Tabri Local Impact Strength of Sandwich Panels

8/2/2019 Tabri Local Impact Strength of Sandwich Panels

1/70

TALLINN TECHNICAL UNIVERSITY

Faculty of Mechanical Engineering

Department of Machinery

Kristjan Tabri

Local Impact Strength of Sandwich PanelsMasters Thesis

Supervisor: Petri Varsta, Professor

Instructors: Jaan Metsaveer, Professor Emeritus

Martin Eerme, Doctor of Philosophy

Tallinn 2003


2/70

AUTHORS DECLARATION

I assure that this masters thesis is a result of my personal work and that no other than

the indicated aids have been used for its completion. Furthermore I assure that all

quotations and statements that have been inferred literally or in a general manner from

published or unpublished writings are marked as such. Beyond this I assure that the

work has not been used, neither completely nor in parts, for the passing of any

previous examinations.

Tallinn, February 7, 2003

Kristjan Tabri


3/70

TALLINN TECHNICAL UNIVERSITY ABSTRACTTitle:

Author:

Place:

Date:

Local Impact Strength of Sandwich Panels

Kristjan Tabri

Tallinn

07.02.2003

Number of pages:

Number of figures:

68

47

Supervisor:

Instructors:

Petri Varsta, Professor

Jaan Metsaveer, Professor Emeritus; Martin Eerme, PhD.

Keywords: Sandwich panels, impact load, bending energy, membrane energy,laboratory experiments, FE simulations, Cowper-Symonds model,

The purpose of the study is to understand the behaviour of I-core steel sandwich panel subjected to a

lateral impact load. Furthermore, the aim is to derive an analytical model describing panels behaviour andthe consequences of impact. Due to the impact, faceplate of the panel is deformed in high velocity. Itmeans that dynamic behaviour of materials should be considered. To verify proposed analytical modeldata is obtained by laboratory experiments and by finite element calculations.

The behaviour of sandwich panels is studied in a series of laboratory tests, where sandwich panels withfour different configurations are tested. General structure of tested panels remains unchanged during thetests and the only changing parameter is the thickness of the faceplate. The effect of core material isinvestigated by filling some of the panels with urethane foam. In the laboratory tests panels are hit by animpact head, which has some predetermined mass and velocity. The most important results of thelaboratory experiments are plastic energy absorption of the panel and the extent of deformation.

In addition to the laboratory experiments, impacts are simulated by finite element method using programLS-Dyna. FE simulations provide a possibility to determine what happens in a sandwich panel during theimpact. The FE simulations are used to obtain information about the velocity of the faceplate and coredisplacements. This analysis gives the transversal velocity profile, which can be approximated by linearline. The decrease of the velocity is shown to be slightly non-linear. Plastic energy absorption and theextent of the deformation are determined also in FE simulations. Several assumptions made in derivationof the analytical formulation are verified by the FE calculations.

Derived analytical model assumes that all the energy is absorbed by the faceplate of the panel, asdisplacements at steel core are small compared to the displacements of the faceplate and can thus beneglected. Furthermore, it is assumed that the panel has infinite length and the global bending of thefaceplate does not occur. The maximum extent of the deformation is assumed to be equal to the span of inner supports. Formulations for energy absorption are derived separately for membrane and bendingenergy. Both elastic and plastic deformation energies are considered. The effect of filling material is takeninto account by using Winklers foundation.

Comparison with laboratory experiments and FE simulations support the purposed analytical model asscatter between the results obtained by different methods is small. In the case of plastic deformationenergy the scatter is at worst 10%. Scatter is slightly larger in the case of total deformation energy. In thatcase the analytical model overestimates the deformation energy in lower deformation values. The reasonfor that is the methodology of calculation of the elastic energy. An improved solution is suggested forfurther research. Guidelines how to describe the behaviour of sandwich panel more precisely and thus howto limit the number of assumptions are also suggested.


4/70

TALLINNA TEHNIKALIKOOL RESMEEPealkiri:Autor:Koht:

Kuupev:

Sandwich paneelide lokaalne tugevus lkkoormuste korral

Kristjan Tabri

Tallinn

07.02.2003

Leheklgede arv:Jooniste arv:

68

47

Jrelvaataja:Juhendajad:

Professor Petri Varsta

Emeriitprofessor Jaan Metsaveer, vanemteadur Martin Eerme

Vtmesnad: Sandwich paneelid, lkkoormus, paindeenergia, membraanenergia,lplike elementide meetod, Cowper-Symondi mudel,

Kesoleva t eesmrgiks on tutvuda sandwich paneelide kitumisega lkkoormuste korral. Paneeli

kitumise ja lkkoormuse mjul tekkinud tagajrgede kirjeldamiseks on tuletatud analtilised valemid.Lkkoormuse tulemusena deformeerub paneeli lemine plaat suurel kiirusel, mis eeldab dnaamilistematerjaliomaduste kasutamist. Analtilise mudeli igsust on kontrollitud laborikatsetest ja lplikeelementide meetodil tehtud arvutustest saadud informatiooni kasutades.

Sandwich paneelide kitumist uuriti laborikatsete abil, kus testiti nelja erineva konfiguratsioonigapaneeli. Paneelid erinesid plaadistuse paksuse ja titeaine poolest. Testitud paneelidest kolm ei sisaldanudtiteainet ja ks oli tidetud uretaanvahuga. Laborikatsetes lasti paneelile kukkuda maral kehal, millel olikindlaksmratud mass ja kiirus. Laborikatsetest saadud thtsamad suurused olid plastnedeformatsioonienergia ja vigastuse ulatus.

Laborikatsetele lisaks simuleeriti kuuli ja paneeli kokkuprget lplike elementide meetodil kasutadesprogrammi LS-Dyna. Lplike elementide simulatsioonid annavad vimaluse jlgida paneeli kitumistkokkuprke ajal. Simulatsioonide abil on vimalik saada informatsiooni paneelis aset leidvate kiiruste jasiirete kohta. Anals osutas, et paneeli lemise plaadi deformeerumiskiiruse pik- ja pikisuunalist jaotustsaab aproksimeerida lineaarsete sirgete abil. Samuti ilmnes, et kiiruse vhenemise kirjeldamiseks ei piisavaid lineaarsest aproksimatsioonist. Sarnaselt laborikatsetele arvutati ka lplike elementide meetodiltehtud simulatsioonide abil plastne deformatsioonienergia ja vigastuse ulatus. Mitmete analtilistevalemite tuletamisel tehtud oletuste igsust on kontrollitud simulatsioonidest saadud informatsiooni abil.

Tuletatud analtiline mudel oletab, et kogu lgist saadud energia neeldub paneeli lemisesplaadis kuna paneeli jigastajates aset leidvad siirded on vikesed vrreldes plaadi siiretega.Samuti on oletatud, et paneel on lpmatu pikkusega ja lkkoormus ei tekita lemises plaadis

laiaulatuslikku lbipainet. Vigastuse maksimaalseks ulatuseks piksuunas on vetud paneelisisemiste jigastajate vahekaugus. Valemid nii elastse kui ka plastse deformatsioonienergiaarvutamiseks on tuletatud eraldi painde- ja membraanenergia jaoks. Uretaanvahu mju on vetudarvesse kasutades Winkleri teooriat.

Laborikatsete, lplike elementide meetodil tehtud arvutuste ja analtilise mudeliga saadudtulemuste kokkulangevust vib lugeda heaks kuna erinevused eri tulemuste vahel on vikesed.Plastse deformatisoonienergia korral erinevus on halvimal juhul 10%. Erinevused on suuremadkoguenergia korral, mil analtiline mudel lehindab neeldunud eneriat vikeste vigastuste puhul.Erinevuse tekib elastse deformatioonienergia arvutamisel kasutatud metoodika. Edasiseksuurimiseks on vlja pakutud parandatud mudel elastse energia tpsemaks kirjeldamiseks. Samuti

on antud soovitusi tuletatud mudeli parandamiseks ja tehtud oletuste mju vhendamiseks.


5/70

Local Impact Strength of Sandwich Panels Kristjan Tabri

3

PREFACE

This work is done for the EU-project entitled: Advanced Composite Sandwich Steel Structures.

This project started on 1.04.2000 and its duration is three years. The SANDWICH project will

develop products utilising sophisticated lightweight steel sandwich panels for primary loadcarrying structures. The Ship Laboratory of Helsinki University of Technology (HUT)

participates in the project as a partner.

I am grateful to supervisor, Professor Petri Varsta, and to the instructors Professor Emeritus

Jaan Metsaveer and Ph.D. Martin Eerme for valuable and essential guidance and

encouragement they gave me throughout the study.

I would like to express my gratitude to Dr.Tech. Pentti Kujala and Lic.Tech. Hendrik Naar for

giving me vital instructions in many fields. I also wish to thank the personnel both in HUT and

in Tallinn Technical University for pleasant and versatile contribution. Last but not least, I

would like to thank Hannele for the support she gave me throughout the study.


6/70


4

CONTENTS

ABSTRACT .................................................... ............................................................ .............................................. 1

KOKKUVTE ......................................................... ............................................................ .................................... 2

PREFACE....................................................... ............................................................ .............................................. 3

CONTENTS..............................................................................................................................................................4

NOTATIONS............................................................................................................................................................6

1 INTRODUCTION ...........................................................................................................................................8

1.1 B ACKGROUND OF THE STUDY ....................................................................................................................8

1.2 R ESEARCH PROBLEMS AND THE PURPOSE OF THE STUDY ...........................................................................8

1.3 L IMITATIONS OF THE STUDY ....................................................................................................................11

2 EXPERIMENTAL SETUP...........................................................................................................................13

2.1 T ESTED STRUCTURES AND MATERIAL PROPERTIES ...................................................................................13

2.2 T EST EQUIPMENT , DATA ACQUISITION AND STORAGE ..............................................................................16

2.3 M EASURED / CALCULATED QUANTITIES ..................................................................................................18

2.3.1 Velocity of the impact head before the impact....................................................................................19

2.3.2 Permanent deflection of the faceplate ................................................................................................20

2.3.3 Deformation energy of the panel ........................................................................................................20

2.4 R ESULTS OF THE LABORATORY TESTS ......................................................................................................21

3 FINITE ELEMENT ANALYSIS .................................................................................................................26

3.1 G EOMETRY OF THE FE MODEL AND THE SIMULATION PROCEDURE ..........................................................26

3.2 M ATERIAL PROPERTIES OF THE MODEL ....................................................................................................29

3.3 R ESULTS OF THE FE ANALYSIS ................................................................................................................30

4 ANALYTICAL FORMULATIONS.............................................................................................................35

4.1 B ACKGROUND AND MAIN ASSUMPTIONS ..................................................................................................35

4.2 A NALYTICAL DESCRIPTION OF THE DEFORMATION SHAPE .......................................................................37

4.3 S TRAIN RATE ...........................................................................................................................................39

4.4 E NERGY ABSORPTION OF THE PANEL .......................................................................................................42

4.4.1 Elastic energy absorbed by bending...................................................................................................42

4.4.2 Elastic energy absorbed by membrane mechanism............................................................................46

4.4.3 Plastic energy absorbed by bending...................................................................................................47

4.4.4 Plastic energy absorbed by membrane mechanism............................................................................49

4.4.5 Energy absorbed by core filling .........................................................................................................49

4.4.6 Approximate solution for membrane energy.................................................................................. .....50


7/70


5

4.5 S OLUTION PROCEDURE ............................................................................................................................54

5 COMPARISON BETWEEN THE LABORATORY TESTS, FE CALCULATIONS AND THE

ANALYTICAL FORMULATIONS ..................................................... ........................................................... .....58

6 CONCLUSIONS............................................................................................................................................64

REFERENCES ......................................................... ............................................................ .................................. 67


8/70


6

NOTATIONS

a, b, c, d constants

B breath

cV ratio between initial and average velocity

C1, C2 constants determining the shape of deformation

CS coefficient used to scale yield stress

D constant describing dynamic behaviour of material

E Youngs modulus, energy

F forceG shear modulus of steel material

k foundation constant

K constant describing material properties

L length

m mass

MP plastic moment

pF support reaction

q constant describing dynamic behaviour of material

r radius

R width of deformation

t plate thickness

v velocity

V volume

w deflection

maximum deflection

angle

S angle of deformation

V angle of velocity profile

strain

strain rate


9/70


7

distributed load

Poisson's ratio of steel material

F compressive strength of filling material

Y static yield stress

D

Y dynamic yield stress

Subscripts

0 initialA average

B bendingEF effective

F filling

I impact body

M membrane

Superscripts

* simplified equationE elasticP plasticD dynamic

Abbreviations

FE Finite Element


10/70


8

1 INTRODUCTION

1.1 Background of the study

The weight of structures has significant importance in ships and in other forms of

transportation. Decreased structural weight allows vessel to transport larger amount of goodsand passengers with a lower expenses. In luxurious cruise ships more and more attractions

should be added to ship in order to keep customers satisfied. All additional recreational

alternatives increase the lightweight of the and in order to keep the buoyancy in same level, the

weight of structures should be increased. The importance of ship buoyancy can be hardly

overestimated as it has straight impact to the resistance and thus also to the energy consumption

of vessel.

Though the weight is one of the most important parameters in design, there are still a lot of

other requirements and demands for structures, which should be satisfied. Especially in marine

structures attention should be paid to strength, noise, vibrations, safety, manufacturing and

installation of structures. Large amount of requirements have made it almost impossible to

satisfy all the demands just by improving conventional structures.

Increasing demand for the lighter and more efficient structures has challenged the engineers toinvent new solutions to improve the structures and satisfy the demands.

1.2 Research problems and the purpose of the study

The weight of structures can be decreased using lighter materials, new constructions or

combining them. In nowadays industry sandwich structures are used to overcome the increaseddemands. General drawing of the sandwich panel is given in Figure 1. Two outer layers, skins


11/70


9

or faceplates, are made of material that gives enough strength and stiffness, abrasive and

corrosive resistance, noise isolation and easy production. In order to increase the thickness of

the panel, and thus to increase the stiffness, without using heavy materials in the skins, a light

core material is placed between the plates. Several criteria should be considered when selecting

the core material. Density, mechanical properties, bond properties, fire isolation are just few

examples.

Figure 1. I-core steel sandwich panel.

Combination of high stiffness and low weight was first used in aircrafts during the Second

World War. Combination of balsa in core and veneer in skins was used because of the lack of

high strength materials. Nowadays sandwich panels are used even in space research industry

where beside the other properties also high impact resistance is appreciated.

Improved welding techniques, especially laser welding, have made it possible to connect very

thin sheets to each other and so to manufacture panels where thin faceplates are welded to steel

core structure. In marine industry the combination of new welding possibilities, material and

strength properties of sandwich panels have made them to be good substitution for

conventional structures. Good examples are balconies, decks and bulkheads where sandwich

panels replace conventional stiffened plating. Figure 2 presents some possible uses for

sandwich panels.

Sandwich panels are efficient in means of global response as panels thickness and sectional

modulus are bigger compared to conventional stiffened plating. Moment caused by bending is

carried by the faceplates while light and low-strength core sustains shear forces. Core

contributes to the global response also in other ways. It makes it possible to increase the span of

the faceplates without loosing local stiffness. Core also supports the faceplates and distributes

stresses to larger area and so prevents the global bending of the faceplate. In other hand thecontribution allows to reduce the thickness of the faceplates and to decrease weight.


12/70


10

conventional ship structure ship structure with sandwich panel

Figure 2. Usage of the sandwich panels in contemporary ship structures.

Weight reduction by using thinner skins introduces a new problem. Though the required global

bending resistance can be achieved by using very thin faceplates, still it can be weakened even

when relatively small body strikes the panel and causes permanent damage. Local deflection in

the faceplate of the panel can decrease the bending resistance significantly. Again core is one

possibility to prevent the local deflections, but also the use of some faceplate coatings or new

steel core structures can be effective to prevent the serious consequences caused by any kind of

impacts on sandwich panels. It should be noted that impact not only causes local deflections to

the faceplate, but may also cause widespread global bending of the faceplate. The global

bending of the faceplate already has crucial effect to the bending resistance and the whole

structure can be close to the collapse. Figure 3 shows the typical local damage of sandwich

panel as a result of strike by a spherical object.

teak coated sandwich panel


13/70


11

Figure 3. Local deflection in sandwich panel.

Above described phenomena indicate that together the global behaviour also the local

behaviour of sandwich panels should be considered. The purpose of the study is to investigate

the local behaviour of the sandwich panels subjected to lateral impact load and to derive

analytical formulations describing the behaviour. More precisely the purpose includes the

following matters:

learn about the local impact behaviour of sandwich panels,

study the influence of the faceplate thickness and material properties,

study the effect of core material.

In one hand formulations are to be simple and easy to use, but still they have to take into

account all the major phenomena concerning the impact event. Attention should be paid to the

strain-rate sensitive behaviour of materials; elastic deformation energy of a panel can be quite

high in a dynamic process and cannot be neglected; the shape of deflection caused by impact

load is different from deflection, which is caused by static load etc. To verify the results of the

analytical formulations, series of laboratory experiment and finite element (FE) simulations are

carried out.

1.3 Limitations of the study

The number of different designs of sandwich panels is large and it is obvious that single study

cannot embrace all of them. This study includes only one design, where material properties and

the thickness of the faceplate are changed. The effect of filling material is studied by urethanefoam. Lateral impact load is caused by spherical impact head, which is used to strike the


14/70


12

panels. Several impacts are simulated using FE program LS-Dyna, but as numerical simulations

are time consuming, number of FE simulations is smaller compared to the laboratory

experiments.

Analytical formulations are derived assuming infinite panel dimensions and limited extent of

deformation. Tearing and global bending of the faceplate are not considered in analytical

model. Strain-rate behaviour of the materials is considered by using Cowper-Symonds

constitutive equation.


15/70


13

2 EXPERIMENTAL SETUP

In order to obtain data to verify the analytical formulations series of laboratory experiments are

carried out. This chapter gives an overview of tested structures and the equipment and methods

used to conduct the laboratory experiments. Also the results of the laboratory experiments are

presented.

2.1 Tested structures and material properties

Altogether 96 impact tests are made for different sandwich panels. General structure of tested

panels remains unchanged throughout the tests and is given in Figure 4. The only changing

parameter is the thickness of the faceplate, which can be 1 to 3 mm with 1 mm spacing.

Figure 4. General drawing of the sandwich panel.


16/70


14

Panels have 4 mm thick I-profiles with 120 mm span as steel core. Material properties of the

steel plates are determined by carrying out tensile tests for specimens cut from the faceplates.

Tensile tests are carried out for the following specimens:

(i) three specimens cut from 1 mm plates,(ii) one specimen cut from 2 mm plate,

(iii) three specimen cut from 3 mm plate.

Information obtained from the tensile tests is gathered into Table 1.

Table 1. Results of the tensile tests.

Name of the

panel

Breath of the

specimen

Thickness0.2

Ultimate

strength

Ultimate

strain

[mm] [mm] [N/mm 2] [N/mm 2] A5, %

5a-6 24.95 3.03 385 485 33.5

I03 24.95 3.08 388 541 33.5

5a-6 25 3.03 370 482 36

N6 12.5 2.02 428 520 285a-13 10.98 1.00 159 288 60

5a-17 10.96 1.00 157 291 64

5a-5 12.4 1.00 179 290 47

Tensile test show that 1 mm thick steel plates are made of material, which yield stress is

significantly lower compared to materials used in 2 and 3 mm plates. For brevity, in following

discussions just low and high yield is used instead of exact values. Obtained yield stress values

are used to predict the strain-rate sensitive behaviour of materials. This behaviour is considered

by using Cowper-Symonds (Jones, 1989) equation, which uses constants D and q to describe

the behaviour. For mild or low yield steel, those constants can easily be found from the

literature. For high strength steels the information about the strain-rate behaviour is scarce and

difficult to get. Some investigations carried out in automotive industry have revealed that high

strain-rate increases the yield stress of high strength steels approximately 20%. According to

that material constants are also calculated for high-yield materials and gathered into Table 2.More detailed description of Cowper-Symonds model is given in Chapter 4.3.


17/70


15

Table 2. Strain rate properties of used steels.

Panel MaterialY [MPa] D q

t=1 [mm] Mild steel 179 40.4 5

t=2 [mm] High-yield steel 428 300000 6

t=3 [mm] High-yield steel 379 300000 6

The effect of core material is investigated by filling some of the panels with 2 mm faceplates

with urethane foam. Mechanical properties of the urethane filling are obtained according to the

measured density from literature (Kolsters; Romanoff, 2000) and a graph given in Figure 5,

which presents the relation between the density and the compressive strength of the urethane

foam /see reference 29/.

Figure 5. Properties of the urethane foam.


18/70


16

Properties of urethane foam are gathered into Table 3.

Table 3. Properties of the urethane foam.

Density Compressivestrength FShear modulus

GYoungs

modulus E

[kg/m 3] [MPa] [MPa] [MPa]

72 0.62 4.8 21

Good overview about the laboratory tests can be given by test matrix, which is presented in

Figure 6.

Faceplatethickness [mm]

Yield stress

Filling

Core type

1 32

High

V o i d

F o a m

Void

Low High

I-core

Void

Figure 6. Test matrix.

2.2 Test equipment, data acquisition and storage

The test equipment mainly consists of a test stand and of a data acquisition system. Aschematic picture of the test stand can be seen in Figure 7. The impact system includes a bar,

supported vertically by rollers to allow sliding movement, a replaceable extra mass, a

replaceable nozzle and three sensors. Impact head with conical nozzle is presented in Figure 8.

The impact head, having some predetermined mass, is dropped on the panel and data is

gathered into a computer and saved as a text file. The energy of the impact head is changed

altering its mass and dropping height. Tested panels are hit by a spherical impact nozzle, which

is made of 25-millimeter bearing ball and is shown in Figure 9.


19/70


17

Figure 7. Test system.

The parameters of the impact system are the following:(i) mass from 4.5 to about 37 kg

(ii) dropping height up to 1250 mm

(iii) velocity at the moment of impact up to 5 m/s,

(iv) potential energy from 2 up to 450 J.

Figure 8. Impact head with conical nozzle.


20/70


18

Figure 9. Spherical impact nozzle.

During the impact three quantities as a function of time are measured:

(i) force acting between the impact head and the panel,

(ii) acceleration of the impact system,

(iii) displacement of the impact system.

The test system uses one sensor for each quantity. General information about the sensors is

gathered into Table 4. Data acquisition system uses one channel per sensor. During the impact,

the system reads a value from one channel and switches to another channel in 7.5 s intervals.

Due to that information from one channel is registered in 22.5 s intervals. This equals to a

sampling rate of a little over 44 kHz.

Table 4. Sensors used in the impact system.

Quantity Manufacturer Model Range Type

Force HBM U9B 50 kN Strain cage

Acceleration B&K 3073 2000 G Piezo electric

Displacement Midori CPP-45 ----- Potentiometer

2.3 Measured / calculated quantities

In the tests, the following information is registered:

(i) dropped mass,

(ii) dropping height,

(iii) displacement, acceleration and force signals,(iv) bouncing height of the impact head,


21/70


19

(v) permanent deflection at the faceplate,

(vi) impact duration,

(vii) impact coordinates.

According to the registered data the following quantities are calculated:

(i) velocity of the impact head before the impact,

(ii) kinetic energy of the impact head before the impact,

(iii) plastic deformation energy of the panel.

In following sections some of the main calculations are explained.

2.3.1 Velocity of the impact head before the impact

Velocity of the impact head before the impact head is calculated from the dropping height

using the energy principle. The kinetic energy of the head just before the impact is certainamount smaller than the potential energy of the impact head before the drop, since some of the

potential energy goes to the revolving motion of the rollers. If all four of the rollers would

follow the movements of the impact head, the rollers would eventually give back their kinetic

energy, but because of the clearance between the rollers and the sliding bar it is assumed that

only two of the rollers follow the bar. The velocity before the impact could also be calculated

by the time-derivative of the registered displacement, but mentioned energy principle

calculations give more accurate results because of the scatter in the displacement measurement.

The calculation is verified by taking the time-derivative of the displacement signal from

repeated trials. The resulting average velocity was then compared to the value obtained by the

energy principle.


22/70


20

2.3.2 Permanent deflection of the faceplate

The permanent deflection of the faceplate or shortly dent depth is measured manually with a

digital dial indicator. The indicator is set to zero at the assumed drop point before every test. Asonly the first hit is under consideration, the impact head is stopped after the first hit to prevent

repetition.

2.3.3 Deformation energy of the panel

Computer programme uses three different methods to calculate the plastic deformation energyof the panel:

(i) difference in potential energies between the dropping and bouncing heights,

(ii) numerical integration of displacement-force curve. Displacement and force are

calculated from the signal received from the acceleration sensor,

(iii) numerical integration of displacement-force curve. Displacement and force are

calculated from the signals received from the acceleration sensor and the force

transducer.

In the first method, it is simply assumed that the elastic deformation energy of the panel returns

to the kinetic energy of the impact head. Due to that impact head bounces from the panel and

the bouncing height is measured. According to the bouncing height elastic deformation energy

can be calculated.

Other two methods employ the similar principle. Since the velocity of the impact head just

before the impact and the acceleration as a function of time are known, the motion of the

impact head during the impact can be calculated. On the other hand also the force acting

between the impact head and the panel is known, which means that the deformation energy can

be calculated by integrating force-displacement curve. Force-displacement curve is shown in

Figure 10 where the plastic deformation energy is the area under the curve. Since also the

displacement sensor is employed the deformation energy could be calculated using the signal

from the displacement sensor, but because of the resolution and the mechanical construction of

the displacement sensor this is considered to be inaccurate.


23/70


21

Force-displacement curve

0

0.5

1

1.5

2

2.5

3

0.2 0.25 0.3 0.35 0.4 0.45 0.5

Displaceme nt [cm]

F o r c e [ k N ]

Figure 10. Force-displacement curve.

In case of the first method the bouncing height is determined by the signal from the

displacement sensor. As the accuracy of the displacement sensor is low the first calculation

method is considered to give imprecise results. In case of the second method where the

acceleration signal is integrated, the electronic filtering of the signal distorts the signal and

causes some error to the calculated value. Electronic filtering is used to smoothen the signal,

which is affected by the high frequency vibrations induced to the impact system due to the

collision. Considering these facts the third method is considered to be the most accurate. The

plastic deformation energy presented in Chapter 2.4 is calculated by using the signals received

from the acceleration sensor and from the force transducer

2.4 Results of the laboratory tests

In laboratory experiments panels are hit to the centre of two middlemost compartments, as can

be seen in Figure 11 and in Figure 12. During the impact panel is lying in the floor, which can

be assumed as infinitely rigid compared to the panels.

Plastic deformationenergy


24/70


22

Figure 11. Permanent deformations.

Investigation of the tested panels revealed that the width of the deformation in the faceplate is

limited by the span of the inner supports as can be seen from Figure 11 and Figure 12. Left

picture in Figure 12 presents the panel where the faceplate of the section subjected to the

impact load is very close to global bending, but plating of the adjacent sections remains

undamaged and no deformations can be observed. Bending of the faceplate is considered to be

global when length of the deformation is large compared to the width of the deformation. Right

picture of the same figure also reveals that width of the deformation does not exceed the span

of the supports. Deformation has circular shape until the faceplate bends globally and the

circular shape is stretched to oval.

Figure 12. The extent of the deformation.

Laboratory tests also showed that the impact does not cause noticeable permanent deformations

in inner supports even when the global bending of the faceplate occurs as depicted in Figure 13.


25/70


23

Figure 13. Global bending of the faceplate.

Results of the laboratory experiments are presented in Figure 14 to Figure 17. In the figures,

single test is marked with rhomb. In addition, to get a better picture of the panels behaviouralso trendlines are presented for every panel type. Initial energy of the impact head or in other

words the total deformation energy of the panel as a function of permanent deflection is

presented by red rhombi and by red solid line, while blue colour presents the plastic

deformation energy. Global bending of the faceplate is marked by a red rectangular. Results for

the panels with 1 mm faceplates are presented in Figure 14. Big scatter of the test results may

be due to the dispersion of material properties as can be seen from Table 1.

Figure 14. Initial energy of the impact head and the plastic deformation energy in case of the panels with 1 mm plating.


26/70


24


Dispersion of the test results is much smaller in case of the panels with 2 mm faceplates and

single tests shows good agreement with the trendline. Both energy levels are significantly

higher compared to the panels with 1 mm faceplates and also the global bending of the

faceplate occurs later.

Figure 16. Initial energy of the impact head and the plastic deformation energy in case thepanels with 2 mm plating and the urethane foam filling.


27/70


25

When the sandwich panel with 2 mm faces is filled with urethane foam the global bending of

the faceplate is prevented and the energy level increases a little, see Figure 16. Shape of the

deformation is similar to one is case of the unfilled panels.


Results of the tests made on panels with 3 mm plates also agree well with the trendline and the

dispersion of the results is small. In case of the panels with 3 mm plating, global bending of the

faceplate was not observed during the laboratory tests.


28/70


26

3 Finite element analysis

Though the laboratory tests provide the verification data, they do not give any information

about the inner mechanics of the sandwich panel during the impact. Finite element simulations

allow to follow the impact process and to obtain the information about the behaviour of the

sandwich panel during the impact. Main purpose of the FE simulations is to verify theassumptions made in the derivation of the analytical formulations.

For the FE analysis four different sandwich panels are modelled. For modelling and three-

dimensional meshing pre-processor LS-Ingrid is used. LS-Ingrid is also used as a translator to

convert a text file into input file for the finite element program LS-Dyna950d. The main

solution method in LS-Dyna bases on explicit time integration. Explicit solution method

exploits the idea that equilibrium equation is always satisfied. At the beginning of the time-step

every node has initial coordinate, velocity and force applied to the system. By the equilibrium

acceleration is found for every node. As the acceleration is known the new velocity and the

displacement of the node can be calculated by using kinematics. New equilibrium force is

calculated by the nodal displacements. Calculated values are used as new initial values for the

next calculation step.

3.1 Geometry of the FE model and the simulation procedure

Though the configuration of the panels is quite simple, it is still not reliable to model the whole

panel as the size of the model also affects the calculation time. Missing part of the panel can be

compensated by boundary conditions.


29/70


27

Figure 18. Geometry and dimensions of the modelled panel.

Lets now consider Figure 18 to understand the use of boundary conditions. In order todetermine which parts of the sandwich panel should be modelled and where the boundary

conditions can be used, several simulations are carried out with different models. Calculations

give that using boundary conditions on sides AB and CD, which are transverse to the inner

supports, causes some overestimation of panels stiffness. To prevent the use of the boundary

conditions on transversal sides panel is modelled on its full length. Also the inner supports are

not fixed at the ends. Lower plate of the panel is not modelled as it does not contribute to the

energy absorption but only supports the inner members. As the supporting of the inner

members can easily be described by the fixed boundary conditions on lines E F and G H, the

modelling of the lower plate is unnecessary. Simulations also showed that the breath of the

modelled faceplate should be at least two times bigger than the span of the inner supports. Too

narrow faceplate causes some overestimation of the panels stiffness. Remaining part of the

panel is compensated by fixing edges A-C and B-D. Furthermore, it is assumed that laser welds

on lines E*F* and G*H* are rigid and do not deform during the impact. It means that the weld

is modelled just by connecting nodes of the faceplate and inner supporting member along the

lines E*F* and G*H*.

When the panel with the urethane filling is under consideration, compressible low-density foam

is modelled inside the panel. To reduce the calculation time only the middle section of the

panel is filled with the foam. As the foam in the other sections prevents the movements of the

inner supports, fixed boundary conditions are used on surfaces E E* F* F and G G* H* H.

Bottom of the foam is fixed to compensate the absence of the lower plate. Information about

the boundary conditions is gathered into Table 5.


30/70


28

Table 5. Boundary conditions of the modelled panels.

STRUCTURAL

ELEMENT SIDE/SURFACE EMPTY PANEL FILLED PANEL

Face late A-B free freeC-D free freeA-C all dof.* fixed all dof. fixedC-D all dof. fixed all dof. fixed

Su ort inner E-E' free freeG-G' free freeF-F' free freeH-H' free freeE-F all dof. fixed all dof. fixedG-H all dof. fixed all dof. fixed

Urethane fillin E-E'-F'-F - all dof. fixedG-G'-H'-H - all dof. fixedE-F-H-G - all dof. fixed

*dof.- degree of freedom

Density of the element mesh depends on the location. Near to the impact zone element

dimensions are the smallest- 1x1 mm. The biggest element dimensions are 4x4 mm. Figure 19

gives a better picture about the mesh and the element sizes. Steel plates are modelled by using

two-dimensional four node shell elements with thickness- known as Belytschko-Tsay elements.

This element type is one of the most commonly used elements in numerical analysis of crash

mechanics of thin-walled structures.

Figure 19. Element mesh.


31/70


29

Element mesh and size of the urethane filling coincides with the mesh of the faceplate, see

Figure 20. Urethane filling is connected to the metal sheets by connecting the nodes, which

have the same coordinates. Urethane filling is modelled by using eight node hexahedron solid

elements.

Figure 20. Modelled panel with urethane filling.

To simulate an impact event, spherical impact head similar to one depicted in Figure 9 is

modelled. Impact head is modelled as a non-deformable rigid body. Energy of the striking body

is given by its mass and by the velocity at the moment of impact.

3.2 Material properties of the model

Steel plates of the sandwich panel are modelled by using LS-Dyna material model no 24

(Piecewise Linear Isotropic Plasticity ). This material model is chosen as it works both in

elastic and in plastic region, capable to use non-linear material properties and can consider

strain-rate sensitive behaviour of the material. In elastic region material behaviour isdetermined by Youngs modulus and by Poissons constant. Material behaviour in plastic

region is determined from the tensile tests. For a purely plastic response without fracture or

plastic localization, it is straightforward to determine the plastic parameters straight from the

tensile tests. Figure 21 presents the results of the tensile test and approximated true stress-strain

curve for LS-Dyna in case of the 3 mm specimens.


32/70


30

Figure 21. Tensile test and approximated true stress-strain curve for LS-Dyna.

Though the tearing of the faceplate did not occur during the laboratory tests, the possibility of

the tearing is still foreseen in FE calculations. The initiation and propagation of fracture in the

structure can be modelled in LS-Dyna by deleting elements from the system once plastic strain

has reached a certain level. To determine that certain level, an equivalent fracture criterion for

the prevailing element is calculated. For the calculation a specimen is modelled and several

tensile tests with different failure criteria are carried out in LS-Dyna. The failure criterion is

evaluated by comparing the real and calculated stress-strain curves. When those two curves

coincide the correct failure criterion is found.

Urethane foam filling is modelled by using material no 14 ( Soil and Crushable Foam with

Failure ). That material model is selected as it provides a simple model for foams whose

properties are not well characterized. Necessary input variables for the selected material model

were given in Table 3.

3.3 Results of the FE analysis

Finite element simulations provide a possibility to obtain information that is hard to get from

laboratory experiments. Addition to deformation energies, the following characteristics are

determined by FE simulations:

(i) velocity profile,(ii) velocity of the faceplate as a function of time,


33/70


31

(iii) shape of the deformation,

(iv) displacements at the core.

Following discussion bases on simulation where the sandwich panel with 2 mm faceplates is hitby the sphere with mass of 20 kg and velocity at the moment of contact is 3.13 m/s. Velocity

profile obtained by the finite element simulations is shown in Figure 22 by blue line and the

shape of the deformation by red line. Profiles in Figure 22 are drawn assuming that initial

contact between the impact body and the panel takes place at the origin.

Figure 22. Shape of the deformation and velocity profile.

Velocity profile is evaluated by analysing velocity time histories for every node between thenodes FP-1 and FP-4, see Figure 24. Profile presents the average velocity values and is made

dimensionless by dividing it with the average velocity of the middle node FP-1. Figure 22

shows that the velocity profile can be approximated by linear line without a significant decrease

in preciseness.

Figure 23. Velocity of a node FP-1 as a function of time.


34/70


32

Velocity as a function of time is presented in Figure 23. Red solid line presents the velocity of

the node FP-1 and red dashed line shows the calculated average velocity. It should be noted

that the same average velocity is used to turn velocity profile into dimensionless mode. Figure

23 also reveals that at the beginning of the impact node FP-1 obtains the same velocity with the

impact head. Velocity starts to decrease but the decrease is not exactly linear, but little

smoother at the beginning and slightly sharper at the end of the impact. Linear approximation is

presented by blue dashed line. Simple operation shows, that ratio between the initial and the

calculated average velocity is approximately 1.5. The same value is later used in analytical

calculations to describe the change of velocity.

Figure 24. Nodes at the cross-section of the panel.

It is obvious that most of the impact energy is absorbed by the faceplate, but the significance of

the steel core displacements should still be investigated. For that the transversal displacementsof the steel core are compared with the displacements of the faceplate. Comparison is done by

carrying out the impact simulation for the panel with 3 mm faces. Panel with 3 mm faceplates

is selected for the investigation as thicker faceplate causes greater displacements of the inner

supporters. Described panel is hit by the sphere with velocity of 3.13 m/s and mass of 30 kg.

Results are presented in Figure 25 and the nodes used in comparison were depicted in Figure

24.


35/70


33

Figure 25. Displacements of the faceplate and core.

In Figure 25 red lines present the transversal displacements of the faceplate nodes and blue line

presents the displacements at the core multiplied by 100. Figure 25 shows that core

displacements are more than hundred times smaller compared to the displacements of the

faceplate and therefore can be ignored.

Initial and plastic energy as a function of permanent deflection are shown in Figure 26 for

panels with 1 and 3 mm plates. Figure 27 presents the results of FE simulations for the

sandwich panels with 2 mm faces.

(a) (b)

Figure 26. Results of FE simulations. Initial energy of the impact head and plasticdeformation energy as a function of permanent deflection in case of the empty panelswith 1 mm (a) and 3 mm (b) faceplates.

e


36/70


34

(a) (b)

Figure 27. Results of FE simulations. Initial energy of the impact head and plasticdeformation energy as a function of permanent deflection in case of the empty panelswith 2 mm plating (a) and the urethane filled panels with 3 mm faceplates (b).


37/70


35

4 Analytical formulations

Aim of the analytical formulations is to provide a possibility to calculate deformations in the

panel when the properties of the striking body are known. Extent of the deformation can be

evaluated by equalizing the kinetic energy of the impact body with the deformation energy of

the panel. As it is easy to calculate the kinetic energy of the striking body the main task is todescribe the energy absorption of the sandwich panel.

4.1 Background and main assumptions

As a result of impact, faceplate of the sandwich panel stretches in all possible in-plane

directions to resist impact loads and can attain large permanent deflections. When plate starts todeform under lateral load, bending plays a major role for small deformations. With an increase

in transversal deformation, the importance of bending diminishes and the membrane force

quickly develops. At sufficient large deformations, the membrane force dominates the

behaviour. This is known as string response.

Furthermore, impact energy is absorbed not only by the faceplate, but also by the inner

supports, lower plate and by the filling if there is any. To consider all the deformation

mechanisms by analytical single model is complicated and even not necessary. The most of the

impact energy is absorbed by the mechanisms where it is done in most efficient way. To

simplify the model several assumptions should be made and verified.

One of the main assumptions is about the displacements of steel core. When inner supports of

the panel are much stiffer compared to the plates and there is no filling inside, most of the

energy is absorbed by the faceplate. Considering the dimensions of the tested panels and the

test matrix given in Figure 6, it becomes obvious that longitudinal bending stiffness of the I-


38/70


36

profile supports is much higher compared to the faceplate. It introduces the first assumption-

inner structure of the tested panels can be considered as rigid and deformation energy is

absorbed only by the faceplate and by the filling. Convenient way to verify that assumption is

to measure core displacements in FE simulations. Measurements showed that displacements at

the core are more than hundreds of times smaller compared to the displacements at the

faceplate.

Second assumption considers the extent of the deformation. As laboratory experiments and FE

calculations have shown the maximum extent of the deformation is equal to the span of the

inner supports. The minimum extent is not limited and should be determined by minimizing the

energy. Furthermore, it is also assumed that the length of the panel is infinite. Assumption

agrees well with the actual use of sandwich panels where one dimension of the panel is often

much larger compared to the others. Importance of the mentioned assumption is that global

bending of the faceplate as can be seen in laboratory tests does not occur and the shape of the

deformation is assumed to be circular. In reality, some global bending of the faceplate occurs

also in the case of infinitely long panels, but the extent of the global bending is small compared

to the panel length. Global bending of the infinitely long panels reveals in deformation shape,

which takes more oval form.

Conclusively the main assumptions are:

(i) majority of the impact energy is absorbed by bending and membrane stresses at the

faceplate as deformations at inner supports and lower plating are small and can be

neglected,

(ii) the maximum width of the deformation is equal to the span of the inner supports,

(iii) length of the panel is infinite, which allows to use circular shape to describedeformation.

Before proceeding to the derivation of energy absorption formulations, analytical description of

the deformation shape is given in Chapter 4.2. Formulations connected to the calculation of

strain rate are given in Chapter 4.3.


39/70


37

4.2 Analytical description of the deformation shape

In order to be able to calculate the energy absorbed by the different deformation mechanisms,

shape of the deformation should be known. Circular shape of the deformation is described bytwo coordinates. Coordinate r is pointed to radial direction and w to the direction of deflection.

The laboratory tests and the finite element simulations presented that it is convenient to divide

the deformation of the faceplate into two parts as shown in Figure 28:

(i) Linear line B-C

(ii) Curve A-B, which can be described by polynomial

Figure 28. Deformation shape.

Extent of the linear line is determined by two constants C 1 and C 2. C1 determines the extent of

the linear line in w direction and C 2 is used to determine the extent of deformation in r-

direction. The linear part is

Rr RC R

r C w

21 1 . (1)

Polynomial part is described by third-order polynomial given and is valid for RC r 20 :

d r cr br ar w 23)( . (2)


40/70


38

Constants a, b, c and d are determined by the following boundary conditions:

d wr 0

,

000

cwr

,

)1( 212 C C w RC r ,

RC w

RC r 12.

(3)

After evaluating constants a and b, shape of the deformation can be written as

,1

0)(

21

223

Rr RC if Rr

C

RC r if r br ar w (4)

where

.

332

22

222

121

332

121

RC

C C C b

RC

C C C a

(5)

Inclination of the deformation shape is determined by taking the first derivative of Eq. (4)

.023)(

21

2

2

Rr RC if R

C RC r if r br ar S (6)

Later, when deriving equations for the energy absorption, also the change in inclination is

needed. It is evaluated by taking the second derivative of Eq. (4)

.0026)(

2

2

Rr RC if RC r if br a

dr r d S (7)


41/70


39

To calculate the energy absorbed by the core filling, compressed volume of the filling material

should be evaluated by using expression

dwwr V 0

2 . (8)

As it is laborious to derive the relation where radius r is given as a function of coordinate w by

using polynomial, two linear lines are used instead and the shape of the deformation can be

written as

!

.111

)(

101)(

1212

2

121

w RC C if C C

w RC

RC C wif RC w

wr (9)

By substituting Eq. (9) into Eq. (8) and carrying out the integration, compressed volume of the

filling material can be calculated by

22122123 C C C C RV

.(10)

4.3 Strain rate

As a result of the impact, the panel is deformed in relatively high velocity and possible effect of

the high strain rate to the material behaviour should be considered. The strain rate sensitivity of the materials is considered by using Cowper-Symonds constitutive equation (Jones, 1989)

given by Eq. (11). Cowper-Symods model simply scales the static yield stress value Y by

considering strain rate and predetermined material constants q and D.

""

#

$

%%

&

'

q

Y DY D

1

1

. (11)


42/70


40

To obtain the formulation for the strain rate, velocity profile over the cross-section of the panel

should be known. The FE simulations gave that it is sufficient to approximate the transversal

velocity profile by linear line, see Figure 29.

Figure 29. Approximated velocity profile.

Inclination of the velocity profile is

R

vV . (12)

Consider cross sectional element of the faceplate (Figure 30) to derive formulations for the

strain rate. Non-deformed length of the element is dr. As the result of the impact the plate

deforms and obtains the deflection that can be calculated as S dr. The engineering strain in the

element is calculated from

dr

dr dr dr S

r

22

. (13)

By expanding Eq. (13) to series and neglecting high order terms, equation takes a following

form:

2

2

2 dr dr

S . (14)


43/70


41

Figure 30. Deformed plate element

Strain rate is obtained by taking time derivative of Eq.(14). It should be remembered, that S dr

describes the deflection of the plate and the time derivative of S dr is the deformation velocity

of the plate. Deformation velocity at any point inside the deformed area can be calculated by

using the maximum velocity value at the point of first contact and the inclination of the velocity

profile. The dimensionless strain rate can be written as

V S

V S

dr

dr dr

222

. (15)

Note that S and V should be used as a dimensionless shape functions and the actual values

for the deflection and the velocity are given by the amplitudes and V 0. V0 is the velocity of the impact body at the beginning of the impact. Velocity time dependence is described by a

single constant c V, which is used to divide the initial velocity to get average velocity

V

Ac

vv 0 . (16)

By substituting Eq. (12) into Eq. (15) the strain rate can be described as

Rc

vr

V

S 0)( . (17)

In Eq. (17) denotes the final permanent deflection of the faceplate.


44/70


42

4.4 Energy absorption of the panel

As stated above, it can be assumed that the steel core and the lower plate do not contribute to

the energy absorption and all the energy is turned into the deformation energy by the faceplateand by the filling material. Formulation are derived both for the absorption of elastic energy as

well as for the plastic energy. Energy absorption in both cases is divided into two parts:

(i) energy absorbed by bending,

(ii) energy absorbed by membrane deformations.

In case of the filled panels also the energy absorbed by the filling is added to the plastic energy.

In analytical calculations it is assumed that the material behaves as elastic, perfectly plastic

material as given in Figure 31. Effect of the high strain rate is considered only in case of the

membrane mechanism.

Figure 31. Elastic, perfectly plastic material.

4.4.1 Elastic energy absorbed by bending

To derive the formulations for elastic energy absorbed by bending, it is assumed that

deformation has circular shape with radius R and deflection w at the middle of the panel

(Figure 32). It is obvious that the bending moment obtains its maximum value at the yield line

where r=R.


45/70


43

Figure 32. Deformed panel.

Amount of the elastic energy is evaluated by equalizing the bending moment at the yield line

by plastic moment M P of the panel:

4

2t M

Y P . (18)

Corresponding force or so-called collapse load E BF and deflectionE

Bw in the middle of the

plate can be evaluated by using plate theory (Ikonen, 1990). Elastic energy can be calculated by

using relation

2

E

B

E

BwF

E . (19)

Deflection of the circular plate subjected to a lateral distributed load can be written as

"#$

%&'

rdr dr r r r

dr

r

dr

K r C C w )(

1221

, (20)

where K describes the material and is equal to

)1(12 2

3

(

t E K . (21)


46/70


44

Distributed load is handled by introducing the concept of the effective radius r EF. Using the

effective radius distributed load can be used as a constant and relation between the force F and

distributed load becomes

2 EF

r

F

. (22)

Using Eq. (22) first integral in Eq. (20) can be evaluated as

EF r

EF

EF EF

F

r

r F r rdr dr r r

0 2

22

222)(

. (23)

Remaining three integrals are evaluated as follows:

)ln(22

1 r F

dr F

r , (24)

22

41

)ln(21

2)ln(2r r r

F rdr r

F

, (25)

1)ln(81

41

)ln(211

2222

r r F

dr r r r r

F

. (26)

Deflection of the plate takes a form

1)ln(81

)(2

21 r r F

r C C r w . (27)

Constants C 1 and C 2 should be solved by using boundary conditions for clamped circular plate:

.0)()(

,0)()(

Rr

Rr

r wdr

d ii

r wi

(28)


47/70


45

From the second boundary condition C 2 can be solved and can be written as

K

Rr F C

1)ln(2

16

1 22 . (29)

By substituting Eq. (29) into the first boundary condition, C 1 becomes

21

161

RK

F C

. (30)

Final form for the deflection is defined as

!) *22 1)ln(2)ln(2161

)( r Rr RK

F r w

. (31)

Bending moment of the circular plates is given by

dr

dw

r dr

wd K r M

2

2

)( . (32)

Derivatives in Eq. (32) are

!)ln(ln41

r RK

r F

dr

dw , (33)

! Rr K

F

dr

wd ln1)ln(

41

2

2

. (34)

Bending moment at the yield line is obtained by substituting Eq. (33) and (34) to Eq. (32)

F r M

Rr

4

1)( . (35)

Now the collapse load E BF can be calculated as


48/70


46

P E

B M F 4 . (36)

Corresponding deflection at the middle of the plate (r=0) when subjected to load E BF is

obtained by replacing Eq. (36) to Eq. (31):

2

16 R

K

F w

E B E

B

. (37)

Elastic energy absorbed by the bending can now be calculated by using Eq. (19) and is obtained

from

242

32 R

K

t E

y E B

. (38)

4.4.2 Elastic energy absorbed by membrane mechanism

To calculate the amount of the elastic energy absorbed by membrane deformation, the same

idea is employed as in case of the elastic bending energy. Stresses in every element inside the

assumed deformed area are equalized by the yield stress of the material. In case of membrane

stress dynamic behaviour of the material plays important role in high strain rates. Due to that

the dynamic yield stress DY should be used. When stress in every point of the panel is known

the deformation energy can be obtained from

A

D

Y

P

M dAt E

'

. (39)

Assuming that Hookes law holds and the relation between the stress and the strain in elastic

region can be expressed as

E Y . (40)

Considering Hookes law in Eq. (39) the elastic energy absorbed by the membrane mechanism

is given by


49/70


47

R

D

Y

E

M dr r

E

t E

0

22

. (41)

4.4.3 Plastic energy absorbed by bending

Concept of plastic hinges is introduced to derive the formulations for plastic energy absorbed

by the bending mechanism. Figure 33 presents a rectangular plate with breath B and thickness t

subjected to a lateral load F.

Figure 33. Plastic hinge.

Due to the load, the panel is deformed and plastic hinge is formed at the point A. Deformation

energy absorbed in forming that plastic hinge can be evaluated from

P

P

BM L E , (42)

where is the angle and L is the length of plastic hinge. Deformation angle as a function of r is

given by Eq. (6). As work is done only in forming the plastic hinge, plastic energy can be

evaluated by using the change of the angle, given by Eq. (7). Absorbed energy is found by

integrating the change over the radius r

2

1

)(2

r

r

SP

P B dr r

dr

r d M E

, (43)


50/70


48

where absolute value of dr

r d S )(

should be used as the energy absorption does not depend on

the direction of deformation angle. Derivation of Eq. (43) is convenient to carry out in two

parts:

(i) energy absorption when 0 < r < C 2 R by using Eq. (43),

(ii) Energy absorption when C 2 R < r < R by using Eq. (42).

When the first part is considered Eq. (43) takes a form

'2

0

262

RC

r

P

P

B dr r br a M E (44)

with constants a and b as given by Eq. (5). By carrying out the integration, Eq. (44) becomes

"#

$%&

'2

3

1 272

)1(2c

d C M E

P

P

B , (45)

where

121 22 C C C c

121 332 C C C d .(46)

Second part of the energy absorption is obtained by using Eq. (42). The angle of the plastic

hinge is R

C 1sin , but as 1C is small compared to R angle can be evaluated as R

C 1

without a deterioration in preciseness. Bending energy absorbed in region C 2R < r < R

becomes

P

P

BM C E

12 .(47)


51/70


49

As all the variables in Eq. (47) are always positive, there is no need to take absolute value. By

adding Eq. (45) and (47), the total plastic energy absorbed by the bending can be calculated

from

2

3

272

12c

d M E P

P B

. (48)

4.4.4 Plastic energy absorbed by membrane mechanism

Plastic energy absorbed by the membrane mechanism is calculated similarly as presented inChapter 4.4.2, but engineering strain is used instead of the relation obtained by the Hookes

law. Consider again an deformed plate element in Figure 30. Due to the impact, element is

stretched and obtains the strain as given by Eq. (13) and (14):

2

222SS

dx

dxdxdx + .

By substituting the strain into Eq. (39) the plastic energy absorbed by the membrane

mechanism is given by

R

S DY

A

S DY

P M dr r r r t dA

r r t E

0

22

)()(2

)()(

. (49)

4.4.5 Energy absorbed by core filling

Effect of the filling is considered by Winklers foundation, where support reaction caused by

the core filling can be written as

)(r wk pF , (50)

where k describes the foundation.


52/70


50

According to the Winklers model the energy absorbed by the core filling can be calculated as a

product of compressed volume V and the compressive strength F of the foundation

F F F C C C C RV E 22221123 . (51)

4.4.6 Approximate solution for membrane energy

Eq. (41) and (49) present the precise solution for energy absorbed by the membrane

deformations. As in Eq. (41) and (49) yield stress of the material is a complicated function of coordinate, it is laborious or almost impossible to carry out the integration. Dependence of the

coordinate of yield function is given by multiplying the yield stress by

q

Dr

r CS

1

)(1)(

. (52)

By finding a new scaling constant CS* without a coordinate dependence material yield stress

becomes also independent from the coordinate and Eq. (41) and (49) can be integrated.

Consider Eq. (41) for the elastic energy absorbed by the membrane deformations. Assuming

that

*)( CSr Y DY

(53)

integration can be evaluated as follows:

.22 22*

0

2*

0

2* R E

t CSdr r CS

E t

dr r E

t E Y

R

Y

R DY

E M

(54)

Integration in Eq. (49) is more complicated and should be carried out in two parts:


53/70


51

,)23(

)()(

2

2 21

0

22*

0

2*

0

2*

""#

$

%%&

'

R

RC

RC

Y

R

SY

R

S DY

P M

dr r R

C dr r xbr aCSt

dr r r CSt dr r r t E

(55)

where

332

121 22

RC

C C C a

22

2

121 332

RC

C C C b .

After the integration and some simplifications absorbed plastic membrane energy is obtained

from

"#$

%&'

3165

116

5 122

1222

** C C C C C

CSt E Y P M

. (56)

Consider Figure 34 and Figure 35 to evaluate the new scaling constant CS*. Figure 34 presents

the effect of the strain rate. In Figure 34a red line presents the absorbed membrane energy

where the effect of the strain rate is considered. Also denotation E(CS) implies the dependence

of strain rate. Blue line is calculated by taking CS=1. In other words it means that the effect of

the strain rate is not considered. Dashed line shows the relation between the two energies.

Figure 34a shows that the ratio of two energies is almost constant and the strain rate effect can

be considered by using a single constant that depends on the initial velocity of impact body and

on the material properties. The same reveals by considering the Figure 34b where first

derivative of membrane energy or briefly energy rate is presented.


54/70


52

(a)

(b)

Figure 34. a) Absorbed membrane energy; b) Rate of energy absorption.

Figure 34b reveals that roughly half of the energy is absorbed by that part of the panel where

the deformation is described by polynomial. In that part of the panel rate of the energy is

several times higher compared to the energy rate in linearly described part. Higher rate of

energy absorption is due to the higher scaling factor CS that depends on strain rate, see Figure

35.


55/70


53

(a)

(b)

Figure 35. a) Strain rate; b) Scaling factor CS.

The new scaling constant CS* should consider both the polynomial and linear part. Considering

an plate element as depicted in Figure 36. Impact causes permanent deflection to the

faceplate.

Figure 36. Stretched plate element


56/70


54

Strain rate in the element becomes

2

L

v A

. (57)

Assuming that the velocity of the faceplate decelerates according to the constant c V, simplified

equation for the evenly distributed strain rate becomes

20*

Rc

v

V

(58)

and the energy can be scaled by a constant

q

V D Rcv

CS

1

20* 1

. (59)

Now the approximate Eq. (54) and (56) can be used instead of complicated Eq. (41) and (49).

4.5 Solution procedure

The shape of the deformation and absorbed energy is determined by equalizing the initial

energy of the impact body with the deformation energy of the panel. Flow chart in Figure 38presents the solution procedure. Flow chart reveals that only the constant C 1 is found by the

minimization. To understand the reason for that consider Figure 39, which presents non-

dimensional shape of the total deformation energy as a function of constants C 1 and C 2. Figure

39a gives that the energy can be minimized respect to C 1 as a local minima can be found.

Behaviour of C 2 is different and minimization tries to use the lowest possible value. To obtain

the best solution constraints should be used. C 1 can obtain any value between 0 and 1, but for

C2

a limit for the lower bound should be evaluated. Lower bound for the C2

is derived from the

geometry given in Figure 37 and can be expressed as


57/70


55

R

r C C C

I +21 11

2 . (60)

At the beginning of the first calculation step initial guess values are given for C 1 and R A.Constant C 2 can directly be obtained by using these guess values.

Figure 37. Theoretical shape of the deformation.

Figure 38. Flow chart of the solution process.


58/70


56

The deformation energy of the panel as a function of deflection is equalized by the energy of

the impact body

I

P

B

P

M

E

B

E

M E E E E E

, (61)

where E I is initial energy of the impact body defined as

2

2 I I

I

vm E (62)

and V I is the velocity of the impact body at the moment of the impact and M I is the mass of the

impact body.

(a)

(b)

Figure 39. Non-dimensional shape of the total deformation energy of the panel as afunction of constants C 1 and C 2.


59/70


57

Permanent deformation is calculated numerically by using Eq. (61). Numerically calculated

deflection and Eq. (61) are now used in minimization process to calculate the constant C 1.

Calculated and C 1 are used as new initial values for the next calculation step. Calculation

loop continues until the values for and C 1 do not change anymore and the equilibrium isfound. Calculated values and C 1 provide sufficient information to determine the shape of the

deformation and to calculate the energy absorbed as the result of the impact.


60/70


58

5 Comparison between the laboratory tests, FE calculations and the

analytical formulations

Comparison between the experiments, FE simulations and analytical formulations is presented

in Figure 40 to Figure 45. Analytical calculations are carried out both by using precise

formulations and also by using simplified expressions derived in Chapter 4.4.6. The results of

the precise formulations are presented by red dots while blue rhomb mark the result of the

simplified equations. Trendlines are drawn in corresponding colour by using second order

polynomial. For readability initial energy of the impact head and plastic deformation energy are

presented in separate figures. Constants and material properties used in analytical calculations

are presented in Table 6.

Table 6. Used constants and material properties for different panels.Name Unit t=1 [mm] t=2 [mm] t=2 [mm],

urethane foam

t=3 [mm]

E [GPa] 210 210 210 210

[-] 0.3 0.3 0.3 0.3

R [mm] 60 60 60 60

t [mm] 1 2 2 3

Y [MPa] 179 428 428 380

D 40.4 300000 300000 300000

q 5 6 6 6

cV* 1.5 1.5 1.5 1.5

F [MPa] - - 0.62 -

* constant c V is obtained by the finite element calculations


61/70


59

The results for the panels with 1 mm plating are shown in Figure 40 and in Figure 41.

Figure 40. Plastic deformation energy in case of the panels with 1 mm plates.

Figure 40 gives that the analytical formulations underestimate the plastic deformation energy

approximately by 10 %. Scatter may be explained by the fact that low yield steels are quite

sensitive to the strain rate effect and the small impreciseness in velocity or in strain rate

calculations may already produce a significant error. Good agreement can be seen between the

simplified and precise analytical equations, which indicates that it is sufficient to save

calculation time and to use simplified expressions.

Consider now Figure 41 for initial energy of the impact head as a function of permanent

deflection. Reveals that LS-Dyna seems to overestimate the energy in case of the high

deflection values. Reason for that may be too stiff boundary conditions that delay the global

bending of the panel, or imprecise material properties, especially yield stress. Analytical

calculations give poor results in low and very good results in high deformation values.

Impreciseness in low deformation values is due to the methodology of elastic energy

calculation. In analytical calculations, the amount of the elastic energy does not depend on the

deformation depth but only on the strain rate effect. If there is no strain-rate effect, amount of

the elastic energy remains constant.


62/70


60

Figure 41. Initial energy of the impact head as a function of permanent deflection in case of the panels with 1 mm plates.

Analytical model always calculates the elastic deformation energy for the predetermined

deformation shape and extent. Model assumes that material inside the predetermined area r=R

is stretched and bent until the end of the elastic region. In some cases, especially in the case of

very small deformations, the latter assumption may not be true as in some areas close to the

boundary, deformations may not reach to the end of the elastic region. According to described

calculation methodology the total deformation energy of the panel can never be zero, but has

some value even when depth of permanent deformation is zero. Such situation may occur when

energy of the impact head is small and the impact causes only elastic deformations. The same

behaviour can also be seen in case of finite element calculations.

Comparison of results obtained by the different methods in case of the panels with 2 mm

plating is presented in Figure 42 and Figure 43. In case of the plastic deformation energy some

scatter between the experimental data and the analytical calculations can be noticed in higher

deformation values. In higher deformation values also the global bending starts to affect the

energy absorption and as the analytical model cannot consider the effect of the global bending,

it may be the source of some impreciseness. Again the simplified analytical formula and

precise formula show very good agreement.


63/70


61

Figure

Date post:	05-Apr-2018
Category:	Documents
Upload:	kristjan-tabri
View:	228 times
Download:	3 times

Tabri Local Impact Strength of Sandwich Panels

Documents