Analysis of Engineering Structuresand Material Behavior

Analysis of Engineering Structuresand Material Behavior

Josip Brnić

University of Rijeka – Faculty of Engineering, Department of EngineeringMechanics, Rijeka, Croatia

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Library of Congress Cataloging-in-Publication Data

Names: Brnić, Josip, 1951– author.Title: Analysis of engineering structures and material behavior / by JosipBrnić University of Rijeka.

Description: First edition. | Hoboken, NJ : Wiley, 2018. | Includesbibliographical references and index. |

Identifiers: LCCN 2017042809 (print) | LCCN 2017048581 (ebook) |ISBN 9781119329107 (pdf) | ISBN 9781119329060 (epub) |ISBN 9781119329077 (cloth)

Subjects: LCSH: Structural analysis (Engineering)Classification: LCC TA645 (ebook) | LCC TA645 .B659 2018 (print) |DDC 624.1/7–dc23

LC record available at https://lccn.loc.gov/2017042809

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“Theory and the real process are referred to each other; theory teachesunderstanding of the process and the process testifies to the real event.”

Let this book be dedicated to the memory of my parents, who instilled in merespect towards people.

Contents

Frequently Used Symbols and the Meaning of Symbols xvPrincipal SI Units and the US Equivalents xxiiiSI Prefixes, Basic Units, Physical Constants, the Greek Alphabet xxvImportant Notice Before Reading the Book xxviiPreface xxixAbout the Author xxxiAcknowledgements xxxiii

1 Introduction 11.1 The Task of Design and Manufacture 11.2 Factors that Influence the Design of Engineering Structures 11.3 The Importance of Optimization in the Process of Design and the Selection of

Structural Materials 31.4 Commonly Observed Failure Modes in Engineering Practice 41.5 Structures and the Analysis of Structures 5

References 5

2 Stress 72.1 Definition of Average Stress and Stress at a Point 72.2 Stress Components and Equilibrium Equations 82.2.1 Stress Components 82.2.2 Equilibrium Equations 92.3 Stress Tensor 102.3.1 Mean and Deviatoric Stress Tensors 102.4 States of Stress 122.4.1 Uniaxial State of Stress 122.4.2 Two-dimensional State of Stress 142.4.3 Three-dimensional State of Stress 182.4.3.1 Stress on an Arbitrary Plane 202.4.3.2 Stress on an Octahedral Plane 212.4.3.3 Principal Stresses and Stress Invariants 222.5 Transformation of Stress Components 24

References 28

vii

3 Strain 293.1 Definition of Strain 293.1.1 Some Properties of Materials Associated with Strain 303.1.1.1 Poisson’s Ratio 303.1.1.2 Volumetric Strain 303.1.1.3 Bulk Modulus 313.1.1.4 Modulus of Elasticity 323.1.1.5 Shear Modulus (Modulus of Rigidity) 323.2 Strain–Displacement Equations 333.3 Strain Tensors 353.3.1 Small Strain Tensor 353.3.2 Finite Strain Tensor 383.3.3 Mean and Deviatoric Strain Tensors 403.3.4 Principal Strains and Strain Invariants 413.3.4.1 Strain Tensor 413.3.4.2 Deviatoric Strain Tensor 423.4 Transformation of Strain Components 433.4.1 Mohr’s Circle 443.5 Strain Measurement 44

References 48

4 Mechanical Testing of Materials 514.1 Material Properties 514.2 Types of Material Testing 524.3 Test Methods Related to Mechanical Properties 524.4 Testing Machines and Specimens 524.4.1 Static Tensile Testing Machine and Specimens 524.4.2 Impact Testing Machine and Specimens 544.4.3 Hardness Testing Machine 544.4.4 Fatigue Testing Machines 564.5 Test Results 564.5.1 Static Tensile Test Results 564.5.1.1 Engineering Stress–Strain Diagram 564.5.1.2 Creep Diagram/Curve 624.5.1.3 Relaxation Diagram/Curve 624.5.2 Dynamic Test Results 634.5.2.1 Tensile, Flexural and Torsional Test Results 634.5.2.2 Toughness Test Results 644.5.2.3 Fracture Toughness Test Results 64

References 64

5 Material Behavior and Yield Criteria 675.1 Elastic and Inelastic Responses of a Solid 675.2 Yield Criteria 675.2.1 Ductile Materials 715.2.1.1 Maximum Shear Stress Criterion (Tresca Criterion) 71

Contentsviii

5.2.1.2 Distortional Energy Density Criterion (von Mises Criterion) 745.2.2 Brittle Materials 765.2.2.1 Maximum Normal Stress Criterion 765.2.2.2 Maximum Normal Strain Criterion 76

References 78

6 Loads Imposed on Engineering Elements 796.1 Axial Loading 796.1.1 Normal Stress 816.1.2 The Principal Stress 826.2 Torsion 856.2.1 Elastic Torsion – Shear Stress and Strain Analysis 866.2.1.1 Prismatic Bars: Circular Cross-section 866.2.1.2 Prismatic Bars: Noncircular Cross-section 956.2.1.3 Thin-walled Structures 966.2.2 Warping (Distortion) of a Cross-section 1016.2.3 Inelastic Torsion and Residual Stress 1036.2.3.1 Residual Stress 1056.3 Bending 1096.3.1 Beam Supports, Types of Beams, Types of Loads 1096.3.2 Internal Forces – Bending Moments (Mf), Shear Force (Q), Distributed

Load (q) 1116.3.3 Principal Moments of Inertia of an Area (I1, I2) and Extreme Values

of Product of Inertia (Ixy) of an Area 1126.3.3.1 Axes Parallel to the Centroidal Axes 1146.3.3.2 Rotation of the Coordinate Axes at the Observed Point

(Rotated Axes) 1156.3.4 Symmetrical Bending 1166.3.4.1 Pure Bending 1166.3.4.2 Nonuniform Bending 1226.3.5 Nonsymmetrical Bending 1266.3.6 Loading of Thin-walled Engineering Elements; Shear Center 1336.3.6.1 Shear Center 1346.3.7 Beam Deflections 1366.3.8 Bending of Curved Elements 1406.4 Stability of Columns 1496.4.1 Critical Buckling Force in the Elastic Range 1506.4.1.1 Pin-ended Columns 1506.4.1.2 Columns with Other End Conditions 1536.4.2 Critical Buckling Stress in the Elastic Range 1556.4.3 Buckling – Plastic Range 1566.4.3.1 Local Buckling of the Column 1576.5 Eccentric Axial Loads 1596.5.1 Eccentric Axial Load Acting in a Plane of Symmetry 1596.5.2 General Case of an Eccentric Axial Load 161

References 164

Contents ix

7 Relationships Between Stress and Strain 1677.1 Fundamental Considerations 1677.2 Anisotropic Materials 1697.3 Isotropic Materials 1717.3.1 Determination of Hooke’s Law – Method of Superposition 1757.3.2 Engineering Constants of Elasticity 1787.4 Orthotropic Materials 1807.5 Linear Stress–Strain–Temperature Relations for Isotropic Materials 184

References 186

8 Rheological Models 1898.1 Introduction 1898.2 Time-independent Behavior Modeling 1908.2.1 Elastic Deformation Modeling 1908.2.1.1 Hooke’s Element (H Model) 1908.2.2 Deformation Modeling after the Elastic Limit 1928.2.2.1 Saint Venant Element (SV Model) 1928.2.2.2 Saint Venant Element–Spring/(SV–Spring) 1928.2.2.3 Saint Venant Element | Spring−Spring/(SV | Spring−Spring) 1928.3 Time-dependent Behavior Modeling 1948.3.1 Newton Element (N Model): Linear Viscous Dashpot Element 1958.3.2 Maxwell Model (M =H−N) 1958.3.2.1 Generalized Maxwell Model 1978.3.3 Voigt-Kelvin Model (K = H | N) 1988.3.3.1 Generalized Voigt–Kelvin Model 1998.3.4 Standard Linear Solid Model (SLS) 2008.3.5 Voigt–Kelvin−Hooke’s Model (K−H) 2018.3.6 Burgers’ Model 2028.4 Differential Form of Constitutive Equations 205

References 207

9 Creep in Metallic Materials 2099.1 Introduction 2099.2 Plastic Deformation – General 2119.2.1 Slip 2119.2.2 Cleavage 2129.2.3 Twinning 2139.2.4 Grain Boundary Sliding 2139.2.5 Void Coalescence 2149.3 The Creep Phenomenon and Its Geometrical Representation 2149.3.1 Creep Deformation Maps and Fracture Mechanism Maps 2169.3.1.1 Creep Deformation Mechanisms 2169.3.1.2 Fracture Micromechanisms and Macromechanisms 2209.3.1.3 Creep Fracture Mechanisms 2219.3.2 Short-time Uniaxial Creep Tests, Creep Modeling and Microstructure

Analysis 223

Contentsx

9.3.2.1 Short-time Uniaxial Creep Tests 2239.3.2.2 Creep Modeling 2259.3.2.3 Microstructure Analysis – Basic 2279.3.3 Long-term Creep Behavior Prediction Based on the Short-time Creep

Process 2289.3.3.1 Extrapolation Methods 2309.3.3.2 Time–Temperature Parameters 2319.3.4 Multiaxial Creep 2329.4 Relaxation Phenomenon and Modeling 234

References 236

10 Fracture Mechanics 23910.1 Introduction 23910.2 Fracture Classification 24010.3 Fatigue Phenomenon 24210.3.1 Known Starting Points 24210.3.2 Stress versus Life Curves (σ–N/S–N), Endurance Limit 24210.4 Linear Elastic Fracture Mechanics (LEFM) 24810.4.1 Basic Consideration 24810.4.2 Crack Opening Modes 25110.4.2.1 Stress Intensity Factor (K/SIF) 25210.4.2.2 Plastic Zone Size around the Crack Tip 26010.4.2.3 Plastic Zone Shape around the Crack Tip 26310.5 Elastic–Plastic Fracture Mechanics (EPFM) 26610.5.1 The J Integral 26710.6 Experimental Determination of Fracture Toughness 27010.6.1 Test Specimens: Shapes, Dimensions, Orientations and Pre-cracking 27110.6.1.1 Shapes and Dimensions of the Specimens 27110.6.1.2 Orientation of a Specimen Made from Base Material 27210.6.1.3 Fatigue Pre-cracking 27410.6.2 Fracture Toughness, KIc and the K–R Curve 27410.6.2.1 R-curve (K–R Curve) 27410.6.2.2 Plane Strain Fracture Toughness (KIc) Testing 27710.6.3 Fracture Toughness JIc and the J–R Curve 27910.6.3.1 R-curve (J–R Curve) 27910.6.3.2 Fracture Toughness ( JIc) Determination/Testing 28010.7 Charpy Impact Energy Testing 28410.8 Crack Propagation 28810.8.1 Introduction 28810.8.2 Fatigue Crack Growth 28910.8.2.1 The Paris Equation 29410.8.2.2 The Walker Equation 29610.8.2.3 The Forman Equation 29710.8.2.4 The Forman–Newman–de Koning Equation 29710.8.3 Creep Crack Growth 29710.8.4 Life Assessment of Engineering Components 298

Contents xi

10.8.4.1 Constant Amplitude Loading 29810.8.4.2 Variable Amplitude Loading 29810.8.5 Crack Closure 29910.8.5.1 Elber Crack Closure Phenomenon 29910.8.6 A Brief Review of Testing of Unnotched, Axially Loaded Specimens 301

References 309

11 The Finite Element Method and Applications 31311.1 The Finite Element Method (FEM) in the Analysis of Engineering

Problems 31311.1.1 Applications of FEM 31311.1.2 The Advantages of Using the FEM 31411.1.3 A Brief Overview of the Historical Development of the FEM 31411.2 Linear Analysis of Structural Behavior 31511.2.1 Formulations of Equilibrium Equations 31611.2.1.1 Variational Formulation of the Finite Element (Equilibrium) Equation 31811.2.2 Structures 33411.2.3 Finite Elements 33411.2.4 Shape Functions – Cartesian and Natural (Dimensionless) Coordinate

Systems 33411.2.4.1 Cartesian Coordinate System 33511.2.4.2 Natural (Dimensionless) Coordinate System 34111.2.5 One-dimensional Finite Elements 34711.2.5.1 Basic 1-D Finite Elements 34711.2.5.2 Finite Elements of Higher Order 35911.2.6 Two-dimensional Finite Elements 36311.2.6.1 Basic 2-D Finite Elements 36711.2.6.2 Finite Elements of Higher Order 37611.2.6.3 Transformation Procedure for the Finite Element Equation 37811.2.7 Three-dimensional Finite Elements 37911.2.7.1 Basic 3-D Finite Elements 38111.2.7.2 Finite Elements of Higher Order 38811.2.8 Isoparametric Finite Elements 39311.2.8.1 Introduction 39311.2.8.2 Isoparametric Representation 39511.2.9 Bending of Elastic Flat Plates 39811.2.9.1 Deformation Theories for Elastic Plates 39811.2.9.2 Finite Elements Based on Kirchhoff Plate Theory 40711.2.10 Basics of Dynamic Behavior of Elastic Structures 41011.2.10.1 Mass Matrix of the Finite Element 41311.2.10.2 Free, Undamped Vibrations of Constructions – Eigenvalues 41411.3 A Brief Introduction to Nonlinear Analysis of Structural Behavior 42111.4 Metal-forming Processes – Brief Overview 42211.4.1 Introduction 42211.4.2 Classification, Variables and Characteristics of Metal-forming

Processes 423

Contentsxii

11.4.2.1 Comparison of Hot and Cold Working Processes in Terms of WorkingTemperature, Shaping Force and Achieved Material Properties 428

11.4.3 Basic Settings Related to the Theory of Metal-forming Processes 42911.4.3.1 Strain-rate Tensor and Data Relating to Yield Criteria 43011.4.3.2 Virtual Work-rate Principle 43311.4.3.3 The Prandtl–Reuss Equations 43311.4.3.4 The Governing Equations of Plastic Deformation 43711.4.3.5 Shape Functions 43711.4.3.6 Strain-rate Matrix 43811.5 The Application of the Finite Element Method in Structural Analysis 43811.5.1 One-dimensional Finite Elements: Finite Element Analysis of Truss Structure

Deformation 43911.5.2 Two-dimensional Finite Elements: J Integral Calculation 44311.5.3 Special Two-dimensional Finite Elements in Shear Stress Analysis 44711.5.3.1 Introduction 44711.5.3.2 Application of General Quadrilateral Finite Elements 450

References 451

Index 453

Contents xiii

Frequently Used Symbols and the Meaning of Symbols

Symbol Meaning

A Cross-sectional area

A0 Initial cross-sectional area

A1 Cross-sectional area after deformation

Ae Finite element area

a Crack length

a, b, c, d, e, t Constants in stiffness matrix

a, [a] Polynomial matrix

[a] Transformation matrix

a0, (ai) Initial crack length

ab, [ab] Polynomial matrix at the boundary of finite element

aeff Effective crack length

af Failure crack length

B Strain-displacement matrix

B, N Parameters

b Width of rectangular

C Constant, contour of considered curve, compliance

C Elasticity matrix (matrix of elastic constants), structural dampingmatrix

C, m Constants in Paris equation(“m” is strain hardening coefficient)

C, n, p, q Experimentally derived constants in Forman-Newman –Koningequation

Cb Generalized elasticity matrix (bending of plate)

CF, mF Constants in Forman equation

Cijkl Fourth-order tensor (elasticity tensor, elastic matrix or stiffness matrix)

CVN Charpy impact energy(specimen with V-notch)

CS Elasticity matrix refers to shear stresses

(Continued)

xv

Symbol Meaning

ce Finite element damping matrix (local coord. System)

D Diameter

D Plate flexural rigidity

D, p, r Parameters

dε Differential operator

dAn Differential area of an arbitrary sloping section (plane)

dAx, dAy, dAz Differential area on x, y, z direction

da Increase in crack length (length of crack: a)

dλ Coefficient

E Modulus of elasticity

Ex, Ey, Ez Young moduli for orthotropic materials

e Position of shear center, distance between the centroid and the neutralaxis, distance

ei Invariants of strain deviator

eij, [e] Deviator strain tensor

F Force (intensity)

F Force, loading

F, [F] Force vector, concentrated force vector

Fcr Critical force

Fi, Mi Nodal forces

Fm Known components of FR

FR Vector of structure nodal forces.tFR Vector of externally applied nodal forces in the considered structure at

time t

Fr Unknown components of FR

FV Vector of volume forcestFσ Vector of nodal forces that corresponds to the element stresses at the

time t

f Yield function, crack opening parameter (in Forman-Newman-deKoning Equation)

fe Finite element nodal forces vector (local coord. system)

fij Dimensionless function

fv, fx, fy, fz Volume force vector and components forces(unit)

G Shear modulus (modulus of rigidity)

Gxy, Gyz, Gzx Shear moduli for orthotropic materials

h Height of rectangular

Imin Minimum moment of inertia

Ip Polar moment of inertia

Frequently Used Symbols and the Meaning of Symbolsxvi

Symbol Meaning

It Torsion moment of inertia

Ix, Iy Axial moment of inertia (area moment of inertia about an in-plane axis)

Ixy Centrifugal/deviation moment of inertia(product of areas)

I1, I2 Principal (principal centroidal) moments of inertia

I1, I2, I3 Stress tensor invariants

imin Minimum radius of inertia

J J-integral (contour integral)

J Jacobi matrix

Je Elastic part of J

JIc Fracture toughness

Jpl Plastic part of J

J1, J2, J3 Invariants of deviator stress tensor

K Bulk modulus, kinetic energy, stress intensity factor

K Global stiffness matrix (structural matrix)

K/SIF, KI, KII, KIII Stress intensity factor, stress intensity factors for three opening modes(I, II, III)

K* Cyclic strength coefficient

Kc, KIc, KIIc, KIIIc Critical stress intensity factor

Keff Effective stress intensity factor

KIc Fracture toughness (Plane strain fracture toughness)

Ktot Total stress intensity factor (as effect of assembled load)

k Constant

ke Finite element stiffness matrix (local coord. system)

kec Condensed stiffness matrix

ke, f

e, [ūe] Finite element stiffness matrix, vector of nodal forces and vector of

nodal displacements in global coordinate system

L Length (of beam, element)

Le Effective (or free) buckling length

Li (i = 1,2,3) Natural coordinates

L0(= G) Gage length

L1 Length of considered element after loading

l Length

l, m, r Direction cosines

li(z), li(Li), li(ξ), li(η), li(ζ) Lagrange interpolating polynomials

M Structural mass matrix

Mf Bending moment (flexural moment)

(Continued)

Frequently Used Symbols and the Meaning of Symbols xvii

Symbol Meaning

Mt Torsion moment(torque)

Mx, My Bending moment (flexural moment) about in-plane axis of cross-section of element (beam), moments in the plate related to the unit ofthe length of plate.

Mxy Twisting moment in plate

me Finite element mass matrix (local coord. system)

N Axial force, number of cycles, shape function

N Axial force (internal)

N, [N] Interpolation matrix(matrix of interpolation functions, shape functionsmatrix)

n Strain hardening exponent (in Holloman-Ludwig equation), normal tothe considered section (plane), degree of polynomial, number of nodes

n* Cyclic hardening exponent

P Larson-Miller parameter

Pn(x) Polynomial

pa Vector of an average stress

pn, pnx, pny, pnz Vector of total stress and its intensity components

Rp, rp Radius of plastic zone around crack tip

Q Shear force, heat

Qxz, Qyz Shear force in plate

q Shear flow, distributed load intensity

qi Eigenvectors (shape vectors)

qv Body force

R Radius, stress ratio

Sij, [S] Deviator stress tensor

Sijkl Fourth-order tensor(compliance tensor)

Sx, Sy First (static) moment of the area with respect to axis x, y

S1, S2, S3 Principal values of deviator tensor

T Temperature

Ti Traction vector

Tm Melting temperature

t Wall thickness, time, thickness

t0, [t0] Basic transformation matrix (rotational matrix)

te Finite element transformation matrix

U Vector of structure nodal displacements

u, [u], {u} Displacement vector

Frequently Used Symbols and the Meaning of Symbolsxviii

Symbol Meaning

u, v, w Displacements, on x, y, z

U Structural velocity vector

[Ü] Structural acceleration vector

Ui, Vi, Wi, Φi Nodal displacements

Um Unknown components of U

Ur Known components of U

U0 Strain energy density

U0D Distortional energy density

ue, [ue], {ue} Finite element nodal displacements vector (local coord. system)

V Volume

V Potential of external load

W Work done by external forces, elastic strain energy density

Wp Polar moment of resistance

Wt Torsion moment of resistance

Wx, Wy Section modulus

x, y, z Cartesian coordinates

α Angle of neutral axis, angle of principal stresses / strain

{α}, α Vector of constants(vector of generalized coordinates)α, β, γ Functions with respect to ratio h / b of rectangular

αT (or α) Coefficient of thermal dilatation

αx, αy Rotation about x,y axes

β Factor

Strain energy release rate (crack extension force or crack driving force).

γ Shear strain, material constant

γxy,….γzx Shear strain components

γR, γρ Angle

ΔKth Fatigue crack growth threshold

ΔK Difference between Kmax and Kmin

ΔL, Δl Elongation

ΔT Change in temperature

ΔV Change in volume

ε Strain

ε Strain rate

ε, [ε], εij Strain tensor

(Continued)

Frequently Used Symbols and the Meaning of Symbols xix

Symbol Meaning

εi Principal strains (dilatations)

εij Strain components

ε0ij, [ε0] Mean strain tensor

εmax Maximum principal strain (dilatation)

εV εv) Volumetric strain

εx, εy, εz Normal strain components-dilatations (directions: x, y, z)

ε0 Mean strain (average dilatation)

ε1, ε2, ε3 Principal strains (dilatations)

η Coefficient of viscosity

θ Twisting angle per unit length

κ Curvature

κx, κy Curvature of the midsurface

λ Slenderness, introduced substitute

λ, μ Lame constants

μ Coefficient of friction

ν Poisson ratio

νxy, νxz, νyx, νyz, νzx, νzy Generalized Poisson ratio (orthotropic materials)

ξ, η, ζ Coordinates, dimensionless coordinates

Π Total potential (total potential energy)

ρ Radius

σ, [σ], σij Stress tensor

σa Stress amplitude

σall Normal stress allowable

σcr Critical stress

σe Equivalent stress

σij Stress components

σmij Mean (spherical) stress tensor

σ0ij, [σ0] Mean (spherical) stress tensor

σm, UTS, US, σUS Ultimate tensile strength, mean stress

σmax, σmin Stress maximum, minimum

σn Normal stress on an arbitrary sloping section (plane)

σoct Normal octahedral stress

σx, σy, σz Normal stress components(directions: x, y, z)

σ0 Mean normal stress, initial stress in rheological models

σ0.2, YS, σYS, σY 0.2 offset yield strength

σ1, σ2, σ3 Principal stresses, stresses in rheological models

τall Shear stress allowable

Frequently Used Symbols and the Meaning of Symbolsxx

Symbol Meaning

τI, τII Extreme values of shear stresses at the directions I, II (plane stress state)

τmax Shear stress maximum

τn Shear stress on an arbitrary sloping section (plane)

τoct Shear octahedral stress

τx,…..τzx Shear stress components

τy Shear stress associated with yielding in uniaxial tension

τ1, τ2, τ3 or τI, τII, τIII Extreme values of shear stresses at three –dimensional state of stress

Φ Prandtl stress function

φ Angle, twisting angle, angle of an arbitrary plane (section) with respectto perpendicular plane

φx, φy, φz Rotational displacement

ψ Reduction in area(contraction of cross-sectional area), radius ratio, S.Venant warping function

ωi Eigenvalues (free vibration frequencies)

Frequently Used Symbols and the Meaning of Symbols xxi

Principal SI Units and the US Equivalents

Quantity SI Unit US Equivalent

length meter (m) 39.370079 inch (in)

meter (m) 3.2808399 feet (ft)

area millimeter2 (mm2) 0.001549907 in2

meter2 (m2) 10.7639104 feet2 (ft2)

mass kilogram (kg) 2.2045855 lb – mass

kilogram (kg) 0.06852177 slug (lb s2/ft)

volume (solid) meter3 (m3) 35.3146667 feet3 (ft3)

meter3 (m3) 61012.8 inch3 (in3)

volume (liquid) liter (l) 0.03531566 feet3 (ft3)

liter (l) 0.26417944 gallon (gal)

force newton (N = kgm/s2) 0.22480894 pound (lb, lbf )

kilonewton (kN) 0.22480894 kilopound (kip)

stress/pressure pascal (Pa = N/m2) 0.855470208 pound/foot2 (psf )

kilopascal (kPa/m2) 0.145037944 pound/inch2 (psi)

megapascal (MPa = N/mm2) 0.145037944 kilopound/inch2 (ksi)

moment of a force newton meter (N m) 8.8507456 pound inch (lb in)

newton meter (N m) 0.73756215 pound foot (lb ft)

energy/work joule (newton meter) 0.73756215 pound foot (lb ft)

velocity meters per second (m/s) 39.37 inches per second (in/s)

meters per second (m/s) 3.2808399 feet per second (ft/s)

density kilograms per cubic meter (kg/m3) 0.06238 pounds per cubic foot (lb/ft3)

stress intensity factorSIF (K); fracturetoughness (KIc)

(MPa m);… (K) 1.099 ksi in…(SIF)

xxiii

SI Prefixes, Basic Units, Physical Constants,the Greek Alphabet

SI Prefixes

Prefix tera giga mega kilo centi milli micro nano pico

SI symbol T G M k c m μ n p

Factor 1012 109 106 103 10−2 10−3 10−6 10−9 10−12

Basic Units

Quantity SI US

length meter (m) foot (ft)

mass kilogram (kg) slug (lb s2 ft)

time second (s) second (sec)

force newton (N= kgm s2) pound (lb)

temperature degree: Kelvin (K)Celsius ( C)

degree: Fahrenheit ( F)Rankine ( R)

TK = T C + 273.15 T F = T R + 459.67

Physical Constants

Quantity SI US

acceleration of gravity(g) 9.80665m/s2 32.1740 ft/s2

density (ρ), unit weight of water(at 4 C = 39.2 F)

1000 kg/m3 62.43 pcf

Normal atmospheric pressure(at)

101,325 kPa0.101325MPa

14.6960 psi

xxv

The Greek Alphabet

α A Alpha ν N Nu

β B Beta ξ Ξ Xi

γ Γ Gamma ο O Omicron

δ Δ Delta π Π Pi

ε E Epsilon ρ P Rho

ζ Z Zeta σ Σ Sigma

η H Eta τ T Tau

θ Θ Theta υ Y Upsilon

ι I Iota φ Φ Phi

κ K Kappa χ Χ Chi

λ Λ Lambda ψ Ψ Psi

μ M Mu ω Ω Omega

SI Prefixes, Basic Units, Physical Constants, the Greek Alphabetxxvi

Important Notice Before Reading the Book

The author and publisher of this book have invested reasonable efforts in its preparation.The book presents material that is the subject of the author’s research and lectures andcovers a wide spectrum of engineering disciplines. It provides a concise written guide totheories and applications, showing the methods for solving particular problems. How-ever, the book contains the author’s interpretations of the above and not facts. The mate-rial in the book is provided as a study aid for the reader and not for business-relatedactivity. Since the book may contain different types of error, the author and the publishermake no warranty of any kind with regards to any of the content of the book or its usage.The author and the publisher shall not be liable in any event for any damages in connec-tion with the usage of any contents of this book. By proceeding to read this book, thereader agrees with the above. This note is a part of this book.

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