SHOCK FAILURE ANALYSIS OF MILITARY EQUIPMENTS BY …

SHOCK FAILURE ANALYSIS OF MILITARY EQUIPMENTS BY USING STRAIN ENERGY DENSITY

A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES

OF MIDDLE EAST TECHNICAL UNIVERSITY

BY

ÜMĐT MERCĐMEK

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF MASTER OF SCIENCE

IN MECHANICAL ENGINEERING

DECEMBER 2010

Approval of the thesis:


submitted by ÜMĐT MERC ĐMEK in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Department, Middle East Technical University by, Prof. Dr. Canan Özgen Dean, Graduate School of Natural and Applied Sciences Prof. Dr. Süha Oral Head of Department, Mechanical Engineering Prof. Dr. F. Suat Kadıoğlu Supervisor, Mechanical Engineering Dept., METU Prof. Dr. Mehmet Çelik Co. Supervisor, Mechatronic Eng. Dept., KTO Karatay University Examining Committee Members: Prof. Dr. Y. Samim Ünlüsoy Mechanical Engineering Dept., METU Prof. Dr. F. Suat Kadıoğlu Mechanical Engineering Dept., METU Prof. Dr. Metin Akkök Mechanical Engineering Dept., METU Prof. Dr. Mehmet Çelik Mechatronic Engineering Dept., KTO Karatay University Asst. Prof. Dr. Yiğit Yazıcıoğlu Mechanical Engineering Dept., METU

Date: 07/12/2010

iii

I hereby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work.

Name, Last Name: Ümit MERCĐMEK

Signature:

iv

ABSTRACT


Mercimek, Ümit

M.S., Department of Mechanical Engineering

Supervisor: Prof. Dr. F. Suat Kadıoğlu

Co-Supervisor: Prof. Dr. Mehmet Çelik

December 2010, 123 pages

Failure of metallic structures operating under shock loading is a common

occurrence in engineering applications. It is difficult to estimate the response of

complicated systems analytically, due to structure’s dynamic characteristics and

varying loadings. Therefore, experimental, numerical or a combination of both

methods are used for evaluations.

The experimental analysis of the shocks due to firing is done for 12.7mm

Gatling gun and 25mm cannon. During the tests, the Gatling gun and the cannon

are located on military Stabilized Machine Gun Platform and Stabilized Cannon

Platform respectively.

For the firing tests, ICP (integrated circuit piezoelectric) accelerometers are

attached to obtain the loading history for corresponding points. Shock Response

Spectrum (SRS) analysis (nCode Glypworks) is done to define the equivalent

shock profiles created on test pieces and the mount of 25mm cannon by means

v

of the gun and the cannon firing. Transient shock analysis of the test pieces and

the mount are done by applying the obtained shock profiles on the parts in a

finite element model (ANSYS).

Furthermore, experimental stress analysis due to shock loading is performed for

two different types of material and different thicknesses of the test pieces. The

input data for the analysis is obtained through measurements from strain rosette

precisely located at the critical location of the test pieces.

As a result of the thesis, a proposal is tried to be introduced where strain energy

density theory is applied to predict the shock failure at military structures.

Keywords: Shock Failure, SRS, Strain Energy Density, Experimental Stress

Analysis, Finite Elements Analysis.

vi

ÖZ

ASKERĐ MALZEMELERDE GER ĐNĐM ENERJĐSĐ YOĞUNLUĞUNU KULLANARAK ŞOK KIRILMA ANAL ĐZĐ

Mercimek, Ümit

Yüksek Lisans, Makine Mühendisliği Bölümü

Tez Yöneticisi: Prof. Dr. F. Suat Kadıoğlu

Ortak Tez Yöneticisi: Prof. Dr. Mehmet Çelik

Aralık 2010, 123 sayfa

Mühendislik uygulamalarında, şok yükü altında çalışan yapılarda sıklıkla metal

hasarları görülmektedir. Yapıların dinamik özellikleri ve değişken yüklerden

dolayı sistemlerin tepkilerini analitik olarak tespit etmek zor olmaktadır. Bu

nedenle, hesaplamalar için deneysel yöntemler, sayısal metotlar veya her iki

yöntem birlikte kullanılmaktadır.

Bu tezde, 12.7mm Gatling silahı ve 25mm top atışları kaynaklı oluşan şokların

deneysel analizi gerçekleştirilmi ştir. Testler esnasında Gatling silahı Stabilize

Makineli Tüfek Platformu (STAMP), top ise Stabilize Top Platformu (STOP)

üzerine yerleştirilmi ştir.

Atış testlerinde, belirlenen noktalardaki şok yüklemesini elde edebilmek için

ICP (integrated circuit piezoelectric) ivmeölçerler yerleştirilmi ştir. Silah ve top

atışları sırasında test parçaları ve top bağlantı parçası üzerinde oluşan şok

yüklerine eşdeğer şok profillerini tanımlayabilmek için Şok Tepki Spektrum

vii

(STS) analizleri (nCode Glypworks) yapılmıştır. Elde edilen şok profilleri ve

sayısal analiz modeli (ANSYS) kullanılarak test parçaları ve top bağlantı

parçası için zamana bağlı şok analizleri gerçekleştirilmi ştir.

Ayrıca, şok yüklemesi için, iki farklı test parçası malzemesi ve farklı

kalınlıklardaki test parçaları kullanılarak deneysel gerilme analizleri yapılmıştır.

Bu analizlerde kullanılan girdiler test parçalarının kritik bölgelerine hassas bir

şekilde yerleştirilen gerinim ölçerlerden alınan ölçüm sonuçlarıdır.

Tezde sonuç olarak, askeri yapılarda şok kırılmasına karar verebilmek için

gerinim enerji yoğunluğu teorisinin uygun olduğu değerlendirmesi ortaya

koyulmaya çalışılmıştır.

Anahtar Kelimeler: Şok Yorulması, STS, Gerinim Enerjisi Yoğunluğu,

Deneysel Gerilme Analizi, Sonlu Elemanlar Analizi.

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To My Family

ix

ACKNOWLEDGEMENTS

I am grateful to my thesis supervisor Prof. Dr. Suat KADIOĞLU and co-

supervisor Prof. Dr. Mehmet ÇELĐK for their guidance, support and helpful

criticism throughout the progress of my thesis study.

I would like to thank my friend Fatih ALTUNEL for his help.

The cooperation and friendly support of my colleagues in ASELSAN during my

thesis study also deserves to be acknowledged.

Thanks to my parents, brother and sister for their unique motivation and

encouragement.

Finally, many thanks to my wife Esin MERCĐMEK for her continuous help and

understanding during my thesis study.

x

TABLE OF CONTENTS

ABSTRACT ........................................................................................................ iv

ÖZ ................................................................................................................... vi

ACKNOWLEDGEMENTS ................................................................................ ix

TABLE OF CONTENTS ..................................................................................... x

LIST OF TABLES ............................................................................................. xii

LIST OF FIGURES .......................................................................................... xiii

NOMENCLATURE .......................................................................................... xx

CHAPTERS

1. INTRODUCTION ...................................................................................... 1

1.1. Mechanical Shock ................................................................................ 1

1.2. Types of Shocks ................................................................................... 3

1.2.1. Classical Shocks........................................................................... 3

1.2.2. Spectrum Shock ........................................................................... 5

1.2.3. Pyroshock ..................................................................................... 5

1.2.4. Seismic Shock .............................................................................. 6

1.2.5. Air Blast ....................................................................................... 7

1.3. Sources of Shocks ................................................................................ 7

1.4. Shock Testing Methods........................................................................ 8

1.5. Overview of the Study ......................................................................... 9

2. LITERATURE SURVEY ........................................................................ 12

3. THEORIES USED IN THE ANALYSIS................................................. 19

3.1. Shock Response Spectrum Theory .................................................... 19

3.2. Theories of Structural Failure ............................................................ 22

3.2.1. Total Strain Energy Theory ....................................................... 23

xi

3.2.2. Distortion Energy Theory .......................................................... 25

3.2.3. Plastic Deformation ................................................................... 25

3.3 Transient Response Analysis ......................................................... 29

3.3.1. Newmark’s Method ................................................................... 30

4. FIRING TESTS AND DATA ACQUISITION ....................................... 32

5. NUMERICAL AND EXPERIMENTAL SHOCK ANALYSIS .............. 39

5.1. Shock Response Spectrum Analysis .................................................. 39

5.2. Experimental Stress Analysis ............................................................ 44

5.3. Numerical (FEM) Analysis of Test Pieces ........................................ 48

5.4. Evaluations of the Results .................................................................. 58

5.5. Application of the Theory to a System in Use ................................... 64

6. DISCUSSION AND CONCLUSIONS .................................................... 72

REFERENCES .................................................................................................. 76

APPENDICES

A. EQUIPMENT USED THROUGHOUT TESTS ...................................... 80

B. STRAIN ROSETTE ANALYSIS ............................................................ 83

B.1 Rectangular Rosette ........................................................................... 84

B.2 Principal Stresses ............................................................................... 85

C. TENSILE TEST OF THE CAST ALUMINUM ...................................... 86

D. EXPERIMENTAL RESULTS OBTAINED BY USING ESAM ............ 88

E. NUMERICAL (ANSYS TRANSIENT) ANALYSIS RESULTS

OBTAINED BY USING ANSYS ............................................................ 94

F. SAMPLE ANALYSIS OF STRAIN ENERGY DENSITY IN ANSYS .....

................................................................................................................ 119

xii

LIST OF TABLES

TABLES

Table 4.1 Visual Inspection Results of Rosette Analysis Tests ........................ 38

Table 5.1 Maximum and Minimum Principal Stresses Results of Rosette

Analysis Tests.................................................................................... 46

Table 5.2 Material Properties of 7075-T7351 Aluminum ................................ 49

Table 5.3 Structural Material Properties for 7075-T7351 Al in ANSYS .......... 51

Table 5.4 Bilinear Isotropic Hardening Properties for 7075-T7351 Al in ANSYS

........................................................................................................... 51

Table 5.5 Results of ANSYS on the Cast Aluminum Test Pieces .................... 57

Table 5.6 Results of ANSYS on the 7075 Aluminum Test Pieces ................... 58

Table 5.7 Material Properties of Impax Steel ................................................... 65

Table 5.8 Material Properties for Impax Steel in ANSYS ................................ 66

Table 5.9 Bilinear Isotropic Hardening Properties for Impax Steel in ANSYS ...

........................................................................................................ 66

Table 5.10 Results of ANSYS Transient Response on the 25mm Cannon Mount

........................................................................................................... 71

Table C.1 Tensile Tests Inputs and Outputs for Cast Aluminum ..................... 86

Table F.1 Strain Energy Density Results of ANSYS for the Sample Bar ...... 123

xiii

LIST OF FIGURES

FIGURES

Figure 1.1 Various Shock Input Pulses ................................................................ 1

Figure 1.2 Response of System to Rectangular Pulses of Varying Duration ...... 2

Figure 1.3 Examples of Mechanical Shock ......................................................... 3

Figure 1.4 Half-Sine (Haversine) ......................................................................... 4

Figure 1.5 Sawtooth Shock .................................................................................. 4

Figure 1.6 Triangle Shock.................................................................................... 5

Figure 1.7 Shock Spectrum .................................................................................. 5

Figure 1.8 Pyroshock Shock Response Spectrum................................................ 6

Figure 1.9 Seismic Shock Time History .............................................................. 6

Figure 1.10 Air blast ............................................................................................ 7

Figure 1.11 Application of a Test Piece on Gatling Gun ..................................... 9

Figure 1.12 A Test Piece used for Experimental Analysis ................................ 10

Figure 1.13 Application of a Mount on 25mm Cannon ..................................... 10

Figure 3.1 Shock Response Spectrum Model .................................................... 20

Figure 3.2 Free-body Diagram of SDOF System .............................................. 20

Figure 3.3 Sample of a Shock Response Spectrum ........................................... 22

Figure 3.4 Strain Energy Density by using Stress-Strain Curve ........................ 24

Figure 3.5 Strain Energy Density - Different Types of Materials ..................... 24

Figure 3.6 Stress and Strain Relation ................................................................. 26

Figure 3.7 Isotropic (left) and kinematic (right) hardening Circle represents the

yield surface ...................................................................................... 28

Figure 4.1 A View of Stabilized GAU19/A 12.7mm Gatling Gun System ...... 32

Figure 4.2 A View of Stabilized KBA 25mm Cannon System ......................... 32

Figure 4.3 ICP accelerometers ........................................................................... 33

Figure 4.4 Locations of the Test Parts and the Accelerometers on them .......... 33

Figure 4.5 Acceleration “g” vs Time “sec” Signal ............................................ 34

Figure 4.6 A Test Piece Equipped with a Strain Rosettes ................................. 35

xiv

Figure 4.7 Rosettes Inputs, Calibration and Balancing Screen .......................... 35

Figure 4.8 Rosette Calibration Screen ............................................................... 36

Figure 4.9 Different Material or Thicknesses of Test Pieces............................. 36

Figure 4.10 A Raw strain data for a gage part of the analyzed rosette .............. 37

Figure 4.11 Cast Aluminum Test Pieces Examples After The Tests ................. 37

Figure 4.12 7075-T7351 Aluminum Test Pieces Examples After The Tests .... 38

Figure 5.1 SRS Block Diagram ......................................................................... 40

Figure 5.2 SRS Graph Property – Time increases ............................................. 41

Figure 5.3 SRS Graph Property – Amplitude increases .................................... 41

Figure 5.4 SRS Graph Property – Classical shock form changes ...................... 42

Figure 5.5 ACC-TDC on Gatling Gun – Test4 (2mm CA Part Test) ................ 43

Figure 5.6 SRS Graph – Gatling Gun Test4 ...................................................... 43

Figure 5.7 Rosette Analysis Input Screen .......................................................... 44

Figure 5.8 Minimum Principal Stresses – SG Measurement (1mm CA) .......... 45

Figure 5.9 Maximum Principal Stresses – SG Measurement (1mm CA) .......... 46

Figure 5.10 Counting Result of Firing Test (3mm 7075 AL) ............................ 47

Figure 5.11 Mean Stress and AES (1mm 7075 AL) .......................................... 48

Figure 5.12 Mesh Model of the Test Piece ........................................................ 49

Figure 5.13 Bilinear Isotropic Hardening Graph for 7075-T7351 Al in ANSYS

......................................................................................................... 51

Figure 5.14 Thickness Definition of the Test Piece ........................................... 52

Figure 5.15 Shell Model of the Test Piece ......................................................... 52

Figure 5.16 Maximum Strain Energy for 300g 1ms ACC (1mm CA) .............. 53

Figure 5.17 Stress-Strain diagram for 7075-T7351 Aluminum ......................... 54

Figure 5.18 SMP Velocity Graph for 300g 1ms ACC (1mm Cast Aluminum) 55

Figure 5.19 SMP Maximum Principal Stresses for 300g 1ms ACC (1mm Cast

Aluminum) ...................................................................................... 55

Figure 5.20 SMP Minimum Principal Stresses for 300g 1ms ACC (1mm Cast

Aluminum) ...................................................................................... 56

Figure 5.21 SMP Equivalent Stresses Graph for 300g 1ms ACC (1mm Cast

Aluminum) ...................................................................................... 56

xv

Figure 5.22 Effect of Material Thickness on Maximum SED (Cast Aluminum)

......................................................................................................... 59

Figure 5.23 Effect of Material Thickness on Maximum SED (7075 Al) .......... 59

Figure 5.24 Effect of Material Thickness on SF of UTS (Cast Aluminum) ...... 60

Figure 5.25 Effect of Material Thickness on SF of UTS (7075 Al) .................. 61

Figure 5.26 Effect of Material Thickness on SF of SE (Cast Aluminum) ......... 62

Figure 5.27 Effect of Material Thickness on SF of SE (7075 Al) ..................... 62

Figure 5.28 Effect of Material Volume on Maximum SED (Cast Aluminum) . 63

Figure 5.29 Effect of Material Volume on Maximum SED (7075 Al) .............. 64

Figure 5.30 Mesh Model of the Mount .............................................................. 65

Figure 5.31 Bilinear Isotropic Hardening Graph for Impax Steel in ANSYS ... 66

Figure 5.32 ANSYS Model of the Mount .......................................................... 67

Figure 5.33 ACC-TDC on the Mount of 25mm Cannon – Test6 ...................... 68

Figure 5.34 SRS Graph – 25mm Cannon Test6 ................................................ 68

Figure 5.35 Maximum Strain Energy for 100g 2ms ACC (Impax Steel) .......... 69

Figure 5.36 Stress-Strain graph for Impax Steel ................................................ 70

Figure 5.37 SMP Velocity Graph for 100g 2ms ACC (Impax Steel) ................ 70

Figure 5.38 SMP Maximum Principal Stresses for 100g 2ms ACC (Impax Steel)

......................................................................................................... 71

Figure 5.39 SMP Minimum Principal Stresses for 100g 2ms ACC (Impax Steel)

......................................................................................................... 71

Figure A.1 IOtech data acquisition system ........................................................ 80

Figure A.2 Traveler Strain Master data acquisition system ............................... 81

Figure A.3 1-axial piezoelectric accelerometer ................................................. 81

Figure A.4 Strain Gage (CEA-13-125UR-350) ................................................. 82

Figure C.1 Tensile Test for the Cast Aluminum ................................................ 86

Figure C.2 Stress-Strain Diagram for the Cast Aluminum ................................ 87

Figure D.1 Minimum Principal Stresses Graph– SG Measurement (Test 6) .... 88

Figure D.2 Maximum Principal Stresses Graph – SG Measurement (Test 6) ... 88

Figure D.3 Minimum Principal Stresses Graph – SG Measurement (Test 10) . 89

Figure D.4 Maximum Principal Stresses Graph – SG Measurement (Test 10) . 89


xvi




Figure D.9 Minimum Principal Stresses Graph – SG Measurement (Test 9) ... 92




Figure E.1 Maximum Strain Energy for 300g 1ms ACC (1mm Cast Aluminum)

......................................................................................................... 94

Figure E.2 SMP Velocity Graph for 300g 1ms ACC (1mm Cast Aluminum) .. 94

Figure E.3 SMP Maximum Principal Stresses Graph for 300g 1ms ACC (1mm

Cast Aluminum) .............................................................................. 95

Figure E.4 SMP Minimum Principal Stresses Graph for 300g 1ms ACC (1mm

Cast Aluminum) .............................................................................. 95

Figure E.5 SMP Equivalent Stresses Graph for 300g 1ms ACC (1mm Cast

Aluminum) ...................................................................................... 96


......................................................................................................... 96

Figure E.7 SMP Velocity Graph for 300g 1ms ACC (2mm Cast Aluminum) .. 97

Figure E.8 SMP Maximum Principal Stresses for 300g 1ms ACC (2mm Cast

Aluminum) ...................................................................................... 97

Figure E.9 SMP Minimum Principal Stresses for 300g 1ms ACC (2mm Cast

Aluminum) ...................................................................................... 98


Aluminum) ...................................................................................... 98

Figure E.11 Maximum Strain Energy for 300g 1ms ACC (3mm Cast

Aluminum) ...................................................................................... 99

Figure E.12 SMP Velocity Graph for 300g 1ms ACC (3mm Cast Aluminum) 99


Aluminum) .................................................................................... 100


Aluminum) .................................................................................... 100

xvii


Aluminum) .................................................................................... 101


Aluminum) .................................................................................... 101

Figure E.17 SMP Velocity Graph for 300g 1ms ACC (4mm Cast Aluminum)

....................................................................................................... 102


Aluminum) .................................................................................... 102


Aluminum) .................................................................................... 103


Aluminum) .................................................................................... 103


Aluminum) .................................................................................... 104


....................................................................................................... 104


Aluminum) .................................................................................... 105


Aluminum) .................................................................................... 105


Aluminum) .................................................................................... 106

Figure E.26 Maximum Strain Energy for 300g 1ms ACC (0,5mm 7075 Al) . 106

Figure E.27 SMP Velocity Graph for 300g 1ms ACC (0,5mm 7075 Al) ....... 107

Figure E.28 SMP Maximum Principal Stresses for 300g 1ms ACC (0,5mm 7075

Al) ................................................................................................. 107

Figure E.29 SMP Minimum Principal Stresses for 300g 1ms ACC (0,5mm 7075

Al) ................................................................................................. 108

Figure E.30 SMP Equivalent Stresses for 300g 1ms ACC (0,5mm 7075 Al) . 108

Figure E.31 Maximum Strain Energy for 300g 1ms ACC (1mm 7075 Al) .... 109

Figure E.32 SMP Velocity Graph for 300g 1ms ACC (1mm 7075 Al) .......... 109

xviii

Figure E.33 SMP Maximum Principal Stresses for 300g 1ms ACC (1mm 7075

Al) ................................................................................................. 110

Figure E.34 SMP Minimum Principal Stresses for 300g 1ms ACC (1mm 7075

Al) ................................................................................................. 110

Figure E.35 SMP Equivalent Stresses for 300g 1ms ACC (1mm 7075 Al) .... 111




Al) ................................................................................................. 112


Al) ................................................................................................. 113





Al) ................................................................................................. 115


Al) ................................................................................................. 115





Al) ................................................................................................. 117


Al) ................................................................................................. 118


Figure F.1 A Simple Square Bar ...................................................................... 119

Figure F.2 The Force Applied on the Bar ........................................................ 120

Figure F.3 The Areas under the Bilinear Isotropic Hardening Graph for 7075-

T7351 Al in ANSYS ..................................................................... 120

Figure F.4 The Time of the Yield Point ........................................................... 121

Figure F.5 The Maximum Strain Energy Value for the Yield Point ............... 121

xix

Figure F.6 The Time of a Plastic Region Point ............................................... 122

Figure F.7 The Maximum Strain Energy Value for the Plastic Region Point . 122

Figure F.8 Force vs Load Point Displacement Graph...................................... 123

xx

NOMENCLATURE

ACC Acceleration

ADF Anelastic Displacement Fields

AES Alternating Equivalent Stress

c Damping

C Matrix of Damping Coefficients

DOF Degree of Freedom

Dɺ Vector of Velocity

e e e1 2 3 Measured Strains

E Young’s Modulus of the Material

fn Natural Frequency of the SDOF System

F Force

F* Gage Factor

F(T) Gage Factor at Test Temperature.

FEA Finite Element Analysis

FEM Finite Element Method

g Gravitational Acceleration

ICP Integrated Circuit Piezoelectric

ISO International Organization for Standardization

k Stiffness

K K Kt t t1 2 3, , Transverse Sensitivities of Strain Gages

L Length

∆L Elongation

m Mass

P Tangent Modulus of the Material

PSD Power Spectral Density

Q Damping of SDOF system

RMS Root Mean Square

xxi

SDOF Single Degree of Freedom

SED Strain Energy Density

SF Safety Factor

SG Strain Gage

SMP Strain Measurement Point

SRS Shock Response Spectrum

ST Shell Thickness

Sy Yield Strength of the Material

t Time

t Thickness

TDC Time Data Collected

u Strain Energy

UTS Ultimate Tensile Strength

w Width

x Displacement

xɺ Velocity

xɺɺ Acceleration

Β, γ Constants

ε Engineering Strain

ε ε ε1 2 3 Calculated Strains

ε εmin max Calculated Maximal and Minimal Principal Strains

ε’ Semi-corrected Strain

ε’’ Indicated Strain

εT/O(T) Thermal Output at Temperature T

ν Poisson’s Ratio of the Material

θ Direction of Principal Strains

σ Engineering Strength

1 2 3, ,σ σ σ Principal Stresses

σ σmin max Calculated Maximal and Minimal Stresses,

aσ Alternating Stress

xxii

mσ Mean Stress

ω Excitation Frequency

nω Natural Frequency

ξ Damping Ratio

1

CHAPTER 1

1. INTRODUCTION 1. INTRODUCTION

1.1. Mechanical Shock

Mechanical shock is classified as a transient phenomenon where the equilibrium

of a system is disrupted by a suddenly applied force or by a sudden change in

the direction or magnitude of velocity. Shock usually contains a single impulse

of energy of short duration and large intensity [1]. A shock input pulse is

described by its peak amplitude A0 expressed in g's (gravitational acceleration),

by its duration t0 expressed in milliseconds and its overall shape which can take

such forms as half sine, triangular, versed sine, rectangular and the form most

likely to occur in nature, a more or less random shaped complex wave form

force and acceleration impulse. Various shock input pulses are shown in Figure

1.1. Acceleration levels and pulse durations vary considerably with the

particular application.

Figure 1.1 Various Shock Input Pulses [1]

The system receives the energy at a high peak value over a short period of time,

stores it and releases it over a longer period of time with a considerably lower

peak value. If the acceleration can be limited, the equipment will survive.

2

In as much as a shock pulse may contain frequency components ranging from

very low to very high; it is not possible to avoid excitation of vibratory process

of the isolator/mass system with its natural frequency. On the other hand, if the

duration of the shock pulse is short, the response of the system may not have

serious consequences. Figure 1.2 demonstrates the comparative response of a

spring mass system to a rectangular pulse whose duration is greater than the

natural period of the vibratory system (I) and to a relatively short impulsive-

type shock (II).

Figure 1.2 Response of System to Rectangular Pulses of Varying Duration [2]

The effects of mechanical shock are so important that the International

Organization for Standardization (ISO) has a standing committee, TC 108,

dealing with shock and vibration; a Shock and Vibration Handbook [3] has been

published and routinely updated by McGraw Hill since 1961; and the

Department of Defense has sponsored a focused symposium on this subject at

least annually since 1947. [4] Figure 1.3 provides several examples of

components or systems experiencing mechanical shock [5].

3

Figure 1.3 Examples of Mechanical Shock [5]

Mechanical shock can be specified in either the time, and/or frequency domains,

or by its associated shock-response spectrum.

1.2. Types of Shocks [6]

There are five classifications of shocks that are described here. They are

Classical Shock, Spectrum Shock, Pyroshock, Seismic Shock and Air Blast.

1.2.1. Classical Shocks

Classical shocks come in many shapes, and tend to be the most common. They

are simple shocks that can be fairly easily reproduced in a test laboratory.

The first is a half-sine (Haversine) as seen in Figure 1.4. It’s the most common

of all Classical Shocks and resembles a half-sine with rounded beginning and

end to make it physically reproducible.

4

Figure 1.4 Half-Sine (Haversine)

Next is the sawtooth shock (Figure 1.5), which resembles a tooth on saw blade.

It has a rising slope followed by a sharp drop off. The shock is also specified in

MIL-STD-810F [7]. A Sawtooth shock is obtained by dropping a cylindrical

lead pellet with conical top on a shock table.

Figure 1.5 Sawtooth Shock

Another classical shock is the triangle shock (Figure 1.6). It is similar to a

sawtooth; however the drop is not quite as severe. The triangle shock is

infrequently used in shock testing.

5

Figure 1.6 Triangle Shock

1.2.2. Spectrum Shock

A shock response spectrum defined by frequency and acceleration level pairs

and uses the Shock Response Spectrum (SRS) for measurement. (Figure 1.7)

Figure 1.7 Shock Spectrum

1.2.3. Pyroshock

Pyroshock is characterized by a high acceleration, short duration shock pulse

Pyroshock also uses the SRS for measurement. (Figure 1.8)

6

Figure 1.8 Pyroshock Shock Response Spectrum

1.2.4. Seismic Shock

Seismic shock is characterized by low acceleration, high displacement, and long

time duration. (Figure 1.9)

Figure 1.9 Seismic Shock Time History

7

1.2.5. Air Blast

Air blast is characterized by sharp rise and longer decay than a pyroshock.

(Figure 1.10)

Figure 1.10 Air blast

1.3. Sources of Shocks [6]

There are three main sources for shock events. They are pyrotechnic excitation,

mechanical excitation and natural phenomena. All of these shock events

commonly occur in the aerospace as well as other applications.

- Pyrotechnic Excitation

a. Point Sources (explosive bolts, pin pullers)

b. Line Sources (linear shape charges, detonating chords)

- Mechanical Excitation

a. Collision impact

b. Handling / drop

c. Evasive maneuvers in aircraft or missiles (gust loading)

d. Ballistic impact

8

e. Aircraft landing

f. Braking

g. Missile / Rocket launching

h. Gunfire

i. High-speed fluid entry

j. Transportation, uneven surfaces, rough terrain

- Natural Phenomenon

a. Earthquake

b. Wind gust

c. Air blast

d. Ocean waves

e. Ice impact

1.4. Shock Testing Methods [6]

Shock testing is commonly performed by the following methods. Each method

imparts the kinetic energy to the system in a different manner.

• Drop

• Hammer / Impact

• Shaker (Electrodynamic / Hydraulic)

• Pyroshock

• Hopkinson Bar

In drop testing ∆V≠0 (change in velocity) and ∆d≠0 (change in displacement),

while others maintain zero overall change in velocity and displacement. Further

details can be found in [6].

9

1.5. Overview of the Study

Shocks can cause structural failures such as cracked hausings, fatigue cracks,

deformation of structures [6], etc. In this study, the shock induced transient

stress analysis of the test pieces for a Gatling gun and the 25mm cannon mount

will be performed to investigate their structural failure (yielding). Test pieces

and the mount of 25mm cannon are integrated on the gun and the cannon

respectively. Test pieces are not the part of the Gatling gun system. They are

just used to obtain data and observe the shock failure without deformation on

the system. Application of the pieces and the 25mm cannon mount on the

systems are shown in Figure 1.11 to Figure 1.13.

Figure 1.11 Application of a Test Piece on Gatling Gun

Test Piece

Gatling Gun

10

Figure 1.12 A Test Piece used for Experimental Analysis

Figure 1.13 Application of a Mount on 25mm Cannon

The fundamental loading for these parts is the shock due to firing. The stress

analysis of the parts under shock loading needs to be performed. While

performing the experimental stress analysis, ESAM [8] software will be used.

Also, the shock loading history for corresponding points will be obtained by

25mm Cannon Mount

11

using ICP (integrated circuit piezoelectric) [9] accelerometers. Shock Response

Spectrum (SRS) analysis (nCode Glypworks) [10] will be done to define the

equivalent shock profiles by using the obtained shock loading history.

Since the implementation of actual shock loading in Finite Element Analysis

takes too much computational time, an equivalent classical shock (Haversine) is

used.

After that, transient shock analysis will be performed to find the numerical

stress values. Numerical and experimental stress values will be compared to

verify the finite element models (FEM). Finally the stress and strain energy

density values will be used to define which values give more accurate results to

define the effect of shock loading on the parts. The details of these studies will

be explained in the following chapters.

12

CHAPTER 2

2. LITERATURE SURVEY

The shock loading problem under consideration in this study is a complex one

due to the following factors:

- It is a rapid phenomenon that excites dynamic (resonant) response of the

material but it causes very little overall deflection,

- It causes multiaxial stress state.

- In a comparatively short time, a moderately high level force impulse is

input to the material.

Shock Response Spectrum (SRS) analysis was developed as a standard data

processing method in the early 1960’s. Firstly SRS was used by U.S.

Department of Defense. Now this signal processing method is standardized by

ISO 18431-4 [11]. Detailed information on SRS is given in Chapter 3.

A brief review of some studies which are related to the work done in this study

is given below.

Biot et al. [12] conceived the shock response spectrum. He defined the SRS as

the maximum response motion from a set of single DOF oscillators covering the

frequency range. He showed how to pick a small number of modes which are

adequate for design. For earthquake applications, he used the traditional

assumption that the ground’s motion is not affected by the dynamic motion of

the building. Later another study demonstrated that this assumption is overly

conservative and leads to over design of equipment. He noted that frequency

peaks in shock spectra from a single earthquake are not constant within the

13

same neighborhood. This led to recommending an envelope approach of all

spectra for design purposes. Relative to loadings on buildings, he found that

stresses calculated using the maximum envelope approach was much higher

than those observed from an actual earthquake. He attributed this to factors such

as damping, plastic deformation, and possible interaction of nearby soil with the

foundation of the building.

Housner et al. [13] developed the first spectra used for seismic design of

structures in the late 1950s. These were obtained by averaging and smoothing

the response spectra from eight ground motion records, two from each of the

following four earthquakes.

- El Centro (1934)

- El Centro (1940)

- Olympia (1949)

- Tekiachapi (1952)

Newmark et al. [14] presented a family of single-step integration methods for

the solution of structural dynamic problems for both blast and seismic loading.

During the past 40 years Newmark’s method has been applied to the dynamic

analysis of many practical engineering structures. In addition, it has been

modified and improved by many other researchers.

Newmark et al. [15] developed an earthquake design spectrum approach based

on amplification factors applied to maximum ground motions. The

amplification factors are listed for different probabilities of occurrence and also

for various levels of damping of the structure. He showed how a spectrum is

developed from the ground motion maxima. The region of amplified response is

between the relatively high and relatively low frequency extremes of the

spectrum. At relatively high frequencies, the shock spectrum level approached

the maximum ground acceleration. This is the aforementioned feature that was

observed in the earlier El Centro earthquake.

14

Kelly and Richman et al. [16] clarified physical descriptions and mathematical

presentation of the shock response spectrum (SRS). This article has been cited

in multiple handbooks on the subject and research articles. The main purpose of

this note was to correct several typographical errors in the Biot manuscript’s

presentation of a recursive algorithm for SRS calculations. These errors were

consistent across all three editions of the monograph. The secondary purpose of

this note was to present a Matlab implementation of the corrected algorithm.

Justin, Andrew and Winfred et al. [17] verified the corrections described in the

preceding [16] by comparing the corrected algorithm and the original algorithm

to an independent SRS code. The independent code used a piecewise-linear

approximation for the base acceleration. The various algorithms were applied to

accelerometer data from the ignition environment of live-fire testing of the

Space Shuttle Reusable Solid Rocket Motor. Specifically, data was evaluated

from the radial channel at station 1479.5 on Technical Evaluation Motor 13.

The data was sampled at 10,000 Hz. The SRS of this acceleration data was

calculated using three different algorithms. The corrected algorithm of

equations was compared with the uncorrected equations from Kelly and

Richman as well as the independent code. For each algorithm, a damping ratio

of ζ = 0.05 was used, and the peak response was calculated for a range of

natural frequencies at one-third octaves up to the Nyquist frequency. The

corrected algorithm and the independent code showed strong agreement with

each other; however, the uncorrected algorithm displayed large differences in

the high frequency regime.

Walter et al. [18] initially clarified what mechanical shock is and why we

measure it. After that, basic requirements are provided for all measurement

systems that process transient signals. High-frequency and low-frequency

dynamic models for a measuring accelerometer were presented and justified.

These models are then used to investigate accelerometer responses to

mechanical shock. The results enabled “rules of thumb” to be developed for

15

shock data assessment and proper accelerometer selection. Other helpful

considerations for measuring mechanical shock were also provided.

Alexander et al. [19] provided a basic overview, or primer, of the shock

response spectrum (SRS). This paper was prepared for the design engineers who

needed to work with the shock response spectrum, and would like to understand

the underlying detail.

Smith and Melander et al. [20] described a study that examined some of the

critical parameters that effect Shock Response Spectrum (SRS) results and

reported on their use by some of the practitioners in the field. They showed that

parameters such as anti-alias filter characteristics, ac-coupling strategies, and

analysis algorithm/strategy can strongly effect the results and that they are not

uniformly applied by system suppliers or users.

Hollowell and Smith et al. [21] discussed the problem further and presents an

analytical procedure that may be applied to achieve agreement between the data

sets acquired and analyzed by different laboratories.

Tuma and Koci et al. [22] presented the method of calculation of the shock

response spectrum, which was corresponding to an acceleration signal exciting

the resonance vibration of substructures. SRS determined the maximum or

minimum of the substructure acceleration response as a function of the natural

frequencies of a set of the single degree of freedom systems modeling the

mentioned substructures. The shock was recorded in digital form, commonly as

acceleration signal. The single-degree-of-freedom systems (SDOF) were

approximated by an IIR digital filter and the filter response to the sampled

acceleration signal was easily calculated. This shock response spectrum shows

how the individual component of the impulse signal excites the mechanical

structure to resonate.

16

Çelik [23] used experimental approach to perform the failure analysis of the

launcher assembly of a military land vehicle. Finite element analysis was

performed to determine the critical locations where strain rosettes were settled

down on the physical prototype. Tests were carried out by performing

operational life profile of the vehicle in the field. Absolute maximum principal

stresses were determined at each rosette location by analyzing the strain data

collected. At the study, functional failures of the electronic equipments in the

system are investigated.

Çelik et al. [24] applied shock and vibration control techniques using spring

isolators to provide dynamic protection of the system units installed on the

vehicles. The Repetitive Shock Response Spectrum (SRS) analysis was

performed to define the gunfire vibration profile and make qualification that the

electronic equipments should withstand. It was also intended to obtain required

stabilization during operation of the platform.

Douglas et al. [25] examined recent efforts attempted to improve the simulation

results of the athwartship (transversely across a ship from one side to the other)

motion including shock spectra analysis, and the reasons behind the disparities

that exist between the simulated values and the actual trial data. He thought that

shock spectra analysis could serve as a design tool as well as a tool for

comparative analysis. Barge testing were used to shock qualify naval equipment

for years, yet using these UNDEX simulations and the shock spectra’s created,

accurate predictions of the frequency response can be achieved. As a

comparative tool, the shock spectra showed that the low frequency response is

very accurately modeled, and in many cases the simulations are more

conservative than the actual trial data.

Parlak et al. [26] did the experimental analysis of repetitive recoil shocks due to

machine gun firing. The machine gun was located on the military Low Level

17

Air Defence System. For the test of shock and vibration on the system, four

different points were determined and ICP (integrated circuit piezoelectric)

accelerometers were located for corresponding points. Shock Response

Spectrum (SRS) analysis was done to define the minimum shock profile that the

electronic equipments should withstand. It was aimed to use these equivalent

simple shock profiles during the shock qualification testing of the equipments.

Rusovici et al. [27] employed high-damping viscoelastic materials in the design

of geometrically complex impact absorbent components. The Anelastic

Displacement Fields (ADF) method was employed to develop new

axisymmetric and plane stress finite elements that were capable of modeling

frequency dependent material behavior of linear viscoelastic materials. The new

finite elements were used to model and analyze behavior of viscoelastic

structures subjected to shock loads. The development of such ADF-based finite

element models offered an attractive analytical tool to aid in the design of shock

absorbent mechanical filters. This work also showed that it is possible to

determine material properties’ frequency dependence by iteratively fitting ADF

model predictions to experimental results.

Carpinteri et al. [28] carried out a study on expected principal stress directions

under multiaxial loading. A theoretical procedure to calculate the Euler angles

from the matrix of the principal direction cosines for a generic time instant was

proposed. The procedure consists of averaging the instantaneous values of the

three Euler angles through weight functions. It was examined how such

theoretical principal directions determined by applying the proposed procedure

are correlated to the position of the experimental fracture plane for some fatigue

tests in the literature. The algorithm proposed is applied to some experimental

biaxial in- and out-of-phase stress states to assess the correlation. From the

results obtained, it was seen that in the case of a small phase angle, the normal

vector to the experimental fracture plane agrees with the expected direction of

the maximum principal stress.

18

Shang et al. [29] developed a new theory for the application of local stress-

strain field intensity to the fatigue damage at a notch. The effects of the local

stress-strain gradient on fatigue damage were taken into account at notches. The

parameters needed for local stress-strain intensity approach, as a fatigue analysis

tool, were calculated from an incremental elastic-plastic finite element analysis

under random cyclic loading.

Consequently, critical conditions and important outcomes found in the literature

are noted to be considered in this study. No studies which are directly related

with the shock failure (yielding) analysis of a mechanical structure by using

SRS could be found.

19

CHAPTER 3

3. THEORIES USED IN THE ANALYSIS

3.1. Shock Response Spectrum Theory

Shock motion in the form of time history is usually not very useful for

engineering purposes. In order to extract useful information, such as the

amount of strain and stress that will be applied on an instrument due to a shock

or to synthesize a shock at laboratory conditions that will have the same

characteristics as that will be experienced in the field, time domain data has to

be reduced to a different form. One of the most commonly used forms of this

reduction is the Shock Response Spectrum (SRS).

Shock response spectrum is the plot of the maximum acceleration of single

degree of freedom (SDOF) systems with different natural frequencies when

excited with a given shock input.

As it was stated above, mechanical shock pulses are analyzed in terms of shock

response spectra. The shock response spectrum assumes that the mechanical

shock pulse is applied as a common base input to a group of independent single-

degree-of freedom systems, see Figure 3.1. The shock response spectrum gives

the peak response of each system with respect to the natural frequency of each

system. Damping is typically fixed at a constant value, such as 5%, which is

equivalent to an amplification factor of Q=10 [30].

20

Figure 3.1 Shock Response Spectrum Model [30]

Newton’s law can be applied to a free-body diagram of an individual system, as

shown in Figure 3.2.

Figure 3.2 Free-body Diagram of SDOF System [30]

In the gun systems “y” is the displacement of the gun and “x” is the

displacement of the part on which the acceleration data is collected. A

summation of forces yields the following governing differential equation of

motion:

mx cx kx cy ky+ + = +ɺɺ ɺ ɺ (3.1)

A relative displacement can be defined as z x y= − . The following equation is

obtained by substituting this expression into equation 3.1.

mz cz kz my+ + = − ɺɺɺɺ ɺ (3.2)

21

Additional substitutions can be made as follows,

2n

k

mω = , 2 n

c

mξω = (3.3), (3.4)

Note that ξ is the damping ratio, and that nω is the natural frequency.

Furthermore, ξ is often represented by the amplification factor Q, where

Q=1/(2 ξ) (3.5)

Substitution of these terms into equation 3.2 yields an equation of motion for

the relative response,

22 ( )n nz z z y tξω ω+ + = −ɺɺɺɺ ɺ (3.6)

Equation 3.6 does not have a closed-form solution for the general case in which

( )y tɺɺ is an arbitrary function. A convolution integral approach must be used to

solve the equation. The convolution integral is then transformed into a series for

the case where ( )y tɺɺ is in the form of digitized data. Furthermore, the series is

converted to a digital recursive filtering relationship (computational process, or

algorithm, transforming a discrete sequence of numbers “the input” into another

discrete sequence of numbers “the output” having a modified frequency domain

spectrum) to expedite the calculation. The resulting formula for the absolute

acceleration is [30],

1 22exp[ ]cos[ ] exp[ 2 ] 2i n d i n i n ix t t x t x tyξω ω ξω ξω− −= − ∆ ∆ − − ∆ + ∆ɺɺ ɺɺ ɺɺ ɺɺ

21exp[ ]{[ (1 2 )]sin[ ] 2 cos[ ]}n

n n d d id

t t t t yωω ξω ξ ω ξ ωω −+ ∆ − ∆ − ∆ − ∆ ɺɺ (3.7)

22

where, ω ω ζd n= −1 2 (3.8)

Equation (3.7) was used to calculate the shock response spectrum in Figure 3.3.

Note that this equation must be used for each natural frequency.

Figure 3.3 Sample of a Shock Response Spectrum [30]

3.2. Theories of Structural Failure

Strain energy is one of fundamental concepts in mechanics and its principles are

widely used in practical applications to determine the response of a structure to

loads.

23

3.2.1. Total Strain Energy Theory [31]

The theory, as proposed by Beltrami, and also attributed to Haigh, is based on a

critical value of the total strain energy stored in the material, and this is a

product of stress and strain.

The work done in elastic deformation or the stored elastic strain energy may be

written as,

1

2u W xδ= (3.9)

or,

1122

W x

Ax

δσε= per unit volume (3.10)

In a three-dimensional stress system, the total strain energy is,

1 1 2 2 3 3

1 1 1

2 2 2TU σ ε σ ε σ ε= + + (3.11)

Now using a stress-strain relationship, the principle strains may be written as,

11 2 3( )

E E

σ υε σ σ= − + (3.12)

22 3 1( )

E E

σ υε σ σ= − + (3.13)

33 2 1( )

E E

σ υε σ σ= − + (3.14)

substituting for 1 2 3, ,ε ε ε and rearranging,

2 2 21 2 3 1 2 2 3 3 1

1( ) ( )

2 2TUE E

υσ σ σ σ σ σ σ σ σ= + + − + + (3.15)

24

Figure 3.4 Strain Energy Density by using Stress-Strain Curve [32]

Figure 3.5 Strain Energy Density - Different Types of Materials [33]

The area under a complete stress-strain diagram gives a measure of a material's

ability to absorb energy up to fracture and is called toughness [33]. The larger

the area under the diagram, the tougher the material. A high modulus of

toughness is important when a material is subject to shock loads.

25

3.2.2. Distortion Energy Theory [34]

Huber, in 1904, proposed that the total strain energy of an element of material

could be divided into two parts, that due to change in volume and that due to

change in shape. These will be termed volumetric strain energy UV, and

distortion or shear strain energy, US. It is rather more simple to determine the

former quantity then the latter, and since the total strain energy has already been

determined, the shear or distortion component can be determined as,

US = UT – UV (3.16)

The distortion energy theory says that failure (yielding) occurs due to distortion

of a part, not due to volumetric changes in the part (shearing causes distortion).

Failure will occur if,

3132212

32

22

1' σσσσσσσσσσ −−−++= ≥ Sy (3.17)

In terms of applied stresses,

( ) ( ) ( ) ( )σ

σ σ σ σ σ σ τ τ τ' =

− + − + − + + +x y y z z xy xy yz zx

2 2 2 2 2 26

2 (3.18)

σ ’ is called the Von Mises effective stress.

The distortion energy theory is used in the simulations of the thesis since

ANSYS which is the program used for the simulations applies this theory.

3.2.3. Plastic Deformation [43]

From mechanics point of view, when tensile load is applied to a specimen of

ductile metal, extension of the specimen will occur and specimen will return to

26

its initial shape when tensile load is removed, this deformation process is called

elastic deformation. Each increment of load is related to corresponding

increment in extension. But when the effect of load makes the tensile stress

exceed yield stress, the specimen will not return to the initial shape after

removing load, this deformation process is called plastic deformation.

Figure 3.6 Stress and Strain Relation

Elasto-plastic material was used in this simulation, which means the

deformation will undergo an elastic deformation process when the stress is less

than yield stress, but afterwards the mixed deformation of elastic and plastic

will appear when the continually increasing stress exceeds yielding point.

Figure 3.6 shows the stress and strain relation from the test of tensile load, x

axis is effective strain and y axis is effective stress. yσ in the figure is the yield

stress and, when the effective stress is below it, the deformation is in elastic

region, when effective stress is above it, plastic flow starts. If the effective strain

is exceeding the fracture point, the material can be sheared off.

27

Work hardening is the strengthening of a material by plastic deformation. As the

material becomes increasingly saturated with new dislocations, more

dislocations are prevented from nucleating (a resistance to dislocation-formation

develops). This resistance to dislocation-formation manifests itself as a

resistance to plastic deformation; hence, the observed strengthening.

In metallic crystals, irreversible deformation is usually carried out on a

microscopic scale by defects called dislocations. At normal temperatures the

dislocations are not annihilated by annealing. Instead, the dislocations

accumulate, interact with one another, and serve as pinning points or obstacles

that significantly impede their motion. This leads to an increase in the yield

strength of the material and a subsequent decrease in ductility.

For hardening materials, the yield surface will evolve in space in one of three

ways. The first form of yield surface evolution is called isotropic hardening. For

isotropic hardening, the yield surface grows in size while the center remains at a

fixed point in stress space. The second form of surface evolution is called

kinematic hardening. For kinematic hardening, the center of the yield surface

translates in stress space, while the size remains fixed. The third type of surface

evolution is called mixed hardening where both isotropic and kinematic

hardening characteristics are evident. For mixed hardening, the orientation of

the yield surface may also change as well. Although isotropic hardening is the

most common form of yield surface evolution assumed in finite element models

for metal forming simulation, it is not necessarily the most accurate. The mixed

hardening model is most likely the most accurate of the three models.

28

Figure 3.7 Isotropic (left) and kinematic (right) hardening Circle represents the

yield surface

Circles in Figure 3.7 represent the yield surface which derives from von Mises

criterion. Isotropic hardening and kinematic hardening are simply distinguished

in two-dimensional figure.

Isotropic hardening was used in all the simulations in this thesis. In order to

derive the formula for hardening, total effective strain is given as,

eff ep E

σε ε= + (3.19)

effε is the effective strain and epε is the effective plastic strain. Then hardening

criterion is given,

exp exp( ) ( )hard eff y ep yE

σσ σ ε σ σ ε σ= − = + − (3.20)

σhard is the yield stress increase due to hardening and σexp is the isotropic

tangent modulus.

29

3.3 Transient Response Analysis [36]

Structural systems are very often subjected to transient excitation. A transient

excitation is a highly dynamic, time-dependent force exerted on the solid or

structure, such as earthquake, impact and shocks. The discrete governing

equation system for such a structure often requires a special solver. The widely

used method is the so-called direct integration method. The direct integration

method basically uses the finite difference method for time stepping to solve the

system equation. There are two main types of direct integration method: implicit

and explicit.

Explicit methods do not involve the solution of a set of linear equations at each

time step. Basically, these methods use the differential equation at time “t” to

predict a solution at time “t + ∆t”. For most real structures, which contain stiff

elements, a very small time step is required in order to obtain a stable solution.

Therefore, all explicit methods are conditionally stable with respect to the size

of the time step.

On the other hand, implicit methods attempt to satisfy the differential equation

at time “t” after the solution at time “t - ∆t” is found. These methods require the

solution of a set of linear equations at each time step; however, larger time steps

may be used. Implicit methods can be conditionally or unconditionally stable.

There exist a large number of accurate, higher-order, multi-step methods that

have been developed for the numerical solution of differential equations. These

multistep methods assume that the solution is a smooth function in which the

higher derivatives are continuous. The exact solution of many nonlinear

structures requires that the accelerations, the second derivative of the

displacements, are not smooth functions. This discontinuity of the acceleration

is caused by the nonlinear hysteresis of most structural materials, contact

between parts of the structure, and buckling of elements.

30

It is the conclusion [36] that only single-step, implicit, unconditionally stable

methods can be used for the step-by-step shock analysis of the structures.

Before discussing the equations used for the time stepping techniques, it should

be mentioned that the general system equation for a structure can be re-written

as,

KD CD MD F+ + =ɺ ɺɺ (3.21)

Where Dɺ is the vector of velocity components, and C is the matrix of damping

coefficients that are determined experimentally. The ANSYS program uses the

Newmark time integration method to solve these equations at discrete time

points.

3.3.1. Newmark’s Method [35]

Newmark’s method is the most widely used implicit algorithm. It is first

assumed that

2 1( ) ( ) [( ) ]

2t t t t t t tD D t D t D Dβ β+∆ +∆= + ∆ + ∆ − +ɺ ɺɺ ɺɺ (3.22)

( )[(1 ) ]t t t t t tD D t D Dγ γ+∆ +∆= + ∆ − +ɺ ɺ ɺɺ ɺɺ (3.23)

where β and γ are constants. Equations (3.22) and (3.23) are then substituted

into the system equation (3.21),

2 1{ ( ) ( ) [( ) ]} { ( )[(1 ) ]}

2t t t t t t t t t t t t tK D t D t D D C D t D D MD Fβ β γ γ+∆ +∆ +∆ +∆+ ∆ + ∆ − + + + ∆ − + + =ɺ ɺɺ ɺɺ ɺ ɺɺ ɺɺ ɺɺ

(3.24)

31

It is grouped all the terms involving t tD +∆ɺɺ on the left and shift the remaining

terms to the right,

residualcm t t t tK D F+∆ +∆=ɺɺ (3.25)

where,

2[ ( ) ]cmK K t C t Mβ γ= ∆ + ∆ + (3.26)

2 1{ ( ) ( ) ( ) } { ( )(1 ) }

2residual

t t t t t t t t tF F K D t D t D C D t Dβ γ+∆ +∆= − + ∆ + ∆ − − + ∆ −ɺ ɺɺ ɺ ɺɺ (3.27)

The accelerations t tD +∆ɺɺ can then be obtained by solving matrix system equation,

1 residualt t cm t tD K F−+∆ +∆=ɺɺ (3.28)

Newmark’s method, like most implicit methods, is unconditionally stable if γ ≥

0.5 and 2(2 1) /16yβ ≥ + . Unconditionally stable methods are those in which the

size of the time step,t∆ , will not affect the stability of the solution, but rather it

is governed by accuracy considerations. The unconditional stability property

allows the implicit algorithms to use significantly larger time steps when the

external excitation is of a slow time variation.

32

CHAPTER 4

4. FIRING TESTS AND DATA ACQUISITION

The tests are performed for acceleration data acquisition at the mounting

location and stress histories of specific locations. In Figure 4.1 and Figure 4.2,

the systems are shown during the firing tests.

Figure 4.1 A View of Stabilized GAU19/A 12.7mm Gatling Gun System

Figure 4.2 A View of Stabilized KBA 25mm Cannon System

33

One-axial Integrated Circuit Piezoelectric (ICP) accelerometers of 5000g and

10000g amplitudes are used for measuring the acceleration level of the desired

locations on the parts. Operation principle of ICP accelerometers is explained in

APPENDIX B. Typical ICP accelerometers are shown in following figure.

Figure 4.3 ICP accelerometers [9]

The locations of the accelerometers to be placed on the systems are determined

according to possible mounting points. In Figure 4.4, the locations of the

accelerometers on the parts are shown. IO-Tech data acquisition system with 72

channels is used for collecting the acceleration histories.

Figure 4.4 Locations of the Test Parts and the Accelerometers on them

Figure 4.5 show the acceleration versus time signal collected at the

accelerometer on a test piece mounting location in the firing axe.

34

Figure 4.5 Acceleration “g” vs Time “sec” Signal

Stress histories of specific locations are required to perform strain analysis of

the structures. After the test pieces are located on the gun, firing tests can be

used directly for this purpose.

ESA Traveller Plus [38] data acquisition system with 32 channels is used for

collecting the strain histories. The locations of the parts in the systems are given

in Figure 4.4.

Initially, the test pieces were equipped with a rectangular rosette, CEA-13-

125UR-350 [37], 350 Ohm strain gage for conducting the experiment. They

were settled next to the slot of the parts, since that portion was under

consideration as a high strain region in this study (Figure 4.6).

35

Figure 4.6 A Test Piece Equipped with a Strain Rosettes

Before the stress tests, the input parameters, the calibration and balancing of the

strain rosettes are set (Figure 4.7) in ESAM software when the test pieces are at

rest, such that each strain rosette has zero reading (Figure 4.8).

ESAM (Electronic Signal Acquisition Module) is a measuring system used for

strain measurements. It consists of a high technology acquisition and

conditioning device and very sophisticated software to control processing data.

It can measure up to 300,000 samples per second.

Figure 4.7 Rosettes Inputs, Calibration and Balancing Screen

36

Figure 4.8 Rosette Calibration Screen

The tests are repeated for different test piece materials and thicknesses as seen

in Figure 4.9.

Figure 4.9 Different Material or Thicknesses of Test Pieces

During the tests, gage readings are recorded. A raw gage reading of the rosette

for the firing axis under shock loading is given in Figure 4.10.

37

Figure 4.10 A Raw strain data for a gage part of the analyzed rosette

During and after the tests, detailed visual inspections are done on the parts and

failures are observed. The following figures show some examples of the test

pieces after the tests.

Figure 4.11 Cast Aluminum Test Pieces Examples After The Tests

38

Figure 4.12 7075-T7351 Aluminum Test Pieces Examples After The Tests

Test visual inspection results for all tests are given Table 4.1.

Table 4.1 Visual Inspection Results of Rosette Analysis Tests

Material of Test Piece

Material Thickness

(mm)

# of Fired Rounds

Visual Inspection

Result

TEST 6

Cast Aluminum

1 10 Broken

TEST 10 2 100 Broken

TEST 12 4 40 Broken

TEST 13

7075-T7351 Aluminum

0,5 40 Deflected

TEST 9 1 100 No

Deformation

TEST 8 3 20 No

Deformation

39

CHAPTER 5

5. NUMERICAL AND EXPERIMENTAL SHOCK ANALYSIS

In this thesis, stress analysis of structures is performed by two basic methods.

These are Numerical Stress Analysis and Experimental Stress Analysis.

Numerical Analysis is used as an assisting study for Experimental Analysis by

means of pointing out the critical locations and it is used to demonstrate that the

strain energy density theory is suited to determine the effect of shock on

mechanical structures. On the other hand, Experimental Analysis is used to get

more accurate stress values than the values obtained in Numerical Analysis. At

the experimental analysis it is assumed that there is no plastic deformation on

the strain gages and on the surface upon which they were bonded. Hence,

experimental stress analysis is valid under the assumption that the material

remains elastic.

5.1. Shock Response Spectrum Analysis

A shock response spectrum (SRS) can be calculated from acceleration time

history data as explained in Chapter 3. Shock Response Spectrum analysis is

done by using nCode Glypworks computer program.

First of all, the acceleration time history is imported to the program. Then, the

data is converted to “*.s3t” format. The block diagram is prepared to obtain

SRS of the data and SRS of a classical shock form (Figure 5.1). In FEM, it is

not possible to apply the original acceleration time history as a load since the

solution of this problem takes too much time (each run takes approximately 1

40

month on Z400 HP workstation). Hence it is tried to obtain a classical shock

form which supplies approximately the same effects of the original data as a

load in FEM.

Figure 5.1 SRS Block Diagram

The SRS of the classical shock form should be as close as possible to the SRS

of the original data . To obtain this, different types, amplitudes and times of

classical forms are tried.

The general rules for the SRS graphs are;

• If the amplitude of the shock form is constant and the time of the shock

form increases, the graph shifts to the left and the peak value does not

change.

41

Figure 5.2 SRS Graph Property – Time increases

• If the amplitude of the shock form increases and the time of the shock

form is constant, the graph shifts up and the peak value increases.

Figure 5.3 SRS Graph Property – Amplitude increases

42

• If the amplitude and the time of the shock form are constant and the

shock form changes from half sine to sawtooth form, the graph shifts

down and the peak value decreases.

Figure 5.4 SRS Graph Property – Classical shock form changes

300 g 1ms half sine classical shock form is found appropriate for 12.7mm

Gatling gun firing tests on the test pieces. One of the original data and SRS

graph of it is shown in Figure 5.5 and Figure 5.6.

43

Figure 5.5 ACC-TDC on Gatling Gun – Test4 (2mm CA Part Test)

Figure 5.6 SRS Graph – Gatling Gun Test4

Test Data SRS

300g 1ms Halfsine SRS

44

5.2. Experimental Stress Analysis

Experimental analysis is essential since the results of it are nearly exact (in the

elastic range) and are used to be compared with the finite element model results.

After the gage readings as explained in Chapter 4, rosette analysis is performed.

Brief information on experimental stress analysis and rosette calculations is

given in APPENDIX C.

ESAM (Electronic Signal Acquisition Module) software is also used for the

experimental stress analysis.

ESAM software requires rosette type, cross sensitivities of the gages, poisson

ratio and modulus of elasticity values for the rosette analysis. The analysis input

sheet is seen in Figure 5.7.

Figure 5.7 Rosette Analysis Input Screen

ESAM software assumes that the strain gages are bonded to a linear-elastic,

homogeneous, isotropic body. This fact must be taken into account while

45

interpreting stresses based on gage readings. The analysis gives the maximal

principal stress, the minimal principal stress, the principal stress direction and

the absolute principal stress. Among the results of the analysis, the maximum

principal stress and minimum principal stress are the main parameters of interest

as an output. It will be used to compare with numerical results. In Figure 5.8

and Figure 5.9 the maximum principal stress and the minimum principal stress

of 1mm cast aluminum test piece graphs are seen. The graphs of all test results

for the maximum principal stress and the minimum principal stress are given in

Appendix E. There are offsets after firing at the stresses. It is believed that the

calibration of the strain gages is affected because of high vibration and shock

values. However, it is also believed that the measured values are not affected

too much. It is also assumed that there is no plastic deformation on the strain

gages. A small amount of plastic deformation, however, might have occurred on

the surface of the test piece where the gage has been bonded. The results can

also be seen in the following table.

Figure 5.8 Minimum Principal Stresses – SG Measurement (1mm CA)

46

Figure 5.9 Maximum Principal Stresses – SG Measurement (1mm CA)

Table 5.1 Maximum and Minimum Principal Stresses Results of Rosette

Analysis Tests

TEST 6 1mm CA

TEST 10 2mm CA

TEST 12 4mm CA

TEST 13 0.5mm 7075 Al

TEST 9 1mm

7075 Al

TEST 8 3mm

7075 Al Time (second) 0,91 7,6 3,3 1,2 7,3 4,68

MAPS (Mpa) 700 392 700 310 330 285 MIPS (Mpa) -30 175 -155 -15 -200 -50 Equivalent Stresses (Mpa)

715 340 789 318 464 313

Time (second) 0,72 3,6 2,9 0,98 11,35 5,6 MAPS (Mpa) 240 265 290 20 135 10 MIPS (Mpa) -196 -20 -200 -245 -445 -560 Equivalent Stresses (Mpa)

378 276 427 256 526 565

The rainflow cycle counting of the absolute principal stress data is performed.

As it is expected for shock loading cases, arrow head type of profile is obtained.

In addition, the mean and alternating stress distribution on the component can

be seen by inspecting the rainflow plot. In Figure 5.10 the rainflow counting

plot of 3mm 7075-T7351 aluminum test piece is given.

47

Figure 5.10 Counting Result of Firing Test (3mm 7075 AL)

An equivalent alternating stress amplitude can be computed by using the below

equation (5.1) with the individual amplitudes obtained by rainflow cycle

counting.

(5.1)

where,

A: Equivalent amplitude

Ai: Individual amplitudes

Ci: Number of cycles of individual stresses

m: Averaging exponent

ESAM software output screen showing the mean and alternating equivalent

stresses for 1mm 7075-T7351 aluminum test piece is shown in Figure 5.11.

48

Figure 5.11 Mean Stress and AES (1mm 7075 AL)

5.3. Numerical (FEM) Analysis of Test Pieces

In this study, elasto-plastic FEM analysis due to shock loading is performed. 3D

model of the test pieces is constructed in an advanced computer aided design

program, I-DEAS. This geometry is automatically imported to the finite element

program ANSYS Workbench [39]. Shell model of it is prepared in ANSYS.

First of all, a simple square bar (Appendix F) is analyzed in ANSYS to show

that the obtained strain energy density results are approximately the same as the

areas under the stress-strain curve.

After that the test piece model is analyzed in ANSYS software with SHELL91

element which is a higher order element and contains 20 nodes. The mesh size

is found as 2mm by performing many runs of FEM until there is no change at

the analysis results below this mesh size. Mesh model of the test piece is given

in Figure 5.12.

49

Figure 5.12 Mesh Model of the Test Piece

The test piece model contains 2430 SHELL elements and 2620 nodes.

The test pieces are made of 7075-T7351 aluminum or cast aluminum. Material

properties of 7075-T7351 aluminum are found by literature survey and entered

into the material library of ANSYS Workbench as seen in Table 5.2 to Table 5.4

and Figure 5.13. On the other hand, material properties of cast aluminum are

found by tensile test. The information of the test is given in Appendix D. From

Appendix D it is observed that CA behaves in a linear elastic manner until

fracture with UTS=104 MPa and rupture strain=0.0015.

Table 5.2 Material Properties of 7075-T7351 Aluminum

7075-T7351

Elastic Region Engineering Strain (εeng_e) 0,006

Elastic Region Engineering Stress (σeng_e) (MPa) 435

Plastic Region Engineering Strain (εeng_p) 0,13

Plastic Region Engineering Stress (σeng_p) (MPa) 505

Density (kg/mm3) 2,81 x 10-6

In ANSYS, all stress-strain input should be in terms of true stress and true (or

logarithmic) strain and the results in all outputs are given also as true stress and

50

true strain. For small-strain regions of response, true stress (logarithmic strain

and engineering stress) and engineering strain data are essentially identical. If

the stress-strain data is in the form of engineering stress and engineering strain

one can convert strain from small (engineering) strain to logarithmic strain by

using equation 5.2,

εtrue = 1n (1 + εeng) (5.2)

and engineering stress to true stress by using equation 5.3,

σtrue = σeng(1 + εeng) (5.3)

This stress conversion is only valid for incompressible plasticity stress-strain

data. In addition to this conversion, bilinear isotropic hardening plasticity

material model is often used in large strain analyses in ANSYS. A Bilinear

Stress-Strain (BISO) curve requires that you input the Yield Strength and

Tangent Modulus. The slope of the first segment in a BISO curve is equivalent

to the Young's modulus of the material while the slope of the second segment is

the tangent modulus.

For elastic region of 7075-T7351 aluminum,

εtrue_e = 1n (1 + εeng_e) = 1n (1 + 0,006) = 0.005982 (5.4)

σtrue_e = σeng(1 + εeng_e) = 435(1 + 0,006) = 437,6 MPa (5.5)

For plastic region of 7075-T7351 aluminum,

εtrue_p = 1n (1 + εeng_p) = 1n (1 + 0,13) = 0.1222 (5.6)

σtrue_p = σeng_p(1 + εeng_p) = 505(1 + 0,13) = 570,65 MPa (5.7)

51

E = σtrue_e / εtrue_e = 73,2 GPa (5.8)

P = (σtrue_p - σtrue_e) / (εtrue_p - εtrue_e) = 1144,8 MPa (5.9)

Table 5.3 Structural Material Properties for 7075-T7351 Al in ANSYS

Young's Modulus 73200 MPa Poisson's Ratio 0,33

Tensile Yield Strength 437,6, MPa

Tensile Ultimate Strength 570,65, MPa

Table 5.4 Bilinear Isotropic Hardening Properties for 7075-T7351 Al in

ANSYS

Yield Strength (MPa) 437,6 Tangent Modulus (MPa) 1144,8

Figure 5.13 Bilinear Isotropic Hardening Graph for 7075-T7351 Al in ANSYS

52

The thicknesses are defined and changed by shell element settings for the test

pieces which have different thickness value.

Figure 5.14 Thickness Definition of the Test Piece

3 kg mass element is connected to shell elements by rigid regions at the fastener

connection holes of the part and boundary hole have fixed boundary condition

in all directions. These are applied on the finite element model as shown Figure

5.155

Figure 5.15 Shell Model of the Test Piece

Connection holes of the mass element

Mass element

Boundary hole

53

The loads obtained by the shock response spectrum analysis as 300g 1ms is

applied to the whole body (mass element + test piece). At the beginning of the

analysis, the substep model analysis is used for the solution but solution can’t be

obtained. After that, the time intervals model is used. The time interval value to

get a solution is found as 1e-5 second by performing many runs for different

time interval values.

The results obtained by applying above conditions in ANSYS transient response

analysis for 1mm cast aluminum test piece are shown in following figures. The

graphs of all test results for ANSYS analysis are given in Appendix F.

Figure 5.16 Maximum Strain Energy for 300g 1ms ACC (1mm CA)

By using the Figure 5.166, the maximum SED is calculated by equation 5.10.

Max. SED = Max. Strain Energy / Volume (5.10)

The result for 1mm thick cast aluminum test piece, Max. SED = 75,67 mJ / (2mm*2mm*1mm) = 18,92 mJ/3mm

54

The allowable strain energy density for the cast aluminum is calculated by

equation 5.11.

SED= (1/2) * σ * ε (5.11)

The limiting value for the cast aluminum is,

E = (1/2) * 104MPa * 0,0015mm/mm = 0,078 mJ/3mm

Figure 5.17 Stress-Strain diagram for 7075-T7351 Aluminum [40]

The allowable strain energy density of 7075-T7351 aluminum is taken from

Figure 5.177 as 2 mJ/mm3 for no permanent deformation and the allowable

strain energy density of 7075-T7351 aluminum is taken from Figure 5.177 as

43.6 mJ/mm3 for no breaking. Here it is proposed that the Max. SED result has

to be below the allowable strain energy density value in order not to have

fracture on the part.

55

Figure 5.18 SMP Velocity Graph for 300g 1ms ACC (1mm Cast Aluminum)

A velocity probe is located on the part in ANSYS as shown Figure 5.155. This

probe is used to determine the total solution time for the analysis is enough or

not. The velocity value has to converge “0” for sufficient solution time.

Figure 5.19 SMP Maximum Principal Stresses for 300g 1ms ACC (1mm Cast

Aluminum)

56

Figure 5.20 SMP Minimum Principal Stresses for 300g 1ms ACC (1mm Cast

Aluminum)

Figure 5.21 SMP Equivalent Stresses Graph for 300g 1ms ACC (1mm Cast

Aluminum)

The maximum and minimum principal stress values are calculated at the critical

point and used to be compared with the experimental results. If the safety factor

The Critical Point

57

of the piece with respect to the ultimate tensile stress goes below “1”, the

maximum and minimum stress values are taken for the time at which the safety

factor is “1”. The equivalent stress values are found to determine safety factor

with respect to the ultimate tensile stress.

All results of ANSYS transient response analysis on the test pieces are given in

Table 5.5 and Table 5.6.

Table 5.5 Results of ANSYS on the Cast Aluminum Test Pieces

TEST 6

1mm CA TEST 10 2mm CA

3mm CA

TEST 12 4mm CA

5mm CA

Maximum Principal Stresses (MPa)

341,7 361,3 226,4 207,8 191,5

Minimum Principal Stresses (MPa)

-65,7 -70,6 -44,5 -40,8 -37,5

Equivalent Stresses (Mpa)

1999,1 1716,5 1452,8 1132,7 977,5

Max. SED (mJ/mm3) 18,92 6,18 3,68 2,61 1,99

Allowable SED (mJ/mm3) for no permanent deformation

0,078

Allowable SED (mJ/mm3) for no crack

0,078

58

Table 5.6 Results of ANSYS on the 7075 Aluminum Test Pieces

TEST 13 0.5mm 7075 Al

TEST 9 1mm

7075 Al

2mm 7075 Al

TEST 8 3mm 7075

Al

4mm 7075 Al

Maximum Principal Stresses (MPa)

68 44,5 520,1 505,7 485,7

Minimum Principal Stresses (MPa)

-424,1 -532,3 -63,9 -62,8 -57,8

Equivalent Stresses (Mpa)

565,7 553,8 534,8 522,3 519,2

Max. SED (mJ/mm3) 32,66 22,64 16,12 10,66 7,10

Allowable SED (mJ/mm3) for no permanent deformation

2

Allowable SED (mJ/mm3) for no crack

43,6

5.4. Evaluations of the Results

Results of both numerical and experimental analysis show the same location as

the most critical region on the test pieces from stress point of view. The location

according to numerical analysis is shown in Figure 5.201, and the crack location

obtained in experimental analysis is shown in Figure 4.11.

It is appropriate to state basic assumptions and the shortcomings of two methods

at this point.

- Stresses due to bolt tightening and at the boundaries are not included in

numerical analysis,

- Shock tests are carried out only one axis, not three axes simultaneously.

59

Although the stress values obtained by numerical analysis are still questionable,

critical location was determined correctly. From Figure 5.222 to Figure 5.2929

all analysis results are shown graphically.

Figure 5.22 Effect of Material Thickness on Maximum SED (Cast Aluminum)

Figure 5.23 Effect of Material Thickness on Maximum SED (7075 Al)

It is seen from the graphs that the maximum strain energy density of the part is

decreasing with the increasing thickness of the material for both brittle and

ductile materials.

60

In the following figures safety factors with respect to UTS and SE are given.

These safety factors are defined as;

Safety Factor (with respect to UTS) = UTS / Equivalent Stress

and,

Safety Factor (with respect to SE) = Allowable SED / Max. SED

Allowable SED = 43,6 mJ/mm3 (for 7075 Aluminum - no crack)

Allowable SED = 0,078 mJ/mm3 (for Cast Aluminum - no crack)

Figure 5.24 Effect of Material Thickness on SF of UTS (Cast Aluminum)

61

Figure 5.25 Effect of Material Thickness on SF of UTS (7075 Al)

As expected the graphs for safety factor with respect to UTS show that the

brittle materials (cast aluminum pieces) have less safety factor with respect to

the ultimate tensile stress than the ductile materials (7075-T7351 aluminum

pieces). Gage readings are taken up to the instant when fracture occurred. Also,

the safety factors of cast aluminum pieces for both of the experimental and the

numerical analysis are under “1”, and this means that the pieces were broken as

what happened. On the other hand, most of the safety factors of 7075 aluminum

pieces for the experimental analysis are above “1” and the safety factors of cast

aluminum pieces for numerical analysis are under “1”. After the visual

inspections, it is known that there is no visually observable deformation on 7075

aluminum pieces under gun shock loading. This means that the numerical

results are more conservative than the experimental results.

In addition, it is seen from the graphs that the safety factor with respect to the

ultimate tensile strength is increasing with the increasing thickness of the

material.

62

Figure 5.26 Effect of Material Thickness on SF of SE (Cast Aluminum)

Figure 5.27 Effect of Material Thickness on SF of SE (7075 Al)

Figures 5.26 and 5.27 show that the brittle materials (cast aluminum pieces)

have less safety factor with respect to the strain energy than the ductile materials

(7075-T7351 aluminum pieces) as expected. Also, the safety factors of cast

aluminum pieces for both of the experimental and the numerical analysis are

under “1”, and this means that the pieces were broken as what happened at the

tests. On the other hand, the safety factors of 7075 aluminum pieces for both

63

experimental and numerical analysis are above “1” and it explains the tests'

visual inspections that there is no crack on this pieces. Allowable strain energy

density value is taken as 43,6 mJ/mm3 for no crack. Based on these preliminary

results, it is proposed that the safety factors with respect to strain energy give

better result than the safety factors with respect to ultimate tensile strength in

predicting fracture based on numerically computed stress and SED. Because,

safety factors based on SE are always greater than 1 whereas those based on

UTS are less than one for some cases. There never was fracture in actual tests

for 7075, therefore predictions of SED are more accurate. Here, of course it is

recognized that evidence is not conclusive and many more controlled tests and

calculations may be required. Such an effort should be considered as a "future

work".

Besides this, the safety factor with respect to the strain energy density of the

part is increasing with the increasing thickness of the material for both brittle

and ductile materials.

Figure 5.28 Effect of Material Volume on Maximum SED (Cast Aluminum)

64

Figure 5.29 Effect of Material Volume on Maximum SED (7075 Al)

The maximum strain energy density of the part is decreasing with the increasing

volume of the material at the critical location for both brittle and ductile

materials.

5.5. Application of the Theory to a System in Use

After the applicability of strain energy theory for shock loading is demonstrated

on the test pieces, it is applied to the real system. A 25mm cannon mount is

tested with respect to the shock loading of cannon by using the proposed theory.

3D model of the mount is constructed in computer aided design program, I-

DEAS. This geometry is automatically imported to the finite element program

ANSYS Workbench. The model is analyzed in ANSYS software with SOLID

186 element which is a higher order element containing 20 nodes. The mesh

size is found as 6mm by performing many runs of FEM up to a point where

there is no change at the analysis results below this size. Mesh model of the

mount is shown Figure 5.300.

65

Figure 5.30 Mesh Model of the Mount

25mm cannon mount model has 9236 SOLID elements and 36359 nodes.

25mm cannon mount is made of impax steel. The material properties of impax

steel are found by literature survey and entered the material library of ANSYS

Workbench seen in Table 5.7 to Table 5.9 and Figure 5.311.

Table 5.7 Material Properties of Impax Steel [41]

Impax Steel

Elastic Region Engineering Strain (εeng_e) 0,00375

Elastic Region Engineering Stress (σeng_e) (MPa) 800

Plastic Region Engineering Strain (εeng_p) 0,175

Plastic Region Engineering Stress (σeng_p) (MPa) 930

Density (kg/mm3) 7,8 x 10-6

for elastic region of impax steel,

εtrue_e = 1n (1 + 0,00375) = 0.003743

σtrue_e = 800(1 + 0,00375) = 803 MPa

for plastic region of impax steel,

66

εtrue_p = 1n (1 + 0,175) = 0.1613

σtrue_p = 930(1 + 0,175) = 1092,75 MPa

E = 214,5 GPa

P = 1839 MPa

Table 5.8 Material Properties for Impax Steel in ANSYS

Young's Modulus 214500 MPa Poisson's Ratio 0,33

Tensile Yield Strength 803 MPa

Tensile Ultimate Strength 1092,75 MPa

Table 5.9 Bilinear Isotropic Hardening Properties for Impax Steel in ANSYS

Yield Strength (MPa) 803 Tangent Modulus (MPa) 1839

Figure 5.31 Bilinear Isotropic Hardening Graph for Impax Steel in ANSYS

67

The cannon is connected as a mass element to the mount by rigid regions at the

pin holes of the mount and boundary hole have fixed boundary condition in all

directions. These are applied on the finite element model as shown Figure

5.322.

Figure 5.32 ANSYS Model of the Mount

100 g 2ms half sine classical shock form is found appropriate for 25mm cannon

firing tests on the mount. One of the original data and SRS graph of it is shown

in Figure 5.333 and Figure 5.344.

Connection holes of the mass element

Mass element Boundary holes

68

Figure 5.33 ACC-TDC on the Mount of 25mm Cannon – Test6

Figure 5.34 SRS Graph – 25mm Cannon Test6

Test Data SRS

100g 2ms Halfsine SRS

69

The curve fit is not very satisfactory since it is tried to close the test data up to

1000 Hz.

The loads obtained by the shock response spectrum analysis as 100g 2ms is

applied to the whole body (mass element + 25mm cannon mount). The substep

intervals model is used. The value of the maximum step number to get a

solution is found as 25 by performing many runs for different maximum step

number values. The results obtained by applying above conditions in ANSYS

transient response analysis for the mount are shown in following figures.

Figure 5.35 Maximum Strain Energy for 100g 2ms ACC (Impax Steel)

By using the Figure 5.355 and the total strain energy theory, the maximum SED

is calculated,

Max. SED = 70,37 mJ / (6mm*6mm*6mm) = 0,33 mJ/3mm

70

Figure 5.36 Stress-Strain graph for Impax Steel

The allowable strain energy density of impax steel is taken from Figure 5.36 as

1,5 mJ/mm3 for no permanent deformation and the allowable strain energy

density of impax steel is taken from Figure 5.36 as 167,4 mJ/mm3 for no

breakage. The Max. SED result has to be below the allowable strain energy

density value for no deformation on the part. The result shows that the mount is

at safe side.

Figure 5.37 SMP Velocity Graph for 100g 2ms ACC (Impax Steel)

A velocity probe is located on the part in ANSYS as shown Figure 5.322. This

probe is used to determine the total solution time for the analysis is enough or

not. The velocity value has to converge “0” for sufficient solution time.

71

Figure 5.38 SMP Maximum Principal Stresses for 100g 2ms ACC (Impax Steel)

Figure 5.39 SMP Minimum Principal Stresses for 100g 2ms ACC (Impax Steel)

All results of ANSYS transient response analysis on the mount are given in

Table 5.10.

Table 5.10 Results of ANSYS Transient Response on the 25mm Cannon Mount

25mm Cannon Mount ANSYS

Transient Analysis Results Maximum Principal Stresses (MPa) 437 Minimum Principal Stresses (MPa) -473 Max. SED (mJ/mm^3) 0.33 Allowable SED (mJ/mm^3) for no permanent deformation

1,5

Allowable SED (mJ/mm^3) for no crack

167,4

72

CHAPTER 6

6. DISCUSSION AND CONCLUSIONS

The shock loading problem is a complex one due to factors of rapid

phenomenon that excites dynamic (resonant) response of the material but causes

very little overall deflection, and a multiaxial stress state. In a comparatively

short time, a moderately high level force impulse is input to the material. It is

difficult to estimate the response of complicated systems analytically, due to

structure’s dynamic characteristics and varying loadings.

Therefore, stress analysis of such structures is performed by two basic methods.

These are numerical stress analysis and experimental stress analysis. Numerical

Analysis is used as an assisting study for Experimental Analysis by means of

pointing out the critical locations and here it is also used to show that the strain

energy density theory is suited to determine the effect of shock on mechanical

structures. On the other hand, Experimental Analysis is used to get realistic

(more reliable) stress values and correct the numerical model.

In this study, the shock induced stress and transient analysis of the test pieces

and the 25mm cannon mount are performed. During the tests, test pieces and the

mount of 25mm cannon are integrated on the gun and the cannon respectively.

Also, the Gatling gun and the cannon are located on military Stabilized Machine

Gun Platform and Stabilized Cannon Platform respectively.

One-axial ICP accelerometers are attached to obtain the loading history for

corresponding points. IO-Tech data acquisition system is used for collecting the

acceleration histories. On the other hand, ESA Traveller Plus data acquisition

73

system is used for collecting the strain histories. Initially, the test pieces were

equipped with a rectangular rosette to obtain stress histories of specific locations

which is required to perform strain analysis of the structures. They were settled

next to the slot of the parts, since that portion was under consideration as a

strain region in this study. In addition, detailed visual inspections are done on

the parts and failures are observed during and after the test.

Furthermore, experimental stress analysis due to shock loading is performed for

two different types of material of the test pieces and different thicknesses of the

test pieces. The input data for the analysis is obtained through measurements

from strain rosette precisely located at the critical location of the test pieces.

ESAM is used for performing the experimental stress analysis and rosette

calculations. Maximum principal stress, minimum principal stress, principal

stress direction, absolute principal stress and Von Misses stress are the results of

the analysis. Among them, maximum principal stress and minimum principal

stress are the main parameters of interest as an output and they are used for

comparison with numerical results.

Shock Response Spectrum (SRS) analysis is done by using nCode Glypworks to

define the equivalent shock profiles created on test pieces and the mount of

25mm cannon by means of the gun and the cannon firing. In FEM, it is not

possible to apply the original acceleration time history as a load since the

solution of this problem takes too much time. Hence we try to obtain a classical

shock form which supplies approximately the same effects of the original data

as a load in FEM. Different types, amplitudes and times values of classical

forms are tried to get the best fitted curve. Classical shock forms are 300 g 1ms

half sine for 12.7mm Gatling gun firing tests on the test pieces and 100 g 2ms

half sine 25mm cannon firing tests on the mount.

Transient shock analysis of the test pieces are done by applying the obtained

shock profiles on the part in a finite element model (ANSYS) to find the

74

numerical stress values. FEM analysis is performed in terms of elasto-plastic

behavior due to shock loading. The material properties of 7075-T7351

aluminum are found by literature survey. On the other hand, material properties

of cast aluminum are found by tensile test. In ANSYS, all stress-strain inputs

are converted true stress and true (or logarithmic) strain and results in all

outputs are given also as true stress and true strain. In addition, bilinear isotropic

hardening plasticity material model is used in large strain analyses in ANSYS.

300g 1ms obtained by the shock response spectrum analysis is applied to the

whole body.

Numerical and experimental stress values are compared to verify the finite

element models (FEM). Both of the numerical and the experimental analysis

show the same location as the most critical region on the test pieces from stress

point of view.

Finally, the stress and strain energy density values are used to define safety

factors. An attempt is made to discover which safety factors give more accurate

results concerning failure under the effect of shock loading on the parts.

The followings are also concluded from the analysis;

- The maximum strain energy density of the part is decreasing with the

increasing thickness of the material for both brittle (cast aluminum

pieces) and ductile materials (7075-T7351 aluminum pieces).

- The numerical results are more conservative than the experimental

results.

75

- The safety factors with respect to strain energy give better result than the

safety factors with respect to ultimate tensile strength in predicting

failure by fracture.

- The safety factor with respect to the strain energy density of the part is

increasing with the increasing thickness of the material for both brittle

and ductile materials.

- The maximum strain energy density of the part is decreasing with the

increasing volume of the material for both brittle and ductile materials. It

is showed that the materials resist to shock loading with their volumes.

After demonstrating the use of strain energy density theory on the test pieces for

determining the shock failure at military structures, it is applied to the real

system. 25mm cannon mount is tested with respect to the shock loading of

cannon by using the theory. 100g 2ms obtained by the shock response spectrum

analysis is applied to the whole body and as a result the mount is found on the

safe side as expected.

As a future work, force response spectrum analysis (FRS) can be applied instead

of SRS while performing the numerical analysis of the gun firing shock in the

finite element model. In the NATO paper [24], it is proposed that FRS analysis

is better way in order supply equivalent force input.

76

REFERENCES

[1] Harris, C.M., Crede, C.E., Shock and Vibration Handbook, 1976.

[2] “A Commercial Guide to Shock And Vibration Isolation”, C.M.T. Wells

Kelo Ltd., May 1983.

[3] Harris, C.M., Piersol, A.G., Hill, M., Shock and Vibration Handbook,

Fifth Edition, 2002.

[4] Pusey, H.C., “Fifty Years of Shock and Vibration History”, The Shock

and Vibration Information Analysis Center (SAVIAC), 1996.

[5] Walter, P.L., “Selecting Accelerometers for Mechanical Shock

Measurements”, PCB Piezotronics, Depew, New York, TN-24, December

2007.

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80

APPENDIX A

A. EQUIPMENT USED THROUGHOUT TESTS

1) Iotech data acquisition system

a. Maximum Aggregate Speed : 500 kHz

b. 72 input channels.

Figure A.1 IOtech data acquisition system

2) Traveler Strain Master data acquisition system

a. Maximum Aggregate Speed : 300 kHz

b. 32 input channels.

81

Figure A.2 Traveler Strain Master data acquisition system

3) 1-Axial 10000g ICP Type Piezoelectric Accelerometers

a. Measurement Range : ± 98000 m/s^2 pk

b. Frequency Range : 0.5-7500 Hz

c. Resonant Frequency : >= 50 kHz

d. Sensing Element : Quartz

e. Sensing Geometry : Shear

f. Nonlinearity <= 1 %

Figure A.3 1-axial piezoelectric accelerometer [5]

4) 1-Axial 5000g ICP Type Piezoelectric Accelerometers

a. Measurement Range : ± 49000 m/s^2 pk

b. Frequency Range : 0.5-7500 Hz

c. Resonant Frequency : >= 50 kHz

d. Sensing Element : Quartz

e. Sensing Geometry : Shear

f. Nonlinearity <= 1 %

5) Strain Gages

a. Grid Resistance : 350 ± 0.4% Ohms

82

b. Temperature Coefficient of Gage Factor : +1.3 ± 0.2 %/100°C

c. For Grid 1:

i. Gage Factor at 24°C : 2.13 ± 0.5%

ii. Transverse Sensitivity : (+0.7 ± 0.2)%

d. For Grid 2:

i. Gage Factor at 24°C : 2.15 ± 0.5%

ii. Transverse Sensitivity : (+0.3 ± 0.2)%

e. For Grid 3:

i. Gage Factor at 24°C : 2.13 ± 0.5%

ii. Transverse Sensitivity : +0.7 ± 0.2%

Figure A.4 Strain Gage (CEA-13-125UR-350)

83

APPENDIX B

B. STRAIN ROSETTE ANALYSIS [8]

Rosettes are used for reducing strain data obtained from measurements made

with three-element or two-element strain gage rosettes.

There are three basic geometry types of strain gage rosettes:

45°- Rectangular : three grids, with the second and third grids angularly

displaced from the first grid by 45° and 90°, respectively,

60°- Delta : three grids, with the second and third grids 60° and

120°, respectively, from the first grid.

T-Rosettes : two mutually perpendicular grids.

Since three element 45°- Rectangular rosettes are used in this study, brief

formulae definitions of the kind will be covered in this section.

e e e1 2 3 : Measured strains for grids 1, 2 and 3 respectively,

ε ε ε1 2 3 : Calculated strains for grids 1, 2 and 3 respectively,

ε εmin max : Calculated maximal and minimal principal strains,

σ σmin max : Calculated maximal and minimal stresses,

K K Kt t t1 2 3, , : Transverse sensitivities of gages 1, 2 and 3,

E : Modulus of elasticity of the material,

ν : Poisson's ratio of the material,

θ : Calculated direction of principal strains.

84

B.1 Rectangular Rosette:

Correction for transverse sensitivity errors,

( ) ( )ε

ν ν1

1 31 1

11 1 3

1 3

=− − −e K K e K

K K

t t t

t t- (B.1)

( ) ( )( ) ( )( )[ ]( )( )ε

ν ν2

2

2

1 31

1

1 1 1 1

1 1

2 2 1 3 3 1

1 3 2

=−

−−

− − + − −

− −

e K

Kt

K e K K e K K

K K K

t t t t t t

t t t

(B.2)

( ) ( )εν ν

3

3 11 1

13 3 1

1 3

=− − −

−

e K K e K

K K

t t t

t t

(B.3)

Maximal principal strain,

( ) ( )εmax =+

+ − + −e e

e e e e1 21 2

2

2 3

2

2

1

2 (B.4)

Minimal principal strain,

( ) ( )εmin =+

− − + −e e

e e e e1 3

1 2

2

2 3

2

2

1

2 (B.5)

Direction (angle) from Grid 1 to the principal axis,

−−−= −

31

3121 2tan

2

1

ee

eeeθ (B.6)

85

B.2 Principal Stresses:

If the material is homogenous and isotropic, Hooke’s Law can be used with the

above equations to calculate the principal stresses, by defining elastic modulus

(E) and the Poisson’s ratio (ν ).

Maximal principal stress,

( ) 106

minmax2max 1

E −×+−

= ενεν

σ (B.7)

Minimal principal stress,

( ) 106

maxmin2min 1

E −×+−

= ενεν

σ (B.8)

In these formulas, it is assumed that maxε and minε are expressed in µm/m.

86

APPENDIX C

C. TENSILE TEST OF THE CAST ALUMINUM

Tensile tests were performed to define the cast aluminum engineering strength

and engineering strain values.

Figure C.1 Tensile Test for the Cast Aluminum

Table C.1 Tensile Tests Inputs and Outputs for Cast Aluminum

Test1 Test2

Specimen thickness (t), mm 2,2 2,1

Specimen width (w), mm 11,45 11,2

Specimen length (L), mm 39,5 39,5

Area A = t x w, mm2 25,2 23,5

Force of rupture (F), N 2470 2585

Elongation (∆L), mm 0,075 0,045

Strength, σ= F / A, N/mm2 98 110

Strain, ε = ∆L / L 0,0019 0,0011

87

It can be taken average of values as the engineering strength and engineering

strain of the cast aluminum material.

σ = (98 + 110) / 2 = 104 MPa

ε = (0,0019 + 0,0011) / 2 = 0,0015

Figure C.2 Stress-Strain Diagram for the Cast Aluminum

88

APPENDIX D

D. EXPERIMENTAL RESULTS OBTAINED BY USING ESAM

Figure D.1 Minimum Principal Stresses Graph– SG Measurement (Test 6)

Figure D.2 Maximum Principal Stresses Graph – SG Measurement (Test 6)

beginning of shooting

end of shooting

end of shooting


89

Figure D.3 Minimum Principal Stresses Graph – SG Measurement (Test 10)



end of shooting


end of shooting

90




end of shooting


end of shooting

91




end of shooting

beginning of shooting end of shooting

92




end of shooting


end of shooting

93




end of shooting


end of shooting

94

APPENDIX E

E. NUMERICAL (ANSYS TRANSIENT) ANALYSIS RESULTS OBTAINED BY USING

ANSYS


Max. SED = 18,92 mJ/ 3mm


95

Figure E.3 SMP Maximum Principal Stresses Graph for 300g 1ms ACC (1mm

Cast Aluminum)

Figure E.4 SMP Minimum Principal Stresses Graph for 300g 1ms ACC (1mm

Cast Aluminum)

96


Aluminum)



97



Aluminum)

98


Aluminum)


Aluminum)

99


Aluminum)



100


Aluminum)


Aluminum)

101


Aluminum)


Aluminum)


102



Aluminum)

103


Aluminum)


Aluminum)

104


Aluminum)

Max. SED = 1.99 mJ/ 3mm


105


Aluminum)


Aluminum)

106


Aluminum)

Figure E.26 Maximum Strain Energy for 300g 1ms ACC (0,5mm 7075 Al)

107


Figure E.27 SMP Velocity Graph for 300g 1ms ACC (0,5mm 7075 Al)

Figure E.28 SMP Maximum Principal Stresses for 300g 1ms ACC (0,5mm

7075 Al)

108

Figure E.29 SMP Minimum Principal Stresses for 300g 1ms ACC (0,5mm 7075

Al )

Figure E.30 SMP Equivalent Stresses for 300g 1ms ACC (0,5mm 7075 Al)

109

Figure E.31 Maximum Strain Energy for 300g 1ms ACC (1mm 7075 Al)


Figure E.32 SMP Velocity Graph for 300g 1ms ACC (1mm 7075 Al)

110


Al )


Al )

111

Figure E.35 SMP Equivalent Stresses for 300g 1ms ACC (1mm 7075 Al)



112



Al )

113


Al )


114




115


Al )


Al)

116



117




Al )

118


Al)


119

APPENDIX F

F. SAMPLE ANALYSIS OF STRAIN ENERGY

DENSITY IN ANSYS

In ANSYS, a simple square bar (Figure F.1) is analyzed to show that the

obtained strain energy density results are approximately the same as the areas

under the stress-strain curve.

Figure F.1 A Simple Square Bar

120

The bar has fixed boundary condition in all directions at the boundary side and a

uniform stress over the cross-sectional area whose resultant is F (Figure F.2) is

applied on the other side of the bar.

Figure F.2 The Force Applied on the Bar

The material of the bar is selected as 7075-T7351 and the properties of it are

entered to the software (Figure F.3).

Figure F.3 The Areas under the Bilinear Isotropic Hardening Graph for 7075-

T7351 Al in ANSYS

121

The times of the equivalent stresses for the yield point and a point at the plastic

region are found from the analysis result (Figure F.4 and Figure F.6).

Figure F.4 The Time of the Yield Point

At this time the maximum equivalent stress value is equal to the yield stress.

The time is 7,834 ms. At the same time the maximum strain energy value is

obtained from the analysis result (Figure F.5).

Figure F.5 The Maximum Strain Energy Value for the Yield Point

122

The maximum strain energy value for the yield point is 137,96 mJ. The mesh

size of the bar is 5 mm. If the maximum strain energy value is divided by the

mesh volume, the maximum strain energy density value is found as 1,1

mJ/mm3.

Figure F.6 The Time of a Plastic Region Point

At the time the maximum equivalent stress value is equal to a stress at the

plastic region. The time is 9,387 ms. At the same time the maximum strain

energy value is obtained from the analysis result (Figure F.7). Near the support,

because of the additional constraint it appears that strain is varying along the

length of the specimen.

Figure F.7 The Maximum Strain Energy Value for the Plastic Region Point

123

The maximum strain energy value for the plastic region point is 1339,8 mJ. The

mesh size of the bar is 5 mm. If the maximum strain energy value is divided by

the mesh volume, the maximum strain energy density value is found as 10,7

mJ/mm3.

The strain energy density results obtained by using FEM simulation are

acceptably close to the areas under the stress-strain curve. All results are given

in Table F.1.

Table F.1 Strain Energy Density Results of ANSYS for the Sample Bar

Yield Point

The Plastic Region Point

Error 100*(B-A)/B

Analysis Result for The Strain Energy Density (mJ/mm3) (A)

1,1 10,7 16%

The Area under The Stress-Strain Curve (mJ/mm3)(B)

1,31 12,31 13%

Figure F.8 Force vs Load Point Displacement Graph

The force for the yield point which is at 7,834 ms is found 236 kN by using

Figure F.2. The work done up to yield point is calculated as 185,1 J by using the

area under the force vs displacement curve (Figure F.8). On the other hand the

force for the plastic region point which is at 9,387 ms is found 282 kN and the

work done up to the point is calculated as 265 J.

The Plastic Region Point

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