Formulas for Mechanical and Structural Shock and Impact BBS

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(2)

Gregory Szuladzi ´nski

Formulas

for

Mechanical

and

Structural

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6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742

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Library of Congress Cataloging-in-Publication Data

Szuladzinski, Gregory,

1940-Formulas for mechanical and structural shock and impact / author, Gregory Szuladzinski. p. cm.

“A CRC title.”

Includes bibliographical references and index. ISBN 978-1-4200-6556-5 (alk. paper)

1. Impact--Mathematical models. 2. Shock (Mechanics)--Mathematical models. 3. Engineering mathematics--Formulae. I. Title.

TA354.S98 2010

624.1’76--dc22 2009013703

Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

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iii

Preface

This book is essentially a collection of formulas describing dynamic responses to shock loads. The presentation is inspired by Roark’s classic Formulas for Stress and Strain, which presents equations and explanatory sketches in a compact manner. The theoretical basis is presented in a concise, although somewhat superfi cial manner, as appropriate for a refer-ence work rather than a text intended to teach the subject. Still, this book is a reasonably self-contained tool. Although it is written for engineers in general, experienced dynamicists may also fi nd some new ideas and approaches.

The objective of this book is to provide a meaningful reference in today’s computer-oriented environment. Nowadays, due to the development and availability of general-purpose structural programs, any large analytical task can be performed by computers. This limits the importance of manual calculations, in general, and of certain analytical methods, in particular, as compared, for example, with the 1960s. Some of the methods in which manual work is and will remain useful in the foreseeable future are

1. Obtaining preliminary fi gures on the anticipated dynamic response of a system that is in an early stage of design and for which a full-scale computation is not practical.

2. The preparatory phase of large-scale calculations where a dynamic model is gener-ated. At this stage, prudent analysts conduct a number of checks to ensure that they neither miss anything important nor incorporate too many unnecessary details, which is wasteful.

3. An indirect verifi cation of computer-generated results by using a simplifi ed model or a calculation method. (This may not only explain some unbelievable results, but may also help to guard against hidden errors, for which there seem to be unlimited opportunities.)

4. Work in preparation for physical testing.

Thus, it is quite clear that the only sensible options for today’s engineers embarking on a hand calculation are the methods and approaches that are concise and relatively easy. Anything more mundane might as well be done by a major fi nite-element (FE) code. This was the major criterion in the selection of problems and methodology. Simplicity is there-fore the main purpose and the keynote of this book. Still, not all situations can be explained in an elementary manner. In such cases, it is preferable to provide a set of two or three simple equations, each of them corresponding to some defi nite concept, instead of building a large, complex, single equation. In many instances, the effort to simplify things goes into offering simple approximate equations instead of more complicated, but also more accurate, solutions. (One should note that terms like “manual” or “hand” calculations in a modern setting mean not only the use of a calculator, as they did in the 1970s, but also the use of such simple programs as a spreadsheet or Mathcad.)

There is another advantage in being able to do simplifi ed, although not very accurate, estimates. This is often evident with regard to extreme loads or structures. For example, one can ask: “Is the peak strain to be expected closer to 5% or to 50%?” with one of the answers

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implying safety and the other indicating danger. To be able to answer such a question early on has a strong impact on the engineering approach to be adopted.

This book is written for engineers and for all others who want to have an insight into how objects and structures respond to sudden, strong impulses that are usually of short duration. Since there is no emphasis on any particular type of structure and the scope is quite broad, those involved in the mechanical aspects of aeronautical, automotive, nuclear, and civil engineering, as well as those in general machine design, may fi nd it equally benefi cial.

Some of the more recent results presented here may be hypothetical or of unproven reli-ability, and must therefore be used with caution. The decision to include such materials was based on two reasons. The fi rst was the lack of better and practical solutions. The second was an old saying that if everyone waited until things could be done perfectly, nothing would ever be done.

In order to benefi t most from this book, the reader should be familiar with the theory of the strength (mechanics) of materials and engineering mathematics. Some familiarity with the concepts of dynamics can be helpful, but this is not an absolute necessity as the introduc-tory part of every chapter should explain the setting.

The author is indebted to several people who helped to bring this book to its present shape. Verl A. Stanford carefully checked most of the material and made many valuable suggestions. Miecia (May) Paszkiewicz carried out the important task of cross-referencing numerous items in the text. Professor Ali Saleh read Chapters 10 and 11 and made useful comments. John Quinn read several chapters and helped to remove some fl aws.

All comments and criticisms regarding the contents of this book can be sent as e-mails to [email protected]. All such comments will be most gratefully received and answered, if possible.

Dr. Gregory Szuladzin´ski

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xiii

Introduction

Every typical chapter of this book is divided into three parts: a theoretical outline, a tab-ulation, and examples. The tabulation is an attempt to present the subject matter more clearly by breaking it down into simple blocks. The examples are an essential part of this book, because it is well known that most people learn best when numerical examples are provided.

Because of a diffi cult subject matter as well as due to style of presentation, there are many cross-references in the text linking the three major parts listed above. The symbols used are listed at the outset.

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xv

Author

Dr. Gregory Szuladzin´ski received his master’s degree in mechanical engineering from

Warsaw University of Technology in 1965 and his doctoral degree in structural mechanics from the University of Southern California in 1973.

From 1966 to 1980, he worked in the United States in the fi elds of aerospace, nuclear engineering, and shipbuilding. He has done extensive work in computer simulations of seismic events and accidental dynamic conditions as related to the safety of nuclear plants and military hardware.

From 1981 until the present time, he has been working in Australia in the fi elds of aero-space, railway, power, offshore, automotive, and process industries, as well as in rock mechanics, underground blasting, infrastructure protection, and military applications. He has a number of publications to his credit in the area of nonlinear mechanics. His fi rst book on the subject, Dynamics of Structures and Machinery: Problems and Solutions, was pub-lished by John Wiley Interscience in 1982.

Dr. Szuladzin´ski has been involved with the fi nite-element method of simulation of structural problems since 1966. In 1978–1979, he worked as the principal analyst for Control Data in Los Angeles in support of fi nite-element analysis (FEA) codes.

Since the early 1990s he has been working on computer simulations of such violent phenomena as rock breaking with the use of explosives, fragmentation of metallic objects, shock damage to buildings, structural collapse, fl uid–structure interaction, blast protection,

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and aircraft impact protection. He has conducted a number of state-of-the-art studies showing explicit fragmentations of structures and other objects.

Dr. Szuladzin´ski is a fellow of the Institute of Engineers Australia, a member of its Structural and Mechanical College, a member of the American Society of Mechanical Engineers and of the American Society of Civil Engineers.

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xvii

Symbols and Abbreviations (General)

SYMBOLS

a acceleration (m/s2_{), a shorter edge length of a rectangular plate (m), inside}

radius (m)

α angular displacement (rad) A area (m2_{), amplitude}

As shear area (m2)

b coeffi cient for bending defl ection of beams and plates, outside radius (m) C damping coeffi cient (N-s/m), torsional constant of cross section (m4₎

Cc critical damping coeffi cient (N-s/m)

c0 speed of propagation, axial waves (m/s)

c1 speed of propagation, twisting waves (m/s)

c2 speed of propagation, lateral wave in a string (m/s)

c3 speed of propagation, shear waves in beams (m/s)

cp speed of propagation, pressure waves in a continuum (m/s)

cpl speed of propagation, wave in plastic range (m/s)

cR speed of propagation, Rayleigh waves (m/s)

cs speed of propagation, shear waves in a continuum (m/s)

d diameter (m)

D energy dissipated in damping (J), plate bending stiffness parameter (N-m), detonation velocity (m/s)

ε

– _{logarithmic strain (m/m)}

e eccentricity, static shaft unbalance (m) eˇ angular acceleration (rad/s2₎

E Young’s modulus of elasticity (Pa) Ec modulus of elasticity of concrete

Eef effective modulus of in presence of lateral constraint (Pa)

Ek kinetic energy (N-m)

Ep plastic modulus (Pa)

f natural frequency (Hz = cycle/s) f(t) function of time

Fy, Fu yield strength, ultimate strength (Pa)

g a gram of mass (kg/1000), acceleration of gravity (9.81 m/s) G modulus of elasticity in shear (Pa)

h height or thickness (m) H(t) unit step function i radius of inertia (m) I = Aρc impedance of a bar (kg/s)

I second area moment (area moment of inertia (m4₎₎

I0 cross-sectional polar moment of inertia (m4)

J mass moment of inertia about axis of revolution (kg-m2₎

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k*, kef effective stiffness (N/m)

k –

stiffness modifi ed by presence of axial force (N/m) kf stiffness of elastic foundation (N/m2)

K stiffness of a rotatory spring (N-m/rad) K_σ geometric stress concentration factor

L length (m)

l length (m)

L work performed (N-m)

Le work performed in elastic straining (N-m) Lp work performed in plastic straining (N-m)

m distributed mass (kg/m or kg/m2₎

M lumped mass (kg)

M* reduced mass (kg)

M bending moment (N-m)

My bending moment at onset of yield (N-m)

M0 moment capacity for perfectly plastic material (N-m)

N one Newton of force

n stress–strain curve parameter, equivalent polytropic exponent

P axial force (N)

Pcr buckling force (N)

Pe Euler force, π2EI/L2 (N)

p pressure (Pa)

q distributed load (N/m), specifi c energy content (J/kg) q0 distributed edge load (N/m)

Q beam or plate shear force (N or N/m), energy content (J) r current radius (m)

R force of resistance (N), (mean) radius (m)

S engineering stress (MPa), true stress (MPa), impulse (N-s)

s coeffi cient for shear defl ection of beams and plates, impulse per unit length of beam (N-s/m)

Sn, St normal and tangential impulse (N-s)

t time (s)

tm loading phase duration (s)

t0 duration (s)

T twisting moment (N-m)

Ty twisting moment at yield (N-m)

Tu ultimate twisting moment (N-m)

u displacement (m)

ust static displacement (m)

uc increase of cavity radius caused by deformation of intact medium (m)

ud dynamic displacement (m)

U strong shock wave velocity (m/s)

v velocity (m/s)

V velocity following rebound (m/s), volume (m3₎

w distributed load (N/m or N/m2₎

W force (N)

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X function of spatial variable x

Y yield parameter (MPa)

α angular displacement (rad), mass-proportional damping coeffi cient β stiffness-proportional damping coeffi cient

γ shear angle

Γ energy expression coeffi cient ω circular frequency (rad/s)

ω– frequency modifi ed by presence of axial force ωd damped circular frequency (rad/s)

δ displacement (m)

ε (engineering) strain (m/m), angular acceleration (rad/s2₎

εy strain at onset of yielding (m/m)

εu ultimate strain, at peak stress (m/m)

εr maximum strain, at rupture (m/m)

ε· _{strain rate (1/s)}

ζ damping ratio

κ coeffi cient of restitution

λ angular velocity (rad/s), elongation of a structural member (m)

Λ half-wave length (m)

ν Poisson’s ratio

ξ distance to moving joint (m)

Π strain energy (N-m)

σ stress, direct (MPa)

σ0 yield stress of EPP material (MPa)

σI incoming wave stress (MPa)

σR refl ected wave stress (MPa)

σT transmitted wave stress (MPa)

ρ specifi c mass or mass density (kg/m3_{), radius of curvature (m)}

τ shear stress (MPa), natural period of vibrations (s) τy shear stress at yield (MPa)

τ0 shear stress at yield for EPP material (MPa)

τu ultimate shear stress (MPa)

τ

– _{natural period modifi ed by presence of axial force (s)} χ damping coeffi cient

ψ coeffi cient of friction, shear defl ection coeffi cient Ω angular velocity (rad/s)

η distance to stationary joint (m)

Ψ energy coeffi cient, material toughness (N-m) Θ work expression coeffi cient

ABBREVIATIONS

2D two-dimensional 3D three-dimensional BL bilinear material model

CC clamped ends

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CG one end clamped, the other guided, center of gravity CL centerline

DF dynamic (magnifi cation) factor DOF degree of freedom

EPP elastic, perfectly plastic material model FE fi nite elements

FEA fi nite element analysis ln logarithm to the base e MDOF multiple degrees of freedom NL nonlinear

PL power law material model PPV peak particle velocity RC reinforced concrete

RO Ramberg–Osgood material model

RSH rigid-strain hardening material model SC one end supported, one clamped SDOF single degree of freedom

SF safety factor

SS simply-supported ends

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1

Concepts and Deﬁ nitions

THEORETICAL OUTLINE

Static or quasistatic loading means that the load is applied slowly. On the other hand, dynamic or shock loading implies fast or abrupt application. Those adjectives are relevant with respect to the natural period of a structural element under consideration. When the load increases to its maximum value over fi ve or six natural periods, it is a quasistatic load. When it does so over a fraction of the period, it is a shock loading. (Broader defi nitions exist, where anything with up to two periods duration is a shock loading.)

When speaking of a shock load, engineers usually have in mind a load that is of a large magnitude, but with a very short duration. But this is not the only circumstance where the term applies. The load can last for a very long time, but if it is suddenly applied or suddenly removed, the term “shock” is appropriate as well. The broadest defi nition seems to be that a shock is any abrupt change of a force, a position, a velocity, or an acceleration affecting the body under consideration, Silva [78].

This book is not a study of periodic motion, but the subject of the natural period (or natural frequency of vibration) is given its due attention. This is because that quantity is an important parameter when assessing a shock response of a structural component.

S

TIFFNESSAND

N

ATURAL

F

REQUENCY

Figure 1.1 presents two simple systems capable of periodic motion: (a) translational oscil-lator and (b) rotational osciloscil-lator. In (a) mass M is attached to the ground by a spring of stiffness k (N/m). In (b) a disk with mass moment of inertia J is connected to the ground by a shaft of an angular stiffness K (N-m/rad). The unit for M is kg and for J it is kg-m2_{. The}

terms stiffness, rigidity, and spring constant can be used interchangeably when describing the resistance capability of an elastic member. The reciprocal of stiffness is called fl exibility although the term compliance is also used by some authors. For a translational motion, stiffness is defi ned as a force that causes a unit (1 m) translation. Flexibility is a displace-ment caused by a unit (1 N) force. In angular motion, the defi nitions are analogous with “moment” replacing “force” and “rotation” instead of “translation.”

According to the elementary vibration theory, the motion of an oscillator in Figure 1.1a, when disturbed from the initial position, may be described by the following equation:

0

Mu+ku= (1.1)

where ü stands for acceleration. The displacement relative to the initial position is given as a solution to the above:

cos( )

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where A and β are constants that depend on the initial conditions and k

M

ω = (1.3)

depends only on system properties. According to this equation, mass M returns to the same point after every interval of time τ called the period of motion:

π

τ =2_ω = π2 M (s)

k (1.4)

The number of periods, or cycles of vibrations per second, is called the frequency or natu-ral frequency and is designated by f:

− ω = =_τ1 _π 1 (s ) 2 f (1.5)

The basic unit of this frequency is one cycle per second, or hertz (Hz). The units of ω are radians per second; ω is also referred to as the natural frequency of the system, but often the term circular frequency is used to distinguish it from f. In translational vibration, ω has little physical signifi cance. It is used because it is convenient in the mathematical descrip-tion of modescrip-tion. The nature of the disturbance dictates the magnitude of the constants A and β in Equation 1.2.

The equation of motion of the rotational system in Figure 1.1b is 0

Jα + α = K (1.6)

in which α is the angle of rotation measured from initial position. This angle of rotation is shown as a straight-line vector throughout this book. (It is distinguished from translation by a double arrow. A curved arrow is used to represent rotation only if the plane of rotation is parallel to the plane of a drawing.) The solution is analogous to Equation 1.2:

cos( )

A t

α = ω − β (1.7)

The circular frequency ω is not any more meaningful than it is for translation, but is often used for convenience:

2 f K J ω = π = (1.8) M W u k (a) J T K α (b)

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T

HE

A

NALOGYBETWEEN

T

RANSLATIONALAND

R

OTATIONAL

M

OTION This analogy is complete in every respect, as shown in Table 1.1.

U

NITS

The discussions relating to the convenience of various systems of units are probably as old as the units themselves. One way to look at this is to say: the most convenient system is such that the quantities, frequently seen in daily life, are expressed by numbers between 1 and 10. This makes the English or imperial system a leader in the fi eld of length measurement. However, that system had been replaced (with the notable exception of the United States) by the metric or SI system. Although m (meter), kg (kilogram), and s (second) are the basic units, the use of subunits is also allowed, which gives the user a fair amount of freedom to choose the most convenient set.

Any unit system can be adopted for calculations, provided it is consistent. That consis-tency can partially be checked by invoking Newton’s law: W = Ma, which must be satisfi ed. The question of convenience can be answered only for specifi c circumstances. It is certainly desirable to see frequently used quantities as numbers that are neither too large nor too small. In the analysis of violent events, system (2) seems to make it possible more often than others. For this reason this “milli” system will be used here for most illustrative examples.

Prior to general acceptance of the SI system in many countries, the unit of kG or kgf (kilogram-force) equal to 9.81 N was used. In older texts a force unit of the dyne = N/105

may be encountered. Also, a unit of pressure, bar = 0.1 MPa, a favorite of weathermen and physicists, may occasionally be found. The gravitational constant may slightly vary

TABLE 1.1

Analogy between Translational (1) and Rotational (2) Vibration

(1) Mass M kg (2) Mass moment of inertia J kg-m2

(1) Force W N (2) Torque T N-m (1) Displacement u m (2) Rotation α rad (1) Velocity v = u· m/s (2) Angular velocity λ = α· rad/s (1) Acceleration a = v· m/s2

(2) Angular acceleration eˇ = λ· rad/s2

(1) Stiffness k N/m (2) Rotational stiffness K N-m/rad (1) Momentum Mv N-s (2) Angular momentum Jλ kg-m2_-rad/s

(1) Kinetic energy Mv2_/2 _N-m

(2) Kinetic energy Jλ2_/2 _N-m

(1) Strain energy ku2_/2 _N-m

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between geographical locations, but g = 9.81 m/s2_{seems to be an accepted level in most}

publications.

Although the examples in this book are typically written in metric units, a need to recal-culate to the English system may often arise.* The format of Table 1.3 will, hopefully, assist that recalculation. If the acceleration of gravity is taken as 9.81 m/s2_{, it corresponds to}

386.22 in./s2_{, while a more accurate, standard 9.80665 m/s}2_{yields close to 386 in./s}2_.

The unit of 1 pound is often used as a unit of force (lbf) and unit of mass (lb). Unfortunately, the latter is inconsistent in terms of Newton’s law, as quoted above. To keep consistency and avoid using artifi cial units, the mass unit in the English system is a quotient of a unit of force and a unit of acceleration, M = W/a. To illustrate one of the ways of changing units, an operation, which is a frequent source of errors, consider a pressure quoted as 1000 lbf/ft2

and its equivalent:

= = = = 2 2 2 4.45 N 1,000 lbf/ft 1,000 47,899 N/m 47,899 Pa 0.0479 MPa (0.3048 m)

*The United States is still using English units in engineering practice.

TABLE 1.2

Metric Unit Systems

(1) (2) (“Milli”) (3)

Length Meter Millimeter Millimeter Mass Kilogram Gram (g) Tonne (1000 kg) Time Second (s) Millisecond (ms) Second (s) Force Newton (N) Newton Newton Pressure or stress Pascal (Pa) MPa (106_Pa) _MPa

Energy Joule (J) N-mm N-mm Acceleration of gravity 9.81 m/s2 _{0.00981 mm/ms}2 _{9810 mm/s}2

Density of steel 7850 kg-m3 _{0.00785 g/mm}3 _{7.85 × 10}−9_tonne/mm3

Young’s modulus of steel 200 × 109_Pa _{200,000 MPa} _{200,000 MPa}

Yield stress, mild steel 250 × 106_Pa _{250 MPa} _{250 MPa}

TABLE 1.3

English to Metric Units (1 g = 1 Gram)

The Unit of Equals to Or to

1 inch (in.) 25.4 mm 0.0254 m 1 foot (ft) 304.8 mm 0.3048 m 1 mile 1609.3 m

1 n. mile (knot) 1852 m 1 av. ounce (oz) 28.35 g

1 pound (lb) 453.6 g 0.4536 kg 1 ton (short) 907.2 kg

1 lbf (pound-force) 4.45 N

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A compact way of stating unit dependence was selected for this book. When, for example, length L is intended to be expressed in meters, we simply say L ∼ m.

I

NERTIA

F

ORCE AND

D

YNAMIC

E

QUILIBRIUM

Newton’s (second) law of motion can be expressed as W = Ma, which means that the applied force W is proportional to the acceleration a, which the mass M is experiencing. In rela-tion to Figure 1.1a, for example, W would be the driving force less the spring resistance. In a more general setting, W is the resultant of all external forces applied to M. The above relationship can also be written as W − Ma = 0. In this form Newton’s law is usually called d’Alembert’s principle as it regards force −Ma as the inertia force. The principle says that an accelerating mass can be viewed as being in a special state of static equilibrium, where the total of the external forces is balanced by the inertia force.

Returning to Figure 1.1a, let W act for a very short time and let us observe the movement of M to follow. Due to spring resistance, M will briefl y stop, in time, and reverse its motion. At that stopping point or stagnation point, there are two forces in equilibrium: the spring force Ku and the inertia force Ma. If the driving force W still acts when a stagnation point is reached, then the expression of dynamic equilibrium is

W−Ku=Ma (1.9)

At that point M is in an instantaneous equilibrium and no kinetic energy is present. This observation is often used when applying energy methods. One can then equate the strain energy of the spring with the energy supplied from outside, usually as the work of the driving force W or the initial kinetic energy arising from the initial velocity of M. This procedure is also extended to large and complex systems, although determination of a stagnation point may then become less than straightforward.

E

QUIVALENT

S

TRESS ANDA

S

AFETY

F

ACTOR

When there are several stress components present, i.e., tension in one direction, com-pression at right angles to it and a shear stress, there is a problem of calculating a single quantity, which would be a combined measure of all the stress components and which could be compared to the (uniaxial) allowable material stress. This is discussed in detail in Chapter 6. With regard to metals, two theories clearly dominate. One is asso-ciated with the name of Tresca and it relates to the maximum difference between the principal stress components. The other one is the energy-of-distortion theory, which is often, but inaccurately attributed to Mises [62], while Huber [37] published it 9 years earlier. For this reason the energy-of-distortion theory is referred to as Huber–Mises theory in this book.

A safety factor (SF) of a structural element is defi ned as a ratio of strength of that ele-ment to the applied load. Instead of using the word strength, one often reads terms like the ultimate load or the breaking load, all having the same meaning. The SF of the whole structure is the smallest factor of any of its components. The need for a SF comes from incomplete knowledge of the designer; the applied loads are known with a limited accuracy only and so are the material properties. The response of a structure to a particular loading is usually a matter to be established by analysis.

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The material supplier provides minimum-guaranteed properties of a structural material. If a sample of that material is tested at random, it will most likely be stronger than what is guaranteed. Mild steel, for example, can have its average yield strength 30% higher and the ultimate strength 20% larger than the “catalog” values. While the statistical averages can often be established, most engineers are content with the minimum-guaranteed values, as this builds an additional SF in their predictions. After all, the objective of the designer is to create safe structures. In aerospace design, for example, the SF can be as small as 1.5. However, using “catalog” values of material constants increases that factor, although in an invisible way.

It is probably quite clear to the reader that loads and material properties should be treated in a statistical manner, because of variations involved. This had been tried, but did not achieve popularity so far, because even the simplest problems have a rather involved for-mulation then. Extending such analyses to nonlinear dynamics may become truly mind-boggling. However, if the statistical input is known, some advanced fi nite-element analysis (FEA) programs offer help in that respect.

The aim of this book is to present methods for determining dynamic response to, typi-cally, transient loading and comparing the result with a dynamic strength of a member in question. If that strength is, in fact, a statistical variable, then using average material prop-erty values offers the most likely outcome in a statistical sense. This outcome is, much of the time, the objective in this book. The application of SFs, as appropriate for a situation at hand, is left to the designer/analyst.

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7

2

Natural Frequency

THEORETICAL OUTLINE

The concepts of stiffness and natural frequency, fi rst introduced in Chapter 1, are being continued here. For a simple oscillator, the equation of motion is the fi rst-order differen-tial equation, and the determination of natural frequency is trivial. When a body has a continuous mass distribution, the description of motion is accomplished by partial differ-ential equations. In this chapter, essdiffer-entially the fi rst natural frequency is of interest, which makes the task relatively simple. Much of the material relates to frequencies of very basic elements, but a few methods combining those are also provided, so that more complex arrangements can be approximated.

D

ETERMINATIONOF

N

ATURAL

F

REQUENCYBYTHE

D

IRECT

M

ETHODANDBY THE

E

NERGY

M

ETHOD

The direct method is based on a simple defi nition. The stiffness k is defi ned as a force that is needed to move a point by one unit of length in the direction of motion. This parameter is used, together with a value for mass, in Equation 1.3. Sometimes the weight, or gravity force, G = Mg, with g = 9.81 m/s2_{, is used instead of mass, as it may be more convenient.}

The symbol W is employed as a designation of force in this book, and its similarity with weight is quite incidental. (If gravity is a part of the picture, a statement is made to that effect and the g symbol placed in tabulations.) For a rotational system, Equation 1.8 is used, with the angular quantities replacing the translational values.

The principle of energy conservation may also be used to fi nd a natural frequency. It says that if there are no damping forces and no external forces involved (beginning at some time point), we have k E C Π + = (2.1) in which C is a constant

Π is the energy of deformation or strain energy

For a spring in Figure 1.1a, the energy of deformation is Π = ku2_{/2, where k is the}

spring rigidity and u is the defl ection from the unstrained position. The kinetic energy is Ek = mv2/2. The incremental form of Equation 2.1 is

k 0

E

ΔΠ + Δ = (2.2)

which means that there is no net increase in the total energy. If gravity forces are involved, we have, in place of the above

k g

E U

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That is, if we consider the system moving from position 1 to position 2, the sum of increases of strain energy and of kinetic energy is equal to the work performed by forces of gravity Ug:

g ( 1 2)

U =Mg z −z (2.4)

in which Mg is the weight of a body under consideration. The terms z1 and z2 denote the

locations of the center of gravity (CG) of mass M. The value of Ug is positive if z1 > z2 (i.e., the

CG moves from a higher to a lower position). Regardless of the gravity force involvement, a vibratory motion takes place between two stationary (zero velocity) positions, for which the kinetic energy is zero. Equation 2.1 has two positive functions on the left side, and if one becomes zero, the other must attain its maximum. Thus, for any stationary point, we have Πm = C. Midway between these extreme points, there is a location at which no strain

exists (neutral position) in the linear-elastic elements, and then Ekm = C. Comparing the last

two equalities gives

m Ekm

Π = (2.5)

which is a restatement of Equation 2.1. Once a particular form of Equation 2.5 is known, it is suffi cient to use the relation, 2 = ω2 2

m m

v u , between the maximum velocity and maximum displacement to determine ω. (Refer to Chapter 1.)

B

ODIESWITH

C

ONTINUOUS

M

ASS

D

ISTRIBUTION Equations of Motion: Bar and Shaft

A segment of an axial bar, shown in Figure 2.1, is in the state of dynamic equilibrium. The external load of intensity, q (N/m), is balanced by the increment of the internal force, P, and by the inertia force of magnitude, mü dx, where m = Aρ is the mass per unit length (kg/m). The rate of change of the stretching force P is ∂P/∂x, therefore its value changes by (∂P/∂x)dx over the segment dx. Projecting all forces on the x-axis, we obtain

d d d 0 P P x P q x mu x x ∂ ⎛ ₊ ⎞ _{− +} ₋ ₌ ⎜ ⎟ ⎝ ∂ ⎠ (2.6)

The elongation of the segment is, by an elementary relation

u P

x EA

∂ =

∂ (2.7a)

FIGURE 2.1 Dynamic equilibrium of a bar element.

dx mÜ dx q dx P u dx P+ ∂P ∂x dx u+ ∂u ∂x x

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or

= ′

P _EAu (2.7b)

If a bar is of variable cross section, A is a function of x. The derivative of P is in this case

( ) P u EA EAu x x x ∂ ₌ ∂ ∂⎛ ⎞ _≡ _{′ ′} ⎜ ⎟ ⎝ ⎠ ∂ ∂ ∂ (2.8)

The equation of motion now becomes

( )

mu− EAu′ ′ =q (2.9)

in which the prime denotes differentiation with respect to x. For a constant axial stiffness, EA = constant, the equation simplifi es to

2 0

q

u−c u′=_m (2.10)

where both u and q are functions of time and the spatial variable x, and c0 can be shown to

be the speed of propagation of a longitudinal elastic disturbance along the bar:

1/2 1/2 0 E EA c =⎛_⎜ _m⎞_⎟ = ⎛ ⎞_{⎜ ⎟}_ρ ⎝ ⎠ ⎝ ⎠ (2.11)

with the last equality being true only if the structural mass and the total mass are equal. Figure 2.2 shows a segment of a shaft in dynamic equilibrium; T is the resultant twist-ing moment at a certain location, while the external distributed load has the intensity, mt

(N-m/m). The moment of inertia about the shaft axis is I0ρ dx, in which I0 is the polar

moment of inertia of the cross section. The independent variable is the angle of rotation, α. There is a complete analogy between a bar and a shaft. Projecting the vectors pertaining to the angular quantities on the x-axis, we obtain

t 0 0 T m I x ∂ + − ρα = ∂ (2.12)

FIGURE 2.2 Dynamic equilibrium of a shaft element. x dx mtdx T T ∂T+∂x dx dx α ∂α+ ∂x α I0ρ ¨ dxα

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If the elementary equation for the torsional deformation is applied to a shaft segment of length dx, one gets

T x GC ∂α = ∂ (2.13a) or T =GCα′ (2.13b) in which

α is the angle of twist

C is the torsional constant of the cross section

Limiting ourselves to shafts with a constant section, we obtain

2 2 T GC x x ∂ ₌ ∂ α ∂ ∂ (2.14)

The equation of motion now becomes

1/2 t 2 1 1 0 0 with m GC c c I I ⎛ ⎞ α − α =′′ =_⎜ _⎟ ρ ⎝ρ ⎠ (2.15)

where c1 is the speed of propagation of twisting disturbances. For circular cross sections,

C = I0.

Although the material to follow refers to a bar, everything stated is also valid for a shaft if the notation is suitably changed. To solve Equation 2.10, we must know the initial condi-tions in the form

0 ( ,0) ( ) u x =u x (2.16a) and = 0 ( ,0) v ( ) u x x (2.16b)

that is, the initial position and velocity of any point of the bar axis at t = 0. Also the end conditions, which are displacements or forces (or combinations thereof) at the ends, must be given. Our concern at this point is limited to solving Equation 2.10 in its homogeneous form, that is, when the right side is zero. A general solution is assumed as

( , ) ( ) ( )

u x t =X x ⋅f t (2.17)

where X is a function of x only and is referred to as a mode shape, while f depends only on time. The detailed expressions for X and f are

1 2 0 0 ( ) cos x sin x X x D D c c ω ω = + (2.18a)

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and

1 2

( ) cos sin

f t =B ω +t B ωt (2.18b)

Substituting the end conditions into this relation gives us a frequency equation for a par-ticular system, with which we can establish the set of natural frequencies, ωi. There are

infi nitely many such frequencies and the like number of natural modes, resulting from Equation 2.18a. In the process, we also establish the relationship between the constants D1

and D2. For example, suppose that we have found D2 = 0.5D1. This allows us to express a

modal shape, i, as 1 0 0 cos i 0.5sin i i x x X D c c ⎛ ω ω ⎞ = _⎜ + _⎟ ⎝ ⎠ (2.19)

Finally, one can set D1 = 1, since a scale factor of a mode shape has no meaning.

The equations of motion for bars and shafts are of the second order with respect to the x variable, which is the reason for describing them as the second-order elements.

Equations of Motion: Cable and Shear Beam

These two also belong to second-order elements. The segment of a cable (string) in Figure 2.3 is subjected to a laterally distributed load, w (N/m), to a stretching force, P, which is assumed constant, and to the distributed inertia force of magnitude, mü dx. The lateral displacement, u, is assumed to be very small in comparison with the length of the cable. By projecting all forces on the vertical axis we obtain

1/2 2 2 with 2 w P u c u c m m ⎛ ⎞ − ′′= =_{⎜ ⎟}_{⎝ ⎠} (2.20)

which is analogous to Equations 2.10 and 2.15. The constant, c2, is the speed of propagation

of lateral displacements. The solution of Equation 2.20, when the right side equals zero, is analogous to that for bars and shafts. But the cable motion is not limited to lateral displace-ments only. The element has a longitudinal stiffness as well, so it may be viewed also as a bar, especially when its initial shape is straight. Accordingly, Equation 2.11 again defi nes the speed of longitudinal waves. In fact, both components of motion are almost invariably

u x P u΄ u P w dx u΄+u˝ dx mÜ dx FIGURE 2.3 Dynamic equilibrium of a cable element.

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coupled in any real cable, although they do not necessarily have the same importance. This matter is treated in greater detail in Chapter 9.

The shear beam is the fourth member of the group. The lateral defl ection of a two-dimensional (2D) beam has two components: bending and shear. The fi rst is due to curving of the axis, while the second results from distortion of elements normal to the axis. When only the latter is accounted for, we speak of a shear beam, as depicted in Figure 2.4.

This model can be thought of as consisting of thin layers capable of shear distortion, but infi nitely rigid in the direction normal to their thickness. The shear angle, θ, and the associ-ated tip defl ection are defi ned by

s Q GA θ = (2.21a) and s QL u L GA = θ = (2.21b) where

G is the shear modulus As is the shear area

When the lateral force is constant along the length, as in Figure 2.4, Q is the same for every vertical slice of the beam.

Although some elements behave nearly like ideal shear beams, the main usefulness of this concept is in visualizing the shear component of defl ection of actual beams. The infl u-ence of shear is likely to be substantial when one or more of the following conditions take place:

1. The beam is short, say, a cantilever whose length is only twice its depth.

2. The cross section is hollow or branched (as opposed to a compact section like a solid circle or a rectangle).

3. The beam is sandwich type with a lightweight core.

By defi nition, the elastic rotation of a cross section cannot take place in a shear beam. From this viewpoint, it makes no difference whether the end section is completely fi xed or merely restrained against translation. The fi xity in Figure 2.4 is needed for rigid-body equilibrium.

u

θ

G, m, As

L FIGURE 2.4 Shear beam deformation.

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The equation of motion is analogous to that of a bar, a shaft, and a cable. The speed of propagation of a shear disturbance is

1/2 s 3 GA c m ⎛ ⎞ = ⎜_⎝ ⎟_⎠ (2.22)

Equations of Motion: Flexural Beam

The beam element in Figure 2.5a is in the state of dynamic equilibrium. The bending stiff-ness, EI (N-m2_{), is assumed constant, and so is the mass per unit length, m = A}ρ_{. The}

equa-tion of moequa-tion is of the fourth order with respect to x:

iv

mu+EIu =w (2.23)

where both u and w are functions of time, t, and position, x. The other important equations for the beam segment are

″ =

EIu _M (2.24)

″ =

EIu _Q (2.25)

Equation 2.24 is the relation between the bending moment and the curvature of the beam axis. Equation 2.25 may be obtained from the previous one by differentiation and by noting that M′ = Q, if the higher order terms are ignored in the angular equilibrium of the beam element. Equation 2.23 may be obtained by differentiation of Equation 2.25 and by writing the equation of vertical equilibrium for Figure 2.5a.

When the equation of motion includes the terms relating to shear deformation and rotary inertia, it is called a Timoshenko beam equation. It is generally agreed that the rotary effects are small in most practical applications, while the additional effects of shearing distortions seem to be best included by applying one of the approximate methods.

The boundary conditions involve defl ection and slope (Figure 2.5b) as well as shear forces and bending moments. The general form of the solution for free vibration (w = 0) is the same as that given by Equation 2.17. Likewise, the solution procedure is the same.

A beam element may be thought of as a cable, additionally endowed with a fl exural stiff-ness. In a conventional design, where small defl ections are involved, the longitudinal and

FIGURE 2.5 Flexural beam element: (a) dynamic equilibrium; (b) displacement and slope. x u u΄ (b) u (a) w dx dx Q Q + Q΄dx x mÜ dx + M M M΄dx

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the geometric stiffness are not very important and are ignored. For intermediate defl ec-tions, a cable-like resistance may become predominant. (Refer to Chapter 11.)

Beams on Elastic Foundation

An elastic foundation means, in effect, a dense row of springs with stiffness kf per unit

length of beam. The springs are independent from one another (much like a railroad track supported on wooden ties, which lie on the ground). As a result, a beam element in Figure 2.5 experiences a downward reaction, kf u dx, and the equation of motion becomes

iv

f

mu+EIu = −w k u (2.26)

where the exponent “iv” means fourth-order integration with respect to u. Addition of the foundation increases the natural frequencies. It can be shown, for example, that for a simply supported beam the circular frequencies can be expressed by

4 2 i i EI m L π ⎛ ⎞ ω = ⎜ ⎟_{⎝ ⎠} (2.27)

for i = 1, 2, 3…, while, after addition of the foundation, the frequencies become [111]

4 f 2 2 f 1 i i k L EI i ⎛ _{⎛ ⎞} ⎞ ω = ω _⎜ + _{⎜ ⎟}_{⎝ ⎠} _⎟ π ⎝ ⎠ (2.28)

while our interest is usually limited to the fi rst mode, i = 1. The foundation modifi es natural frequencies, but has no infl uence on the mode shapes. The relation between natural fre-quencies with and without a foundation, expressed by Equation 2.28, applies to beams with other end conditions as well. A more detailed discussion of beams on elastic foundations is available in Chapter 10.

Continuous Elastic Medium

When a rigid block is attached to the surface of a semi-infi nite medium, it can be treated as supported by elastic springs. The constants of these springs are given in Cases 2.48 and 2.49. The basic formula for a natural frequency can then be used with an appropriate mass (be it for translation or rotation) and a prescribed spring constant. This, however, is only the fi rst approximation. The medium also has some inertia, which gives rise to the increase in the effective mass of the block. The reader can fi nd more information in Ref. [17].

T

YPESOF

S

TRUCTURAL

A

RRANGEMENTS

There are a variety of structural arrangements, that is, ways in which the elements can be connected to each other. The two cases that are found most often are referred to as in series

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and in parallel. In the fi rst instance, the load is the same in all members, and therefore, the defl ections (due to fl exibilities) are additive. The resultant fl exibility, 1/k*, is thus

1 2

1 1 1 _... 1

* n

k = k + k + + k (2.29)

with the summation extended to all n members of the system. In a parallel connection, the displacement of all members is the same and their loads and rigidities (stiffnesses) are addi-tive. The resultant stiffness of a parallel connection is

1 2 ...

* n

k = + + +k k k (2.30)

As the cases presented later show, structural elements in a series connection are often placed along one continuous line, while in a parallel connection they are located side by side, as in Figure 2.6. The appearance, however, may sometimes be misleading, and the defi nitions given above must always be kept in mind.

A T

WO

-M

ASS

O

SCILLATOR

A two-mass oscillator represents an important system, to which many real confi gurations are reducible. The two masses in Figure 2.7 are connected with a weightless elastic bar of stiffness k. There is no external force applied, therefore each mass is subjected to only the resistance of the elastic element during vibrations. If, at any instant, the stretching force is Ps, then one has

1 1 s

M u =P (2.31a)

FIGURE 2.6 Structural arrangements: (a) parallel and (b) in series.

k1 k2

(b)

k2

k1

(a)

FIGURE 2.7 A vibrating two-mass oscillator. (A1 and A2 stand for amplitudes of displacement.)

A1

A2

M1

u1 u2

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and

2 2 s

M u = −P (2.31b)

This indicates that the ratio of accelerations is inversely proportional to the ratio of masses, that is 1 2 2 1 u M u = −M (2.32)

For a system vibrating with a frequency ω, one can put

1 1sin

u =A ωt (2.33a)

and

2 2sin

u = −A ωt (2.33b)

From this and Equation 2.32, after differentiation, the ratio of amplitudes becomes

1 2

2 1

A M

A = M (2.34)

as illustrated in Figure 2.7. In this way, the system is equivalent to two oscillators, both having the base at the stagnation point determined by Equation 2.34. If the system is initially at rest, then not only accelerations, but also displacements and velocities are proportional to the same ratio of masses. For a series connection, Equation 2.29 holds, so one can fi nd, for the left spring, for example, k1 as a function of k and M1/M2, which leads to the

determina-tion of the natural frequency as

1 2 1 1 2 1 1 1 where * * k k M M M M M ω = = = + (2.35)

The resulting effective mass, M*, is smaller than any of the two component masses. This system has two degrees of freedom (DOFs), therefore two natural frequencies, at least nom-inally, must be present. One of them is associated with vibrations, as determined above, and the other one relates to rigid-body motion, which means ω = 0.

G

EOMETRIC

S

TIFFENING

There is an important effect associated with slender structural elements, namely, elements that have (at least) one dimension considerably smaller than the other two. If the loading of a member of this type is carried out in two steps, the magnitude of defl ections in the sec-ond step may depend on the magnitude of loads in the fi rst. This effect is called geometric stiffening, and is best illustrated by the behavior of a rod that has one end pinned and the

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other end free. As long as the element has no axial loading, it can be freely rotated. If, however, a horizontal stretching load is applied fi rst and an attempt is then made to defl ect the free end in the direction perpendicular to the initial force, a resistance is encountered. This axial force, which remains here parallel to the original direction, is called the pre-load or prestress.

Consider a rigid, pin-ended bar shown in Figure 2.8, which illustrates one possible arrangement. When the horizontal force P0 is zero, the angular stiffness of the system is K.

The presence of P0 makes it more diffi cult to rotate the bar because of the additional

moment, P0Lα, opposing the rotation. The total apparent stiffness is thus

0 g

K= +K P Lα = +K K (2.36)

where Kg is called the geometric stiffness. When the axial force is directed opposite, so that

it places the bar in compression, the apparent stiffness K– is less than the elastic stiffness K. If the magnitude of the compressive force is suffi cient, K– may become zero, which means that no resistance is offered to rotation. We then say that the system becomes unstable and write cr 0 K−P L= (2.37a) or cr / P =K L (2.37b)

where Pcr is the critical force or the buckling force. Using this concept, we can put the

expression for the apparent stiffness in a different form:

0 cr 1 P K K P ⎛ ⎞ = +_⎜ _⎟ ⎝ ⎠ (2.38)

in which positive P0 means tension. When J is the mass moment of inertia of the bar

with respect to the pivot point, the natural frequency in the absence of P0 is calculated from

ω2_{= K/J. Taking the axial force into account merely modifi es the stiffness; therefore,} 0 2 2 cr 1 P P ⎛ ⎞ ω = +_⎜ _⎟ω ⎝ ⎠ (2.39) M α P0 L K M

Formulas for Mechanical and Structural Shock and Impact BBS

Gregory Szuladzi ´nski

Formulas

for

Mechanical

and

Structural

Contents

Preface

Introduction

Author

Symbols and Abbreviations (General)

SYMBOLS

ABBREVIATIONS

1

Concepts and Deﬁ nitions

THEORETICAL OUTLINE

S

N

F

T

A

T

R

M

U

I

F

D

E

E

S

S

F

2

Natural Frequency

THEORETICAL OUTLINE

D

N

F

D

M

E

M

B

C

M

D

T

S

A

A T

-M

O

G

S