NUMERICAL AND EXPERIMENTAL STUDY OF BENDING BEHAVIOR OF
THIN WALLED BEAMS FILLED WITH METALLIC FOAMS.
Eng. Gustavo José Cazzola
National Technological University
General Pacheco, Buenos Aires 1617
Dr. Eng. Francisco Aparicio Izquierdo & Eng. Teresa Vicente Corral
Superior Institute of Investigation of Automobile (INSIA)
Polytechnic University of Madrid
Madrid, 28031
ABSTRACT
The purpose of this work is to know the behaviour of big rotation of thin walled beams filled with metallic foam under bending
loads. Potential applications of filled sections with metallic foam are to improve on the passive safety of vehicles, for example
to the improvement of coaches structures under lateral rollover. The filling acts as a slowdown of the collapse by beams wall
buckling, consequently more energy can be absorbed.
Introduction
The lateral resistance of the vehicle structures are conditioned by the bending resistance and capability of energy
absorption of beams used. The mentioned beams of low thickness to obtain the high inertias with low weight, which behaviour
is conditioned to the fold formation at the compression flange characterized by a moment-rotation response that displays as a
basic parameters : the maximum resistant moment and the negative slope of decresing resistance based on the plastic
rotation, , this fact is in opposition to the previous one of different construction with beams of low thickness being due therefore
to reach solutions that keep a difficult balance between resistance and weight. This behaviour had been studied by several
investigators [1-2] within whom it is possible to mention Dr. Andrés García Thesis, professor of the Polytechnic University of
Madrid.
Recent developments in the relation cost-benefit from the processes for the production of metallic cellular materials of low
density, such as the metallic foam [4], position them like an alternative of special interest for the application like elements of
absorption of energy to reinforce structures. The filling with metallic foam can be more efficient in terms of optimization of
weight compared with the increase of structural columns thickness[6-11].
The application of metallic foam filling in structures of vehicles is beginning to be used in certain zones of the tourism
vehicles, being totally novel any attempt of application in structures of coaches and buses .
Based on previously exposed Advanced Pore Morpholgy (APM) foams [12] has been used in this study, developed by
Fraunhofer for Institute Manufacturing and Advanced Materials, like filled material for sections buses bodies.
IFAM has developed and patented the process of pulvi-metallurgy for the foamed metals FOAMINAL ® . In contrast to
FOAMINAL® process the general concept of technology APM is to separate the process in two parts:
1-) Foam expansion
2-) Foam conformed in parts
The parts of foam with pore morphology outpost consist of metallic foam elements of small volume which are expanded
production volume /mass. When united with others in a separated process elements they form foam APM.
The objective of this paper is to analyze the crushing behavior of thin –walled beams in the deep bending collapse mode.
The strengthening method with aluminium foam are compared to the conventional method of wall thickening. The
effectiveness of the lightweight core is assessed by examining the ratio of energy absorption to the column weight.
The numerical models and tests presented in this work, were made in the facilities of the Superior Institute of Investigation
of Automobile (INSIA) of the Polytechnical University of Madrid.
Model of finite elements
The geometric model and the model of finite elements were made with the commercial code of finite elements ANSYS.
Beams were modelled with plate elements of 4- node shell element with six degrees of freedom, for each node, displacement
and rotations in the three directions of the space, and is applicable for great rotations problems and/or great nonlinear
deformations. The filled material of metallic foam was modeled with solid elements of 8- node with three degrees of freedom,
displacement in the three directions of the space, and is applicable to analyses that includes plasticity, to hyper elasticity,
creep, great displacements and deformations.
In order to prevent the interaction between the walls with the metallic foam material , contact elements were used. The filling
is defined as the “Master Element” , the walls of the beams are defined as the “Slave Element”. These contact elements are
based on Penalti formulation, where the geometric interaction between the surfaces in contact are penalized by forces of
opposite sense that they are proportional to the depth of penetration.
In the presence of adhesive, bonding between metal filler and the column wall is modelled with the model of friction of
Coulomb included in the contact element, in which a shear ι tension is defined, for which the sliding of the surfaces begings
like a fraction of the contact pressure p (ι = µ*p + COHE, where µ is the friction coefficient and factor COHE specifies the
resistance to the sliding by cohesion). Once the shear tension is exceeded the two surfaces will relatively slide respect to the
other.
Figure. 1- Behavior Modeled of the adhesive of the metallic foam
Material Modelled
The behaviour of the thin plate element for the material of the beams are based on the elastic-plastic Von Mises model with
isotropic algorithm of plasticity. The plastic hardening is based on the definition of the polygonal curve, in which both tangent
modules and plastic tensions must be adjusted to make the validation tests on the unfilled beams. The material of the columns
is steel St-42 type, with a Young modulus E = 210 Gpa, initial yield stress σy = 260 Mpa and Poisson´s ratio ν = 0,3.
Advanced Pore Morphology Aluminum Foam (APM aluminum Foam)
Most of metallic foam properties can be approximate on the basis of its relative ρfoam/ ρsolid (density of the foam ρfoam ,
divided by the density of the solid material ρsolid) and a constant for the considered property. These properties can be
calculated according to the following law [4-12]:
Property foam = Constant property*(ρfoam / ρsolid )n
Figure 2 is a good example of the general behaviour of metallic foam under compression loads. An initial approximately linear
regime is followed by and extended plastic plateau, truncated by a densification response at high strains during which the
stress again increases steeply.
Based on these characteristics, the behaviour of metallic foam is characterized by the elastic module E*, the plastic tension of
collapse σ*pl, the shear plastic tension ι*pl , the shear modulus G *, the densification strain εD . These parameters depend
strongly on the density of foam ρ*.
Figure. 2- Behaviour to compression of metallic foam
The mechanical properties of the modelled metallic foam have been determined by the following mathematical expressions[4]:
Parameter
Young’s Modulus E* / GPa
Plastic tensile collapse stresses σpl*
/ MPa
Equation
E*= Const. Young’s modulus*(ρ*/ρs )n
Const.mód. de Young = 80 GPa
ρs = 2.7 g/cm3 , n = 1.85
σpl* = Constcomp * ( ρ*/ ρs )n
Const.comp.. = 361 MPa
ρs = 2.7 g/cm3 , n = 2.27
Shear modulus G* / Gpa
G* ≈
Plastic shear collapse stresses ιpl* /
Mpa
Tensile strength σt / Mpa
0.5 * σ pl
Densification strain εD
(2)
(3)
3
* E*
8
(4)
*
1.1 * σ pl
(1)
(5)
*
(1 − 1.4 * (ρ * / ρs ) + 0.4 * (ρ * / ρs ) )
3
ρs = 2.7 g/cm3
(6)
Experimental and numerical results
Preparation of the test beam filled with metallic aluminium foam APM.
In order to filled up, an aluminium plate was placed at the end of beams. Later, the metallic foam was placed by gravity
through the top of beams until covering the height anticipated in each test beams.
The following step consisted of curing the adhesive with which the metallic foam spheres APM are covered, producing the
bonding among the metallic spheres and as well to the walls of the beam. The curing of the adhesive was made in a climatic
camera in which the cured time depended the temperature.
INSIA has a climatic camera in which a temperature of 150 ºC, can be reached that´ s why the cured time of the test beam
with the metallic foam filling was of 3 hours.
Figure 3- Test beam in climatic camera
Figure 4- Cured Foam into beam
Static bending test .
Description of the test method.
The following figure is the scheme of the test device. The force in the cable is measured by means of a loading cell. Next to
the pulley the displacement sensor mounts. The end of the cable of the displacement sensor is joined to a thin cable which is
joined in the other end to the screw fixed to beam that holds to the cable that contains the loading cell. The cables of
connection of the loading cell and the displacement sensor are connected to a signal amplifier. The amplified signals of
displacement and force pass through a conditioning, which is an analogical-digitalis converter. Finally the signals of originating
force and displacement of the conditioning arrive at the computer through the bus. The bus is connected to the computer data
acquisition card.
Test Parameters and quantities.
Previous measures to the test.
L0: Measurement from the point of the pulley tangency to the screw axis of which it is used to pull the beam.
b: Test beam length. Measurement from the screw axis of which it is used to pull to the midpoint of the inferior longitudinal
beam.
c: Measurement from the point of the pulley tangency to the midpoint of the inferior longitudinal beam.
Figure 5- Scheme of assembly and data acquisition of the test
In order to make the experimental validation of the mathematical models, five bending test were made in the following
sequence:
Test nº
Test tube with profiles of rectangular section
Specimens test code
1
Beam 80.40.3 empty
Prb_80403_sr
2
Beam 80.40.3 with 70 mm of foam filling length.
Prb_80403_cr150
3
Beam 80.40.3 with 200 mm of foam filling length.
Prb_80403_280
4
Beam 80.40.3 with 400 mm of foam filling length.
Prb_80403_480
5
Beam 80.40.4 empty
Prb_80404_ sr
For the mathematical models validation bending static tests were made on rectangular beams sections more frequently used
on the coaches structures. As rectangular representative section, beam 80,40 mm was used in thickness of 3 and 4 mm.
Figure 6- Deformation pattern of filled beam with APM foam.
Figure 7- Numerical simulation and experimental results of foam filling and empty beam.
Code
Height
of
filling
(mm)
Peak
moment
rotation
Model
(degrees)
5.83
Peak
moment
rotation
Error
%
-14.51
Peak
moment
Test
(Nm)
Peak
moment
Model
(Nm)
Peak
moment
Error
(%)
---
Peak
moment
rotation
Test
( degrees )
6.82
80403_SR
4457
4444
-0.29
80403_CR150
150
10.01
7.98
-20.28
4786
4940
+3.2
80403_CR280
280
9.87
7.83
-20.67
4929
4954
+0.5
80403_CR480
480
10.01
7.828
-21.8
5000
4947
-1
80404_SR
---
9.87
7.8
21
5566
6273
+18
Table 1- Comparative analysis of peak moment –rotation response obtained in the test and model of beam 80.40 mm
Plastic moment –rotation response
(Nm)
Beam
M(0º)
4444
4940
4954
4947
6273
80.40.3_sr
80.40.3_cr150_04
80.40.3_cr280_04
80.40.3_cr480_04
80.40.4_sr
M(5º)
3800
4425
4441
4433
5341
M(10º)
3330
4209
4237
4232
4900
M(15º)
2980
4058
3943
3975
4547
Falling rate
%
M(20º)
2700
3812
3726
3742
4218
M(25º)
2310
3538
3494
3543
3910
48
28.4
29.5
28.4
37.6
Table 2- Plastic moment-rotation response and falling rate between peak and minimum moment of beam 80.40 mm.
Plastic energy –rotation response
(Joule)
Beam
E(0º)
0
0
0
0
0
80.40.3_sr
80.40.3_cr150_04
80.40.3_cr280_04
80.40.3_cr480_04
80.40.4_sr
E(5º)
368
408
408
410
506
E(10º)
675
792
794
797
949
E(15º)
947
1141
1146
1151
1360
E(20º)
1196
1487
1481
1489
1750
%
E(25º)
1421
1774
1761
1774
2046
0
24.8
23.9
24.8
43.9
Table 3- Plastic energy-rotation response and variation rate respect to the test beam 80.40.3_sr
Specific energy –rotation response
(Joule/Kg)
Beam
80.40.3_sr
80.40.3_cr150_04
80.40.3_cr280_04
80.40.3_cr480_04
80.40.4_sr
E(0º)
0
0
0
0
0
E(5º)
141
147
140
130
147
E(10º)
259
286
272
252
276
E(15º)
364
412
392
364
396
E(20º)
460
536
507
471
509
%
E(25º)
546
640
603
559
595
0
17.2
10.4
2.3
9
Table 4- Specific plastic energy-rotation response and variation rate respect to the test beam 80.40.3_sr
Results Analysis
It is observed a high correlation between the results of the tests and the models of moment-rotation response. Respect to the
peak moment in the models, in the 3 mm thickness beam the peak rate error respect to the test is 3,2 percent. In the 4 mm
thickness beam a difference of 18 % over the value obtained in the test is observed (table 1).
This is because the models of finite elements have in the zone of elasto - plastic behaviour, a behaviour with a greater rigidity
to the real one. Thus, the angle in which it is reached the maximum resistance, is smaller in the models than in the tests. In the
zone of plastic collapse, the models reproduce beams behaviour with greater fidelity.
The plastic moment-rotation response, it is observed in the test beam (table 2) an increase of the peak plastic moment of 13.2
% and a decrease in the loss of resistant capacity of around 20 % of the foam filled beam respect to empty. The filled test
beam with 4 mm thickness presents an increase of the peak plastic moment of 25 % and a decrease in the loss of resistant
capacity of around 10 % respect to the empty beam of 3 mm thickness.
Plastic energy absorbed at a 25 degree rotation (table 3), was 24.8 % higher in the filled beam than the empty beam with
3mm thickness. In the beam of 4 mm thickness, an a increase of 43.9 % of the plastic energy was obtained respect to the 3
mm tickness beam.
Specific plastic energy absorbed (table 4), bring us information about efficiency of filling material efficiency like a energy
absorption element. With a minimun height of filling (150 mm ) , the specific plastic energy was 17.2 % greater than the
unfilled beam with 3 mm of thickness. Specific plastic energy of the unfilled beam with 4 mm of thickness was 9 % greater than
the unfilled beam with 3 mm of thickness
Conclusion
From the analysis of the results obtained in the present work, it is observed that the contribution of the metallic foam filling
respect to the capacity of absorption of energy in plastic deformation, was obtained increasing the pick moment so as
decreasing the negative slope of resistant moment. In the test beam unfilled foam with thickness of 4 mm the absorbed plastic
energy it is greater than in the test beam with foam filled. In the test foam filled beam with minimum foam filling, the specific
energy is greater than in the test beam empty with thickness of 4 mm. This indicates that the element of absorption of energy,
foam filling is more efficient than the increase of the thickness beam.
Another aspect to consider, is that the height of filling, with the density of foam used in the present study (0.4 g/cc) didn`t
influence in the response of energy absorption.
It is highly recommendable to deepen the study of the influence of the metallic foam filling in thin walled beams, like an
element of absorption of energy in vehicle structures. This will allow improve the structures behaviour as opposed to the
absorption of energy no increasing the weight of beams ( greater thickness). It is important to mention that the filling with
metallic foam produces changes in the process of manufacture of structures, due to the cured of foam inside the beams. This
will have to be considered for a future incorporation of metallic foam in the productive process of vehicles structures.
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