A VARIATIONALAPPROACHTO STRUCTURALANALYSIS

DAVID V. WALLERSTEIN

A Wiley-Interscience Publication

JOHN WILEY & SONS, INC.

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A VARIATIONALAPPROACHTO STRUCTURALANALYSIS

A VARIATIONALAPPROACHTO STRUCTURALANALYSIS

DAVID V. WALLERSTEIN

A Wiley-Interscience Publication

JOHN WILEY & SONS, INC.

Copyright 2002 by John Wiley & Sons, New York. All rights reserved.

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To Christina and Edward

vii

CONTENTS

PREFACE xi

1 INTRODUCTION 1

2 PRELIMINARIES 7

2.1 Variational Notation / 72.2 The Gradient / 102.3 Integration by Parts / 112.4 Stokess Theorem / 132.5 Greens Theorem in the Plane / 152.6 Adjoint Equations / 162.7 Meaning of 2 / 192.8 Total Differentials / 202.9 Legendre Transformation / 21

2.10 Lagrange Multipliers / 242.11 Differential Equations of Equilibrium / 272.12 Strain-Displacement Relations / 292.13 Compatibility Conditions of Strain / 332.14 Thermodynamic Considerations / 35

Problems / 38

viii CONTENTS

3 PRINCIPLE OF VIRTUAL WORK 40

3.1 Virtual Work Definition / 403.2 Generalized Coordinates / 413.3 Virtual Work of a Deformable Body / 423.4 Thermal Stress, Initial Strain, and Initial Stress / 473.5 Some Constitutive Relationships / 483.6 Accounting for All Work / 513.7 Axially Loaded Members / 533.8 The Unit-Displacement Method / 603.9 Finite Elements for Axial Members / 65

3.10 Coordinate Transformations / 713.11 Review of the Simple Beam Theory / 743.12 Shear Stress in Simple Beams / 923.13 Shear Deflection in Straight Beams / 953.14 Beams with Initial Curvature / 993.15 Thermal Strain Correction in Curved Beams / 1123.16 Shear and Radial Stress in Curved Beams / 1143.17 Thin Walled Beams of Open Section / 1213.18 Shear in Open Section Beams / 1473.19 Slope-Deflection Equations / 1553.20 Approximate Methods / 165

Problems / 173

4 COMPLEMENTARY VIRTUAL WORK 196

4.1 Complementary Virtual Work Definition / 1964.2 Complementary Virtual Work of a Deformable

Body / 1974.3 Symmetry / 2104.4 The Unit Load Method / 2174.5 Force Elements / 2354.6 Generalized Force-Displacement Transformations / 239

Problems / 242

5 SOME ENERGY METHODS 261

5.1 Conservative Forces and Potential Functions / 2615.2 Stationary Potential Energy / 2715.3 Castiglianos First Theorem / 2745.4 Complementary Energy / 277

CONTENTS ix

5.5 Stationary Complementary Potential Energy / 2805.6 Engesser-Crotti Theorem / 2825.7 Variational Statements / 2875.8 The Galerkin Method / 2905.9 Derived Variational Principles / 300

Problems / 3056 SOME STATIC AND DYNAMIC STABILITY CONCEPTS 318

6.1 Linear-Stability Analysis / 3186.2 Geometric Measure of Strain / 3226.3 A Beam with Initial Curvature Revisited / 3306.4 Thin Walled Open Beams Revisited / 3376.5 Some Stability Concepts / 3496.6 Energy Criterion of Stability / 3506.7 Stiffness / 3536.8 Stiffening and Unstiffening Models / 3606.9 Bifurcation Analysis / 369

6.10 Imperfection Analysis / 3726.11 Circulatory Dynamic Stability / 3776.12 Instationary Dynamic Stability / 384

Problems / 388REFERENCES 396

INDEX 401

xi

PREFACE

My objective in writing this book has been to provide a discourse on the treat-ment of variational formulations in deformable structures for upper-level under-graduate through graduate-level students of aeronautical, civil, and mechani-cal engineering, as well as engineering mechanics. Its self-containment is alsodesigned to be useful to practicing professional engineers who need to reviewrelated topics. Emphasis is placed on showing both the power and the pitfallsof virtual methods.

Today, analysis in structures, heat transfer, acoustics, and electromagnet-ics currently depends on the finite element method or the boundary elementmethod. These methods, in turn, depend on virtual methods or their generaliza-tion represented by techniques such as the Galerkin method.

The notes for this book come from a graduate aerospace course in struc-tures that I have taught at the University of Southern California for the lasteighteen years. The students come from such diverse backgrounds as mechani-cal engineering, structural engineering, physics, fluid dynamics, control theory,and, occasionally, astronomy. At least half of an average class is made up ofgraduates working in the industry full-time. The majority view the course asa terminal course in structures, but only about half have a firm understand-ing of structures or plan to make a career in the subject. All, however, needan understanding of virtual methods and structures to survive in the industrialworld. The control theory expert must be able to convey his or her needs to thestructural-modeling group. It is highly embarrassing to find out three years intoa project that there has been a lack of communication between design groups.

Without the necessity of memorizing numerous formulas, virtual meth-ods provide a logical, unified approach to obtaining solutions to problems

xii PREFACE

in mechanics. Thus students have a readily understood method of analy-sis valid across all areas of mechanics, from structures to heat transfer toelectricalmechanical, that will remain with them throughout their careers.

Students often want to be taught the finite element method without hav-ing any basic understanding of mechanics. Although my notes touch in anextremely basic way on finite element methods (FEM), I try to impress on stu-dents that FEM codes are nothing more than modern slide rules that are onlyeffective when the underlying mechanics is fully understood. All too often, Ihave observed in my twenty-odd years as a principal engineer for the worldslargest FEM developer that even in industry, there is a blind use and acceptanceof FEM results without consideration of whether the model is a good mathe-matical representation of the physics. That said, however, I should also statethat I have encountered clients with a far better understanding of the use of thecode and its application to physics than the code developers.

The text is divided into six chapters, with discussion basically limited tosprings, rods, straight beams, curved beams, and thin walled open beams (whichrepresent a specialized form of shells). Springs and rods were selected becausethe reader can easily understand them, and they provide a means in nonlineardiscussions to get closed solutions with very general physical interpretations.The three classes of beams were chosen for several reasons. First, my yearsin aerospace and my current interaction with both auto-industry and aircraft-industry clients indicate that beams are still a very important structural com-ponent, as they are used extensively in both automobile and aircraft structuresand to model such complex nonlinear problems as blade-out conditions on air-craft engines. Second, beams can be used to clearly demonstrate the ease withwhich virtual methods can be applied to determine governing equations. Third,beams can be used to clearly demonstrate when and why virtual methods willyield inconsistent results. Fourth, beams yield equations that can generally besolved by the student. And fifth, the physics of the solution can be discussedin relationship to what an FEM code can perform.

The first two chapters are mainly introductory material, introducing the varia-tional notation used and reviewing the equilibrium and compatibility equationsof mechanics. Since variational methods rely heavily on integration by partsand on the variational operator functioning in a manner similar to a total dif-ferential, these techniques are discussed in great detail. Though the text itselfdoes not extensively use the concept of adjoint operators, a section is includedhere in an attempt to bridge the gap between basic mechanics courses, whichmake no mention of the subject, and advanced texts, which assume extensiveknowledge of the subject. Legendre transformations are introduced for later usein establishing the duality between variational methods and complementary vir-tual methods. Lagrange multipliers are discussed, and a possible physical inter-pretation is given.

The third chapter covers virtual work. It uses kinematical formulations forthe determination of the required strain relationships for straight, curved, andthin walled beams. The importance of accounting for all work is emphasized,

PREFACE xiii

and it is pointed out that if a particular strain assumption (such as the theoryof plane sections) does not contain certain strain components, virtual work willlose information. It is also pointed out, however, that if additional assumptionsconsistent with the fundamental constraints are included, then virtual work canyield more general results. Examples of this are discussed with the straightbeam theory, where the assumption that the tangent to the deflection curve isnormal to the plane section is relaxed, yielding a shear deflection theory forstraight beams. For curved beams, the requirement that curved beams do notdistort in the plane of the cross section under thermal loading is relaxed; theremoval of this restriction is required to get correct thermal stress results forcurved beams. In thin walled beams of open section, the fundamental relation-ship is that the shear strain of the median surface is zero. By defining a shearstrain consistent with this requirement, the Saint Venant torsion theory for thinwalled beams of open section is automatically included via virtual work. Addi-tionally, this chapter shows how easily virtual work concepts can produce theresultant curvature-stress relationships or the differential equations for deflec-tion curves. One observation made in teaching the material of this chapter isthat the homework problems that seem to pose the most difficulty when it isasked to have them solved with the use of virtual work are those types usuallyfound early in a strength course involving stress and strain with axial load-ing. In these problems, the student really has to think in selecting a virtual dis-placement.

The fourth and fifth chapters cover complementary virtual work and energymethods. These are problem-solving chapters. For complementary virtual work,emphasis is put on the selection of virtual loads that meet the requirement ofstatic equilibrium and answer the question: Does the selected virtual set yieldresults useful to the solution of the problem at hand? Virtual load diagramsare used extensively for complementary virtual work solutions. In Chapter 5,stationary potential energy, Castiglianos first theorem, and the Engesser-Crottitheorem are derived and used in the solution of problems. Variational statementsare discussed and generalized with the introduction of the Galerkin method.Here, the concept of single- and multipoint constraints is introduced. Also intro-duced are derived variational principles, which are important because even insupposed displacement FEM analysis, mixed methods are often used in elementformulations and can also have utility in optimization methods.

The sixth chapter discusses some static and dynamic stability concepts. Var-ious geometric measures of strain are introduced. Straight, curved, and thinwalled beams are revisited and studied in a deformed geometry. For curvedbeams, different forms of curvature arise from different parametric represen-tations of curvature. These are shown to be equivalent. For thin walled beamsof open section, there is no conjugate variable for transverse shear. Thus it isshown that a blind use of virtual work yields incorrect stability equations. Clas-sic equilibrium techniques are then used to correct the equations obtained byvirtual work. General stability concepts are then introduced. Two of the mostimportant influences on stability are the type of loading and the concept of stiff-

xiv PREFACE

ness. Instationary and stationary loading is discussed, with the latter brokendown into gyroscopic, dissipative, circulatory, and noncirculatory categories.Stiffness is broken down into that provided by the material and that providedby the internal loading. Generalized stiffness leading to secant methods of anal-ysis are also introduced. Then, classic rigid-link mass-spring models are usedto gain an understanding of the various forms of stability.

I would like to thank Mrs. Julie Suva of Beyond Words for doing the figuresfor this book. Also, it has been a real pleasure working with Mr. Bob Hilbertof Wiley, who kept the whole process going.

DAVID V. WALLERSTEIN

1

CHAPTER 1

INTRODUCTION

The object of this course is to provide an introductory treatment of variationalformulations in structural analysis. A knowledge of these formulations alone,however, is not adequate in reaching engineering solutions for large or complexstructures. Currently, such solutions are most commonly obtained by means ofa matrix method classified as the finite element method. Since most modernfinite element methods are formulated via variational methods, their inclusionfollows in natural fashion.

The power of the variational formulations is that they yield a systematic wayof deriving the governing equations and corresponding boundary conditions thatrelate to the behavior of a structure under loading. Indeed, the governing equa-tions they yield are in general nothing more than the equations of equilibriumor the equations of compatibility.

The boundary conditions are usually classified as two types:

1. Forced boundary conditions, which are mathematical expressions of con-straint.

2. Natural boundary conditions, which are necessary conditions for station-ary requirements.

The boundary conditions for a free edge of a plate is a classic example inwhich a variational procedure routinely yields the correct results, while bound-ary conditions obtained by manipulation of the fourth order governing differ-ential equations yield inconsistent results. Figure 1.1 shows a rectangular platethat is clamped on three edges and free of restraint on the fourth edge. Thestress resultants shown on the free edge are My, Myx, and Qy, which repre-

2 INTRODUCTION

QyMy

Myx

x

y

z

Figure 1.1 Plate with free edge.

sent the bending moment normal to the y axis, the twisting moment, and thetransverse shear on the free edge, respectively.

The natural assumption is that on the free edge, My 0, Myx 0, and Qy 0. However, these conditions will not generally satisfy the differential equationof the plate. A variational approach yields the correct boundary conditions.

My 0 (1.1)

Qy +Myx

x 0 (1.2)

Just as variational formulations yield a systematic procedure in the derivationof the governing equations and their boundary conditions, matrix methods vis-a-vis the finite element method yield a systematic procedure in the solutionof these equations. These methods are based on the concept of replacing theactual continuous structure by a mathematically equivalent model made upfrom discrete structural elements having known properties expressible in matrixform.

The range of application of the variational formulations and hence the finiteelement formulations fall into various categories, such as the following:

Equilibrium problemstime independent Displacement distributionstructures Temperature distributionheat Pressure or velocity distributionfluid Eigenvalue problemssteady state Natural frequencies

INTRODUCTION 3

1 2

1 2f1,u1 f2,u2

f1 f2k u2 u1( ) k u2 u1( )

k

Figure 1.2 Spring element and nodal equilibrium.

Structural stability Lamina flow stability Propagation problemstime domain

The finite element method itself yields a major bonus to problem solving.

Example 1.1 Consider first a simple elastic system discretized into a compos-ite of simple spring elements. Figure 1.2 shows one typical such element. ByHookes law, the force f in the spring is related to the spring elongation ue bythe following relationship:

f kue

where k is the spring stiffness. The elongation ue is related to the end displace-ments at ends 1 and 2 by the following expression:

ue u2 u1

Equilibrium of a free body at end 1 (node 1) requires that the applied force f 1at end 1 be related to the end displacements by the following relationship:

f 1 ku1 ku2

At end 2 (node 2), equilibrium requires the following:

f 2 ku1 + ku2

These two expressions for equilibrium may be combined into a single matrixequation of the following form:

k [ 1 11 1 ] { u1u2 } { f 1f 2 }The 2 2 matrix is called the element-stiffness matrix.

4 INTRODUCTION

x

A = Constant

21

T T x( )=

H1, T1 H2, T2

Figure 1.3 Heat flow through a rod.

Example 1.2 As a second example, consider the heat flow through the rodshown in Fig. 1.3. Fouriers law of heat conduction gives the following expres-sion:

h k(d T/ dx)where

h is the steady-state heat input per unit area ( joule/ m2)k is the thermal conductivity ( joule/ m/ C)T is the temperature (C)A is the area (m2)l is the length (m)

H1, h1A is the total heat input at node 1 (joule)H2, h2A is the total heat input at node 2 (joule)

Consider a linear temperature distribution of the following form:

T a + bx

where a and b are constants. Evaluating the equation for temperature distribu-tion at end 1 (x 0) and end 2 (x l), we obtain the following relations:

a T1

and

b (T2 T1)/ l

where T1 and T2 are the rod-end temperatures. The temperature distributioncan then be expressed as follows:

T 1 x/ l x/ l { T1T2 }

INTRODUCTION 5

with d T/ dx expressed as

d T/ dx 1/ l 1/ l { T1T2 }Fouriers law of heat conduction then takes the following form:

h (k/ l ) 1 1 { T1T2 }where is a row matrix and { } is a column matrix.

The total heat input at node 1 and node 2 can then be expressed as:

Ak/ l [ 1 11 1 } { T1T2 } { H1H2 }The 2 2 matrix is called the element-conductivity matrix.

While the two element matrices formed in these two examples have quitedifferent physical meanings, they are identical in mathematical form. Further-more, their assembly into a global system is identical. This characteristic assem-bly procedure is one of the great advantages of the finite element procedure.The equation solver used in the finite element method need never know thephysical system being solved.

Variational methods and finite element methods are applicable to generalcontinua with both geometric and material nonlinearities. When solving struc-tural problems with these methods, however, there are additional requirementsthat must be met. For example, when using these methods to solve the problemof torsion of bars, the strain-displacement assumptions and the ramifications ofthese assumptions on the nonvanishing components of stress must be knownbeforehand. For the bar shown in Fig. 1.4, the Saint Venant torsion assumptions[1] are given as follows:

ex ey ez exy 0

exz w/ x vyeyz w/ y + vxw vw(x, y)

j xz Gexzj yz Geyz

where

6 INTRODUCTION

y

z

x

m

m

Figure 1.4 Torsion in a circular bar.

es are the strainsj s are the shear stresses

G is the torsion modulusw, a function of (x, y), is the warping functionv , a constant, is the rate of twist produced by constant end twisting

momentsw is the component of displacement along the z axis

Similarly, for beams, the Bernoulli-Euler hypothesis, the more generalVlasov hypothesis, or even your own hypothesis must be decided in advanceof any variational method.

7

CHAPTER 2

PRELIMINARIES

2.1 VARIATIONAL NOTATION

Variational principles are in their most general form covered by the methodsof the calculus of variations [2]. Their application to mechanics in general isfound in its most elegant form in The Variational Principles of Mechanics byLanczos [3].

We will in general discuss pertinent aspects of the calculus of variations asneeded. The outstanding difference between calculus and variational calculuswhen we seek stationary values is that, in the former, we seek the stationaryvalues of a function, whereas in the latter, we seek the stationary value of afunctional, which is usually a definite integral of the following form:

V x1

x0

F(x, y, y, y, . . . , y(n)) dx

To discuss variational principles, it is necessary to discuss variational nota-tion. The most important aspect of this notation is the d process.

Let f be a function of the independent variable x. Figure 2.1 shows a plot ofy f (x), and df is a change along the curve f (x) from a change in x. Define anew function as follows:

f *(x) f (x) + e h(x) f (x) + df (x)

8 PRELIMINARIES

dy df fdx= =

y f x( )=

f* x( )

y

f x( )

dx

x

Figure 2.1 Variational notation.

where h (x) is a function with at least continuous first derivatives and e is asmall parameter. We define the variation in f (x) as follows:

df (x) dy

f *(x) f (x)

e h(x)

where df is a change to a different curve with the independent variable x fixed.For the one-dimensional function shown in Fig. 2.1, the variation is the verticalmovement dy.

Consider the commutative properties of the d operator:

d(dy)/ dx d[ f *(x) f (x)]/ dx d[e h(x)]/ dx e h(x)

Next, take the derivative of f *(x) and f (x) and define as follows:

d(dy/ dx) f *(x) f (x) [y(x) + e h(x)] y(x) y(x) + e h(x) y(x) e h(x)

Thus, on comparing the above two expressions for e h(x), we may write thefollowing:

VARIATIONAL NOTATION 9

ddydx

ddydx

(2.1)

This shows that the derivative of the variation is equal to the variation of thederivative. In general, the properties of d are similar to the properties of theoperator d.

Next, consider the behavior of the d operatorwith integration:

d f (x) dx f *(x) dx f (x) dx [ f *(x) f (x)] dx df (x) dx (2.2)

Thus, we see that the d operator may be taken in or out of the integral at ourconvenience.

There is often a restriction placed on h (x): namely, h (a) h (b) h (a) h (b) 0. These restrictions impose the forced boundary conditions. Relaxingthese restrictions results in the so-called natural boundary conditions.

In our discussion of d, we treated d as an operator. This implies that thereis a function f that can be operated on by an operator d to give df . There arevirtual quantities for which we can define, say, da as a symbol, but with theunderstanding that a has not been defined. In this case, d does not operate on a,but rather da is a complete, self-contained symbol and the d cannot be separatedfrom the a. For example, in dynamics we may define a virtual rotation [4] dasuch that

da n

r 1

wqr dqr

In this equation, wqr are the nonholonomic rates of change of orientation andqr are the generalized coordinates.

While

d(dqr)/ dt d(dqr/ dt)

would be appropriate, no function a is defined; hence

d(da/ dt) d(da/ dt)

10 PRELIMINARIES

dr n d+ C dC+=

C=

Figure 2.2 Surface contour.

2.2 THE GRADIENT

We may define the gradient by the following expression:

f grad f (df/ dn)n (2.3)where n is the unit normal to the surface f shown in Fig. 2.2 and grad isthe del operator.

Equivalently, the gradient [5] may be defined as follows:

df f . dr (2.4)To see how Eqs. (2.3) and (2.4) arise, consider again Fig. 2.2, from which we

observe that dC is the actual increase of f from the first surface to the secondsurface. The rate of increase depends upon the direction and is expressed as

dCdr . dr

The expression

dr . dr has a minimum value for dr || n and a magnitude ofdn. Then, for unit normal n, n dn is the least value for dr, and (dC/ dn) nrepresents the most rapid rate of increase from the first surface to the secondsurface. Thus we may write

dCdn

n dfdn

n

as the defining equation for the gradient. If we note that dn n . dr and thendot Eq. (2.3) with dr, Eq. (2.4) follows immediately.

From the above definitions for gradient, it should be observed that the gra-dient is independent of any coordinate system. Also, the del operator is inde-pendent of any coordinate system. In a rectangular cartesian coordinate system,the del operator takes the following form:

i x

+j y

+ k z

(2.5)

INTEGRATION OF PARTS 11

In general, the del operator is defined as

limDV 0

1DV V dS

where DV is a simply connected volume enclosing the point P about which thedel operator is applied, and dS is the bounding surface to the enclosed volume.

A useful mnemonic for the del operator based on the above definition is

/ x ds/ dS dS/ d V (2.6)

In the mnemonic, ds is a measure of arc, dS(dS ) is a measure of area, and dVis a measure of volume.

2.3 INTEGRATION OF PARTS

For one-dimensional analysis, if the integrand can be expressed as the productof two functions u(x)v(x) u(x) dv(x)/ dx, then

p2

p1

u(x)v(x) dx u(x)v(x)||||

p2

p1

p2

p1

u(x)v(x) dx (2.7)

is the formula for integration by parts.For two dimensions and three dimensions, we start with the divergence the-

orem. Consider the two-dimensional case shown in Fig. 2.3. The divergencetheorem [by Eq. (2.6)] may be written as

n

dS ndS=n

ds

t

N t n=

boundary

ds ds N=

dr ds t=

dr

Figure 2.3 Closed surface and boundary.

12 PRELIMINARIES

S . A dS S A . ds (2.8)Define A as

A fw

Then, . A becomes

. A f . w + f2w

Substitute the above expressions for A and . A into Eq. (2.8) with the fol-lowing result:

S f . w dS + S f2w dS S fw . ds (2.9)Define

l N . i cos(N,i )

m N . j cos(N,j )

Then Eq. (2.9) becomes

S [(f/ x) (w/ x) + (f/ y) (w/ y)] dS S f2w dS + S f[l(w/ x) + m(w/ y)] ds (2.10)

For three dimensions, replace dS with dV and ds N ds, with dS n dS, toobtain

V [(f/ x) (w/ x) + (f/ y) (w/ y) + (f/ z) (w/ z)] d V V f2w d V + V f[l(w/ x) + m(w/ y) + n(w/ z)] dS

(2.11)

where

STOKESS THEOREM 13

l n . i cos(n,i )

m n . j cos(n,j )

n n . k cos(n, k)

Equation (2.11) is often called Greens first theorem. It can be obtained directlyfrom Eq. (2.7) by term-by-term integration by parts of the left side of Eq. (2.11).For example, consider the first term as follows:

Vfx

fx

dx dy dz V xfx

wx

l dS dx

where l dS dy dz. Let u (w/ x)l dS and let dv (f/ x) dx; then applyEq. (2.7) to obtain

V fwx

l dS V f2wx2

d V

which represents the x terms of the right side of Eq. (2.11). The other termsfollow in a similar fashion.

In three dimensions, V A . n dS is called the flux (or flow) of A throughthe surface and represents

the average normal component of A the surface area

In addition, . A represents the rate per unit volume per unit time that A isleaving a point. Then, V . A is the total flow from the volume and V A . dSis the total flux through the surface.

2.4 STOKESS THEOREM

Stokess theorem relates to a closed curve and any surface bounded by thiscurve. The curve and surface are orientated by positive tangent direction t andpositive normal direction n according to the right-hand rule. In the plane of dS,an element of surface area

ds N ds

represents any element of arc on the boundary of dS. The unit vectors (t, n, N)form a right-handed orthogonal system, with

N t ndr t ds

14 PRELIMINARIES

representing an element of the closed curve bounding the surface. These rela-tionships are shown in Fig. 2.3.

We are interested in determining a relationship for the line integral of A. Fig-ure 2.3 shows the surface divided into smaller regions with common boundariesof neighboring regions. Along these boundaries, all line integrals appear twicein both directions. Thus by summing up all smaller line integrals we find thatthe net result is the line integral of the boundary curve. Therefore we need onlyconcentrate our attention on the triad (dr, dS, ds ) on the subregions along theboundary curve.

Using the second form for , as shown in Eq. (2.6), and taking the vectorcross-product with a vector A, we may write

A dsdS

A

Multiplying through the dS and by integrating, we get

S A dS S ds ANext, by dotting both sides with the unit vector n normal to the surface shownin Fig. 2.3, we obtain

S n . A dS S n . ds A S n ds . A S t . A ds

Thus we may write

S A . dS S A . dr (2.12)The S A . dr in Eq. (2.12) is called the circulation and represents

the average tangential component of A the distance around the contour