range of pf = (0 - 20%)p,. With pj =0, the problem reduces to a non-filled one. Both
foam-filled and non-filled beams are optimized using the SQP algorithm, and the
173
optimum solutions are listed in Table 5-3. In all cases, the final tip displacement is
assumed to be uf = 300mm.
Table 5-3 Optimum solutions for empty and foam-filled S-frames
Variables
Constraints
b (mm)
No constraints
t (mm)
0.7-3.0
Pf
0-0.54, (0-20% relative density)
3
)
(g/cm 3
25
)
(kNm
2
F
E
F
E
F
E
F
E
F
E
b* (mm)
91.5
73.7
91.2
89.3
91.0
104.4
90.4
119.1
90.1
133.5
t- (mm)
0.70
3.0
0.70
3.0
0.70
3.0
0.70
3.0
0.70
3.0
0.111
0.0
0.131
0.0
0.150
0.0
0.169
0.0
0.187
0.0
1.846
2.747
2.028
3.326
2.197
3.888
2.356
4.437
2.506
4.975
1.084
0.728
1.233
0.752
1.365
0.772
1.486
0.789
1.597
0.804
Solutions
Pf
)
(g/cm 3
m* (kg)
SEA
(kJ/kg)
Note: F=Foam-filled; E=Empty
A few observations can be made based upon the above optimization results
*
The optimized specific energy absorption (SEA, energy absorption per unit mass) of
foam-filled S-frame is on average about 75% higher than that of non-filled one for the
specified target energy absorption levels, see Fig.5-67. This substantiates the
argument that the foam-filled members are superior to non-filled ones in the light of
weight-effective energy absorption.
174
*
The optimum wall thickness of a filled section is generally smaller than that of a nonfilled section. In the above examples, the lower bound of the wall thickness (0.7mm)
is chosen as the optimum value for the foam-filled sections, while the upper bound of
the wall thickness (3mm) is chosen for the non-filled sections.
" Another consequence of foam filling is the reduced sectional width compared to nonfilled sections. Therefore, in addition to the significant weight saving, considerable
volume reduction can also be achieved by utilizing aluminum foam filler.
For the target energy absorption levels considered in calculation, the optimum
solutions of filled section yield relatively low foam density (about 5% relative
density).
2.0
Foam-filled
-
1.5
C:
0
0~
1.0
2)
Empty
C
w
0.5
I
0
CL
-
*
C,)
0.0
2.0
2.5
3.0
3.5
4.0
Target Energy Absorption (kJ)
Fig.5-67 The specific energy absorptions of optimized foam-filled and empty sections
175
5.4 Numerical Optimization of Aluminum Foam-filled Front Side Rail
Generally, the improvements in car design have been based on engineering judgment or
on a trial-and-error approach. Neither guarantees that an optimal design is obtained.
Automatic optimization techniques that directly apply the FE crash model, such as the
usage of evolutionary-based optimization [31,78,79] require more than hundreds of FE
simulations, which cannot be performed in a reasonable time and cost. The Design Of
Experiment (DOE) technique with a few carefully chosen sample points in the design
space is considered by the industry as the best way to solve this dilemma in the field of
crashworthy design of structures [70]. With these sample points one can create a
surrogate or response surface model to perform optimization in all subsequent steps. A
systematic computer-based DOE followed by a response surface based optimization is
the dominant method in the field of crashworthiness optimization in terms of achieving a
balance between delivering the desired accuracy and time and cost for the simulation or
experiment [71,72]. The flow chart of the entire process is shown in Fig.5-68.
In this thesis, the optimization process described above is applied to the design of
aluminum foam-filled front side rail and subassembly structure of a mid size passenger
car (Ford Taurus) with the objective of minimizing the structure weight. Six design
variables, the thickness of two outer panels and four densities of aluminum foam in
different locations are used in the optimization process.
The front side rails are the most important structural member in frontal collision [73,81].
They absorb about 50% of crash energy during the collision process. The design of front
rail has to satisfy both crashworthiness and weight efficiency requirements, and therefore
has become a focus of activity in the crashworthiness design of full car structure.
Previous work on filling metal tube with lightweight metallic core such as aluminum
foam or honeycomb showed that large increase of energy absorption could be achieved
while maintaining high weight efficiency [59,74]. Advantages of reinforced members
over their empty counterparts are present in all three major loading cases encountered in
automotive structures, which are axial compression, bending, and torsion [59]. Thus, the
176
front side rail filled with lightweight metallic core can be an innovative way of
reinforcing the automotive structure without gaining considerable weight [80]. The
original non-filled front side rail structure with stiffening members made of sheet metal
steel is taken as a base-line model and will be compared with the optimized aluminum
foam-filled front side rail (without original sheet metal reinforcements) in terms of
energy absorption and weight efficiency.
Define optimization problem
Objective, constraints, design variables
Experimental Design
- Sampling process
Construction of the response surface (RS)
based on the results of simulation or experiment
-
Running Simulation or experiment
of the sample points
Numerical optimization based on RS
- Finding min or max points
confirmation run for optimal design
-
____
___No
ccQAccura cy/Convergence?
Accu o
Add new point to
*
rcntutR
reconstruct RS
Yes
Fig.5-68 General Procedure of the Crash Optimization
The crash simulation is performed by means of the nonlinear dynamic explicit code
PAM-CRASH. This study was carried out jointly with Safety Optimization and
Robustness (SOAR) Group in Ford Motor Company. The commercial optimization code
iSIGHT and an in-house optimization code of FordMotor Company were used.
177
5.4.1 Overview of the Model
Shown in Fig.5-69 is the original front rail model. Two main outer panels are assembled
by spot-welding, and the stiffeners are attached inside of the rail. The thickness of the
main panels is 1.9mm. In the front part, triggers in the form of corrugated flanges are
placed to reduce peak load and avoid the bending collapse in this part [82]. The
approximate dimension of the front end section is 100x130 (mm). The subframe or cradle
is attached to the rail by a rigid body constraint. The overall length of the model is
approximately 1390mm. A total of 4825 shell elements are used to construct the finite
element model.
Fig.5-69 Front side rail and subassembly structure
Simulation of the collision process into a rigid wall with the initial velocity of 31mph was
conducted for the original empty model. This model has the internal stiffeners made of
sheet metal. The termination time is taken to be 30msec, and the added mass of 713.93kg
is attached to the end part of the front rail to compensate for the mass of entire car
structure. As this model represents a half of the front rail and sub-frame, symmetry
boundary conditions are applied to the nodes of the cradle on the plane of symmetry.
Two empty models were considered in this study, with original trigger and without the
trigger. Every other dimension, and condition were same for both cases. The wall force
response is presented in Fig.5-70, and the deformed shape of the model with the trigger is
178
shown in Fig.5-71. The overall characteristics are similar in both cases. During the initial
10msec, the axial collapse of the front part takes place, and after that the force drops as
the global bending collapse is initiated. The transition from initial axial collapse to global
bending collapse is smoother in the model without trigger. The peak forces are slightly
lowered in this model with trigger, but the difference is small. One can see that there is a
room for improving the triggering mechanism in the considered structure. Re-designing
of the trigger will be presented in the following section. During the global bending
process, the force stays almost constant around 20kN.
100
trigger
-with
----- without trigger
------
I
60
-
------ .--.-
--
-- - -------------
-
80
2
cc
40
20
0
0
5
10
15
20
25
30
Time (msec)
Fig.5-70 Wall force response of original front rail structure
Fig.5-71 Deformed shape of original model with trigger
179
5.4.2 Redesigning of the Trigger
The trigger or initiator is commonly used in the design practice of front side rails to lower
the peak load, induce progressive folding in the axially collapsing part, and to avoid the
global bending mode. As seen in the previous section, the trigger imposed on the original
design does not work well. A new design of trigger is proposed in this study. The new
trigger is based on the elastic buckling mode of the cross-section. Concave and convex
buckles are applied on each pair of opposite column walls (See Fig.5-72). The depth of
dent was chosen to be 2mm.
A simulation of quasi-static crushing of front end part is carried out with the new trigger.
Three models are considered. The first one is the model without trigger, the second one is
the original model, and the third is the model with the newly designed trigger. The
thickness of 1.9mm is used for the column wall of all models. The force response is
shown in Fig.5-74. The "spike" of initial force response is successfully removed by a
simple design of the new trigger.
Fig.5-72 Elastic buckling mode shape of the cross-section
(a) w/o trigger
(b) original trigger
(c) new trigger
Fig.5-73 Front end parts for quasi-static crushing
180
100
-------
--
6--
w/o trigger
riniiIc1c~f
rnn ld s n
0
0
20
60
40
80
100
Crushing distance (mm)
Fig.5-74 Force response of crushing of front end parts
5.4.3 Description of Foam-filled Model
As seen from the deformation modes of empty front rail, there are four zones where
severe deformation occurs. The first zone is the front of the rail, where the axial collapse
takes place. The second, third, and fourth zones are the regions of global bending
collapse. Considering the space needed for the suspension, wheel housing and engine
compartment, the middle part of the rail is very narrow and has very high aspect ratio.
The dimension of the middle part (zone 2 in Fig.5-75) is approximately 40x90 (mm), and
for the rear end part (zone 3 in Fig.5-75) about 40x205 (mm). In this study, these four
parts are reinforced separately by aluminum foam filling. The newly designed trigger is
imposed on the front portion of the model. The foam filling locations are indicated in
Fig.5-75. Note that the internal sheet metal reinforcements in the original design are all
removed in the foam-filled model.
The aluminum foam model used in this study represent the HYDRO aluminum foam with
plastic flow stress of bare foam material o-o = l5OMpa. The idealized mechanical
behavior of the foam under compression is described in Fig.5-76. The mechanical
properties for the input data of foam material definition such as Young modulus E, first
181
tangent modulus Etj, shear modulus G, densification strain e,, foam crushing strength af,
foam shear crushing strength Tf are calculated using Eq.(5-19)-(5-24) [59]. Note that all
the mechanical properties of foam are determined by the foam relative density d = pf/ps,
where ps is the mass density of solid cell wall of the foam and pf is foam density. In this
study the relative density of foam is used as a parameter determining the foam strength.
Fig.5-75 Aluminum foam-filled front side rail
aY
E
2
a
Second Tangent Modulus
First Tangent Modulus
Crushing
Strength
E Young Modulus
Densification strain
Fig.5-76 Mechanical properties of aluminum foam (compression)
E = E S(P
As
182
(5-19)
Et = 0.02E
(5-20)
G=E
(5-21)
1.5
af =
(5-22)
P
\1.5
Tf
= 0 .5 0o
I ~(5-23)
( P)
6C =1-1.4 P(5-24)
where E, is the Young's modulus of solid cell wall of the foam, respectively.
5.4.4 Numerical Optimization
Prior to the optimization process, the Design Of Experiments (DOE) and Response
Surface Method (RSM) are employed to explore the design space of the foam filled front
rail. DOE provides a systematic and formal way of defining design matrix and studying
the effect of design variables. In this study, Latin Hypercube Sampling (LHS), which is
capable of capturing the higher order of nonlinearity using a larger order of levels with
less design points, is employed to explore the design space.
The optimal Latin Hypercube Sampling, first theoretically described in 1979 by McKay
et al [75], has become popular in many computer simulations for optimal design and
robustness analysis. Unlike the conventional factorial DOE, the LHS is capable of
capturing the higher order of nonlinearity using larger number of levels with fewer design
points [76]. The LHS is suitable for the case when one does not have any prior
knowledge of the parametric form of the model, and it ensures uniform coverage of the
entire design space [76]. For these reason the LHS is regarded as the best sampling
183
technique for structural crashworthiness design. A brief description of the LHS is given
below.
The main feature of LHS is that it stratifies all variables simultaneously. Let p, k, and N
denote the number of design variables, the number of levels for each design variable, and
the number of experiments or computer runs to be made, respectively. Each variable is
divided into k strata of equal marginal probability 1/k and the sample in each stratum is
drawn at the midpoint of each stratum. The components of each variable are matched at
random to construct a matrix configuration for experimentation. The design of
experiment matrix consists of N runs (rows) x p variables (columns). Other similar LHS
matrices can be generated by simply performing element permutations in each column.
To assure a uniform coverage of the entire design space, an optimization is performed
instead of random matching among all variables by minimizing the bias part of Mean
Square Error (MSE). The algorithm for optimal LHS construction is explained as follows.
Let m be one possible case of Latin Hypercudes M, s(m) be a pairwise exchange
operation between elements in a column of m among all possible column-pairwise
operations S(m). A subset T of S is constructed whose elements are called forbidden or
"taboo" exchange moves. The elements of T contain historical information from the
exchange process extending up to t previous exchanges. The incorporation of "taboo"
search provides a mechanism to guide an exploration of new local minima without falling
back into a local minimum from which it visited previously. The iterations continue until
a chosen number of total iterations has elapsed or no more improvement in m is obtained
for a chosen number of iterations.
In this study, the sampling size for optimal LHS is chosen to be 4n for six continuous
design variables xi, i = 1,2,...,6 (2 gage thickness and 4 relative foam densities) of a front
rail with aluminum foam-filled crash problem, where n is the number of design variables.
Namely, total of 25 computer experiments including the baseline are constructed. The
number of levels of each design variable in LHS is set to be 10 to capture the nonlinearity
of the responses. CAE crash simulations are then carried out based on the LHS matrix
generated to compute the deterministic responses.
184
Table 5-4 shows the optimal LHS design matrix and the corresponding responses such as
weight, energy absorption (IBE) and initial average force (IAF), where IE is the total
energy absorbed, IAF is the mean value of the force in the initial 7 msec. t1 and t2 are the
thickness of two outer panels, and d1 , d2 , d3 , and d4 are the relative aluminum foam
densities for each filling zone described in Fig.5-75, xi = { tI, t 2 , di, d2 , d3 , d4 1. Using the
LHS matrix the "efficient-to-compute" surrogate model of response surface functions
was constructed. Among the response surface methods, Stepwise Resgression (SR) was
chosen for its accuracy and robustness for FEA crash problems [77].
Table 5-4 Optimal LHS design matrix and FE results summary
Baseline
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
tI(mm)
t 2 (mm)
1.9000
1.2111
1.4667
1.2111
1.9778
2.2333
3.0000
1.4667
2.7444
3.0000
2.4889
1.4667
1.7222
3.0000
0.7000
2.2333
0.7000
0.9556
2.7444
1.7222
1.9778
0.7000
2.2333
0.9556
2.4889
1.9000
1.4667
2.7444
2.7444
1.7222
1.9778
1.2111
0.9556
2.4889
1.2111
2.4889
2.2333
2.2333
3.0000
0.7000
3.0000
1.4667
1.7222
2.2333
0.7000
1.4667
1.9778
0.9556
3.0000
0.7000
d2
dW
d3
WeightIAF
d4 (kg) IE (J) (A)
(kg)
________
0.0300
0.1667
0.0000
0.0833
0.0000
0.1944
0.0833
0.1667
0.2500
0.0556
0.0833
0.1667
0.2500
3.0000
0.1111
0.0556
0.2222
1.7222
0.2500
0.0000
0.0278
0.1389
0.1944
0.2222
0.1389
0.1800
0.1389
0.1944
0.0833
0.2500
0.0000
0.0833
0.0000
0.1667
0.1111
0.0278
0.1944
0.2222
0.2222
0.1667
0.1111
0.0556
0.0278
0.0556
0.1667
0.0000
0.2500
0.2500
0.0833
0.1389
0.2000
0.0000
0.0278
0.1667
0.1944
0.1944
0.0556
0.0556
0.0833
0.1111
0.0000
0.0000
0.2222
0.1111
0.1667
0.2222
0.2500
0.1389
0.1389
0.0278
0.2500
0.1667
0.0833
0.0833
0.2500
0.2000
0.0000
0.0833
0.2500
0.1667
0.0000
0.0000
0.1667
0.0278
0.2500
0.1389
0.2500
0.1944
0.1944
0.2222
0.0556
0.1111
0.0278
0.2222
0.1111
0.1667
0.0556
0.1389
0.0833
0.0833
26.648
22.175
25.024
25.503
26.706
26.903
26.983
22.221
29.952
28.206
27.198
25.385
28.164
31.940
21.076
28.914
22.509
21.474
29.884
22.346
25.339
23.303
26.390
24.720
26.811
______
14570
9338
11220
15010
15390
14380
13230
11120
14090
15520
12980
12830
16120
17040
11070
16160
10570
9770
16990
12460
13000
11650
13880
12040
14120
* Original empty model : weight = 26.22 1kg, IE = 12110 J, IAF = 55.922 kN
185
_____
47.384
43.510
51.277
61.029
40.743
68.050
65.309
41.489
81.271
61.800
71.532
58.883
65.817
86.069
27.776
76.188
39.822
34.933
79.446
20.625
40.960
44.693
57.505
61.516
57.753
In stepwise regression, the model is constructed recursively by adding or deleting the
independent predictions one at a time. When the model is built up, the procedure is
known as forward selection. The first step is to choose one predictor, which provides the
best fit. The second independent predictor to be added to the regression model is that
which provides the best fit in conjunction with the first predictor. Given the other
predictors already in the model, further optimum predictors are then added at each step in
a recursive fashion. Alternatively, backward elimination can be used. After certain
predictors have been added in the model, the predictors are dropped one at a time. The
predictor that has the least effect on the fit of the model is dropped at the stage. The
stepwise regression model is built by combining the techniques of forward selection with
backward elimination.
In this study, the function of the component weight is constructed with the linear basis
function while the quadratic basis function is selected to construct the internal energy and
initial average force response functions. The actual response
surface functions
constructed are given in Eq.(5-25), (5-26) and (5-27).
Weight [kg ]=14.22+ 3.133t, +-1.654t 2 +5.352d, +6.077d 2 +7.110d 3 +3.201d
IE [J ] = 6046 + 2615t +16020d 2 -1886td
+ 2956td3 -4080t 2 d 2
+7107t2 d3 + 4587t2 d4 -12740d d2 + 26080d d3 - 210.6t2
4
(5-25)
2
(5-26)
- 41410d2
(5-27)
IAF [kN] = 25.1+62.96d, +6.661tlt 2
where t, and t2 are in mm.
The Sequential Quadratic Programming (SQP) is selected as the optimization technique
to solve crash optimization problems. The forward finite difference with 5% step size is
used to calculate the objective and constraint sensitivities. After each approximate
optimization, a CAE confirmation run is performed to check the accuracy and
186
convergence of the response surface and optimization process. Fig.5-77 illustrates the
flowchart of the approximate optimization process.
.
MOMMIL
* Gages
* Relative
densities
Weight
DOE Sampling
(4n Optimal LHS)
Tota internal ener
CAE Crash
Initial average force
Simulation
I
Stepwise
Regression
RSM Based
Optimization
CAE Confirmation JRun
for Optimal Design
Accuracy/
No
Add new design
Convergence ?
Yes
STOP
Fig.5-77 The approximate optimization process
5.4.5 Optimization Results
Six parameters that have the significant impact are identified as the design variables,
namely two gages and four relative foam densities. The lower bound and upper bound for
each design variable are listed in Table 5-5.
187
Table 5-5 Design variable and design space
Lower
Bound
Baseline
Upper
Bound
Thickness of outer panel 1 (tI)
0.7
1.9
3.0
Thickness of outer panel 2 (t 2 )
0.7
1.9
3.0
Relative foam density of foam filling zone 1 (d1 )
0.0
0.03
0.25
Relative foam density of foam filling zone 2 (d2)
0.0
0.18
0.25
Relative foam density of foam filling zone 3 (d3 )
0.0
0.2
0.25
Relative foam density of foam filling zone 4 (d4)
0.0
0.2
0.25
Design Variable
Three single objective optimization problems are investigated to achieve a more balanced
design between weight and internal energy in this study. They are explained in Table 5-6.
Table 5-6 Optimum Designs
Target
Optimum Design 1
Optimum Design 2
Optimum Design 3
to minimize weight
to maximize IE
to minimize weight
IE
Constraint
IAF
17000 J
Weight 5 26.2 kg
IAF
55 kN
55kN
IE
IAF
12110 J
40kN
The upper bound of IAF in both problems is targeted at 55KN to control the peak force
response. Note that the weight constraint 26.221kg in Optimum Design 2 and lIE
constraint 121 10J in Optimum Design 3 are the weight and the energy absorption of the
original empty model.
The optimization problems are solved using SQP. Three iterations of confirmation run are
made for all optimization problems. The predicted optimum design based on the response
surface approximation, exact CAE confirmation run result, and the error percentage of
the three iterations are shown in Table 5-7, 5-8, and 5-9.
188
Table 5-7 Result summary for Optimum Design 1
Iteration 1
Prediction
Confirmation
Weight (kg)
26.8
26.8
0.0
IE (J)
17000
16269
4.5
IAF (kN)
53.8
52.9
1.7
%
Error
Iteration 2
Prediction
Confirmation
Weight (kg)
28.1
28.1
0.0
IE (J)
17000
17097
-0.6
IAF (kN)
55
56.4
-2.5
%
Error
Iteration 3
Prediction
Confirmation
Weight (kg)
28.0
28.0
0.0
IE (J)
17000
16660
2.0
IAF (kN)
55
55.7
-1.2
%
Error
Table 5-8 Result summary for Optimum Design 2
Iteration 1
Prediction
Confirmation
Weight (kg)
26.2
26.2
0.0
IE (J)
16594
15851
4.7
IAF (kN)
52.2
49.7
5.0
%
Error
Iteration 2
Prediction
Confirmation
Weight (kg)
26.2
26.2
0.0
IE (J)
16141
16454
-1.9
55
54.9
0.2
IAF (kN)
%
Error
Iteration 3
Prediction
Confirmation
Weight (kg)
26.2
26.2
0.0
IE (J)
16310
16534
-1.4
55
54.7
0.5
IAF (kN)
189
%
Error
Table 5-9 Result summary for Optimum Design 3
Iteration 1
Prediction
Confirmation
Weight (kg)
22.9
22.9
0.0
IE (J)
12110
11417
6.1
IAF (kN)
40.0
40.5
1.2
%
Error
Iteration 2
Confirmation
Weight (kg)
22.3
22.3
0.0
IE (J)
12110
12737
-4.9
40
39.3
1.8
IAF (kN)
Error
%
Prediction
Iteration 3
Error
Confirmation
Weight (kg)
22.1
22.1
0.0
IE (J)
12110
11861
2.1
40
45.0
-11.1
IAF (kN)
%
Prediction
The numbers marked bold in Table 5-7-9 are the finally chosen value for each
optimization process. Note that 26.2kg and 12110J are the weight and energy absorption
of original empty model with internal sheet metal stiffeners. In Optimum Design 1, the
energy absorption is increased by 41% (from 12110 to 17097J) and the weight is
increased by 7% (from 26.2 to 28.1 Kg) compared to the original non-filled model. But
the constraint of the LAF is violated. More iteration is required for the optimization
process to converge. In Problem 2, the optimization problem is converged after 3
iterations. The optimization results show that he internal energy is increased by about
37% (from 12110 to 16534 J) while the weight remains unchanged compared to the
original model. About 15% of weight is reduced in the Optimum Design 3 maintaining
the same energy absorption of the original model.
From the results above it is clearly seen that for the axially collapsing part, low strength
of foam is preferable, and high strength foam is better for global bending zone. The wall
190
force response is shown in Fig.5-78. In all optimized results, the peak force is lowered
considerably, and the resistance in bending is almost doubled.
100
Original empty mode
with internal sheet metal stiffener
---- Foam-filled Optimum Design 1
....... Foam-filled Optimum Design 2
---- Foam-filled Optimum Design 3
Zi
80
----
60
.-
------
-
-
-
--
I'
0
0Z
40
-------- --- --- -- -
-
--- - - --- -
.. . . . . .-
20
0
0
100
200
300
400
Displacement (mm)
Fig.5-78 Wall force response of Optimized aluminum foam-filled front side rail
5.4.6 Weight Efficiency
The measure used to assess the weight efficiency of the model is Specific Energy
Absorption. Up to 37% increase of S.E.A. is achieved in the optimized designs of
aluminum foam-filled front rail compared to the original non-filled model with internal
sheet metal stiffeners (see Fig.5-79).
191
0.7
0.6 - ---
c 0.5 -
0.3
c 0.2
C.
----
-
-- ---
~0.1
CL,
C.)
0.0
Original Design
with sheet
metal internal
Aluminum
foam-filled
optimal design 2
Aluminum
foam-filled
optimal design 1
Aluminum
foam-filled
optimal design 3
stiffener
Fig.5-79 Comparison of the specific energy absorption
original design vs foam-filled optimum designs
18000-
.
Optimal Design 1
16000 -
-
Optimal Design 2
.
-
-
_0- - Optimal Design 317
10
S14000 -
~ ~
-
(D 12000 ......................-.......-.....
-rrrji
.0
U
.
.with
10000
.
20
Ad0-0
22
.
24
26
28
.
-
Original empty model
internal sheet metal stiffener
30
Weight (kg)
Fig.5-80 Relation between weight and energy absorbed
192
32
Shown in Fig.5-80 is the relation between the weight and the absorbed energy. The
results of all computer runs are plotted as individual points. It is clearly seen that as the
weight increases the absorbed energy increases. The lower and upper bounds of the
distribution are plotted as dashed lines. The points of the optimal designs are located on
or above the upper bound, which means highest weight efficiency. Thus, one can see the
optimization process with the objective of minimizing weight is successfully carried out.
Also the result of the original empty model with stiffeners made of sheet metal is plotted
as an empty square. This point is placed at the lower bound. Thus one can see that the
reinforcing the front side rail by foam-filling gives either
1) reduced weight for the same level of energy absorption (horizontal arrow)
2) increased energy absorption for the same weight (vertical arrow)
In either case, considerable improvement in the performance in the S-frame is achieved.
193
Chapter 6. Conclusions and Recommendations
6.1 Conclusions
Throughout the extensive study on the crushing behavior of thin-walled structural
member subjected to complex loading cases, a sound understanding of mechanics of
complex crushing process was achieved. The analytical solution of the crushing
resistance of the three-dimensional "S" shaped frame was derived and the weight
efficient way of strengthening three-dimensional "S" shaped frame was found with
several types of proposed optimum designs. The accuracy of the analytical results was
established by comparison with the results of numerical simulations.
Biaxial Bending Collapse of Thin-walled Beams
The mechanics of biaxial bending up to deep collapse range was studied for the thinwalled rectangular and square cross-section beams. It was shown that the initial failure
locus ( interaction curve between two bending moments ) is composed of two intersecting
parabolas with a distinct point of slope discontinuity at the critical orientation angle
which distinguishes between two deformation modes of the biaxial bending. The initial
and subsequent interaction curves were constructed from the stress and strain distribution
analysis assuming rigid, perfectly plastic material behavior. Also, the normality rule was
shown to hold in the biaxial bending collapse case. The normality rule allows one to
predict the components of velocity and displacement vectors in beams subjected to
biaxial bending process.
Crush Behavior of Thin-walled Prismatic Column under Combined Bending and
Compression
The numerical and analytical study on the crushing characteristics of prismatic thinwalled beam subject to combined compression and bending loading was carried out to
give exhaustive answers to the questions about the form of the complex interaction curve.
194
The first failure locus under combined loading was constructed analytically using the
analysis of stress distribution over the cross-section considering the position of the
neutral axis. The predicted initial failure locus correlated well with the numerical results.
The failure locus was observed to shrink as the deformation progresses, and over most of
the range the shape of the locus is linear. Using the Shanley spring model and the concept
of the "Superbeam element", the analytical prediction of the interaction curve between
axial force and bending moment was presented and the accompanying finite element
analysis verified the results.
Analysis of Crushing Response of S shaped Frame
The theories developed on the biaxial bending collapse of prismatic beams and collapse
under combined bending and compression were applied to a practical problem of car
body design. Analytical solutions of the crushing resistance were derived for three
structural models with increasing degree of complexity : simple one-hinge pin-pin
supported prismatic thin-walled model, planar "S" shaped model, and a threedimensional "S" shaped model. All the analytical derivations correlated well with the
finite element results. It was shown that the simplified calculation routines can be
developed applicable to an early stage of design of car bodies based on the present
analysis.
Strengthening of Three-dimensional "S" Shaped Frame
Two types of internal stiffeners, i.e. diagonally positioned reinforcing member and the
ultralight foam-filler, were proposed and analyzed extensively for the strengthening of
"S" shaped member. The reinforcing was applied to both aluminum extruded member
and spot-welded hat-type cross-section member. It was shown that with the proper crosssectional shape and the type of reinforcing member as well as the triggering mechanism
the specific energy absorption could increase up to 267%. As the systematic process for
the design of structural member, the optimization technique based on "Sequential
Quadratic Programming" was applied to the aluminum foam-filled "S" shaped frame and
195
the combined optimization process of "Design of Experiment" and "Response Surface
Method" was carried out for the design of aluminum foam-filled front side rail structure.
It was shown that both approaches could be successfully incorporated in the design of car
body structures.
6.2 Future Research
It is suggested that further research should be conducted on the following issues:
" Three-dimensional interaction surface for three major loading components (two
components of bending moment and axial force) should be constructed for
general collapse of thin-walled structure.
* The analytical derivation for square and rectangular sections should be extended
to the arbitrary cross-section members
" The finite element methods were used to get an insight into the complex
deformation process and to verify the theories developed. Physical testing can be
developed for further verification of the present findings.
*
It was assumed that no fracture would take place throughout the present study.
The mechanism of the fracture initiation and propagation in the complex loading
should be developed for the more realistic modeling and application.
*
The assessments of the presented design methodologies are confined at the
component level. The extension of the present findings to the full-scale structural
models should be made.
196
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APPENDIX : Calculation of Sectional Force and Moment
using Shanley Spring Model
Case 1. Both Compression
~2
~i
6
j
Fig.A-1 Case 1 : Both Compression
The sectional force and moment are,
F=f1 + f
C
--= - +>
~C
2
1
H
8 33 33
M -f
beg
(f2
2 1
f
2)
3
4
H
3b C
/
Wii
1
41
1
1
C____I
0
3
4
3
4i
4
I__
__
4
3
4
(A-1)
(A-2)
4
Normalizing force and moment,
F
o-A
205
(A-3)
M
be
m=
(A-4)
2
Then the Eq.(A-1) and (A-2) are expressed as
Crob[
1
b-
C
1
+
3/4)7+3/4
+
nH
n=-A
V3/4)7-3/4
1
o-0A b
3
4
1_
+3/4
]
_
(A-5)
(A-6)
3/4--3/4_
where, according to Eq.(3-39)
C
u-0 A
2H
bO
1.9493( t
5 6
/ KC
3
3 j 2 +J1/
(A-7)
477
V kb
Case 2. fi - compression, f2 - tension
Subcase 1 : 51 is in postbuckling range, and 62 is in elastic range
f
8
8
82
Fig.A-2 Case 2: fi - Compression, f 2 - tension, subcase 1
The force is expressed as,
F = f, + f 2
206
CVIH
AE
2H
(A-8)
The moment is,
M
beg
b
2
2
efff
f2 =
beffCI
2 79=
b AE
2----
2 2H
21
(A-9)
Note that the b12 is used for the moment arm for the tensile part(f2 ), and bej/2 for
compressive part(f,).
Normalizing force and moment as (A-3) and (A-4), the Eq.(A-8) and (A-9) become,
CIi
E
g
22H
-OA
C -i'i 4
OAf9
(A-10)
E
3 2orH
(A-11)
Adding Eq.(A-10) and (A-11),
7 CVI
(A-12)
+
4
m = -- n
3
3 o-AJ3
Subcase 2 : 8 1 is in postbuckling range, and 62 is in plastic range
f
3
82
Fig.A-3 Case 2 : fi - compression ,f2 - tension, subcase 2
207
The force and moment are calculated as,
C
F = f, + f2 =
(A-13)
HO
beff
b
2
2
beff C
-2I-+2 Si
b
A2
(A-14)
2
H
After normalizing force and moment as Eq.(A-3), (A-4), the normalized force and
moment are expressed as,
n=
CH
c 0A
M=
C
cOA
--
Si
1
H4
- +31
3
(A-15)
(A-16)
Adding Eq.(A-15) and Eq.(A-16),
4
3
7 C
3 aA Si
(A-17)
Note that Eq.(A-17) is the same equation as Eq.(A-12). From Eq.(A-12) and (A-17)
one can see that the failure locus in bending dominant range is linear line with the
slope of -4/3 intersecting m axis at m = (7C/3arA) H / 9.
From Eq.(3-39) and (3-43), the displacement of spring one 81 can be uniquely related
to the parameter k and r7.
208
2 k 3+37
9 +16272
2
(A-18)
At the same time because Eq.(A-12) and Eq.(A-17) are straight lines with a slope 1 =
0.75, from the normality rule one can infer that
(A-19)
81 = bk -1.24
2
Thus, the shrinking yield condition is given finally by
m=
3
n+ - C
H
bk -1.24
2
3 u0A
(A-20)
which reduces to
4
(t)
m = -- n + 4.084 -
3
b
209
1
F
(A-21)