[Solid Mechanics and Its Applications] Elementary Continuum Mechanics for Everyone Volume 194 ||...

E. Byskov, Elementary Continuum Mechanics for Everyone,Solid Mechanics and Its Applications 194, DOI: 10.1007/978-94-007-5766-0_33,Ó Springer Science+Business Media Dordrecht 2013

Chapter 33

Budiansky-HutchinsonNotationIn many instances it proves convenient to utilize a more compact notationthan one which entails the use of indices. There are several such notationsavailable, and here we choose the so-called Budiansky-Hutchinson Notation(Budiansky & Hutchinson 1964) which is introduced below. Another, quitepopular, notation utilizes dyads and polyads, see e.g. (Malvern 1969).33.1

33.1 Linear, Quadratic and Bilinear Opera-

torsNote that the Lagrange Strain as given by (2.19) is a the sum of a termwhich is linear in the displacements (and their gradients) and a term whichis quadratic in the displacements (and their gradients)

Lagrange strain

γmnγmn = 1

2 (um,n + un,m) + 12uk,muk,n (33.1)

or, with a change of notation33.2

εmn = 12 (um,n + un,m) + 1

2uk,muk,n (33.2)

If our concern is not so much the individual components of the strain,or for the purpose of deriving general statements,33.3 we may write (33.2)in the following form Generalized

strain-

displacement

relation

ε = l1(u) +12 l2(u) (33.3)

which may, at a first glance, seem to contain almost no real information, but,as we shall see later, it is part of a convenient basis for general statements.We shall also see how it can be interpreted for various kinds of structures.

In (33.3), u denotes all components of the generalized displacements,33.4 Generalizeddisplacement u

Generalized strain ε33.1 For our present purposes, I have judged that it offers too little in compensation fora more involved set of rules, but acknowledge that under other circumstances it certainlyhas its important virtues.33.2 In most of our applications the strain tensor is called εmn and not γmn, whether itis a linear or nonlinear measure.33.3 In this book focus is on the general statements.

August 14, 2012 Continuum Mechanics for Everyone Esben Byskov

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Mathematical Preliminaries

l1 is a Linear Operator working on u, l2 is a Quadratic Operator on u, andε signifies all components of the generalized strains.33.5

The linearity of l1 is given by the following rules

Linear operator l1 l1(ku) = kl1(u) and l1(ua + ub) = l1(ua) + l1(ub) (33.4)

where k is a constant, and ua and ub are two different displacement fields.

The operator l2 shows its quadratic property in the following rule, whichis the analogy of (33.4)

Quadratic

operator l2l2(ku) = k2l2(u) (33.5)

In order to establish a rule for the quadratic operator l2, which corre-sponds to (33.4b), it proves necessary to introduce a Bilinear Operator l11by

Bilinear operator

l11l2(ua + ub) = l2(ua) + 2l11(ua,ub) + l2(ub) (33.6)

where it is obvious that l11 is symmetric

l11 is symmetric l11(ua,ub) = l11(ub,ua) (33.7)

and that

l11(u) = l2(u) l11(u,u) = l2(u) (33.8)

The bi-linearity of l11 shows itself in the relation

l11(kaua, kbub) = kakbl11(ub,ua) (33.9)

where ka and kb are two constants.

We may easily see that the original formula (33.2) behaves in accordancewith the rules given above for the operators l1, l11 and l2. In order to doso, we write (33.2) in the more explicit form

εmn(uaj + ubj) = + 1

2 (uam,n + ubm,n + uan,m + ubn,m)

+ 12 (u

ak,m + ubk,m)(uak,n + ubk,n)

(33.10)

where the labeling of the two displacement fields now is given by upperindices a and b.

33.4 Sometimes the generalized displacements include rotations which is the case forTimoshenko beam theories, see Sections 7.4 and 7.6 as displacements.33.5 For instance, the generalized strains may include axial, shear (in Timoshenko beams)and curvature strains in beams, and in plates membrane, shear (in Mindlin plates),bending and torsional strains in plates, see Part II.

Esben Byskov Continuum Mechanics for Everyone August 14, 2012

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Budiansky-Hutchinson Notation

Elementary manipulations on (33.10) provide

εmn(uaj + ubj) = + 1

2

(uam,n + uan,m

)+ 1

2uak,mu

ak,n

+ 12

(ubm,n + ubn,m

)+ 1

2ubk,mu

bk,n

+(uak,mu

bk,n + ubk,mu

ak,n

)(33.11)

and, clearly, this is equivalent to

ε(ua + ub) = + l1(ua) +12 l2(ua)

+ l1(ub) +12 l2(ub)

+ l11(ua,ub)

(33.12)

For more interpretations of l1, l2 and l11, see Part II, where expressions forthese operators are given for beams and plates.

33.2 Principle of Virtual DisplacementsThe Budiansky-Hutchinson notation proves particularly convenient in con-nection with principles such as the Principle of Virtual Displacements, seeSection (2.4), where the principle is written in (2.97)

Principle of

Virtual

Displacements for

δui = 0, xj ∈ S0u

∫

V 0

tijδγijdV0 =

∫

S0T

τiδuidS0 +

∫

V 0

qjδujdV0

∀ δuj = 0, xj ∈ S0u

(33.13)

see Section 2.4.1 where the Budiansky-Hutchinson Notation is also men-tioned.

In order to write this equation in a compact fashion we introduce twomore symbols in addition to u and ε. The first is σ and denotes the gener- Generalized stress σ

Generalized load Talized stresses.33.6 The second is T and signifies the generalized prescribedloads, which may include point loads, line loads, surface tractions and bodyforces.

Further, let a dot ( · ) between two fields indicate a (generalized) Inner A dot ( · ) implies

inner productProduct of the fields. In the present, three-dimensional case the dot impliesvolume integrals over the body of the internal virtual work and of the workof the body forces, while it signifies surface integrals over the relevant partsof the surface of the work of the surface loads (and in the case of (2.95),which does not assume δuj = 0 on S0

u, also the work of the reactions onS0u).

With the Budiansky-Hutchinson Notation in hand we may write eitherof the principles of virtual displacements (2.95) or (2.97) in the very short

33.6 For instance, the generalized strains may include normal and shear (in Timoshenkobeams) force and bending moments in beams, and in plates membrane and shear (inMindlin plates) stresses as well as bending and torsional moments, see Part II.


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form, see also (2.98)Principle of

Virtual

Displacements

σ · δε = T · δu (33.14)

The interpretation of the dot may require some further explanation, sohere is a list which is not complete:33.7

• In some cases the dot just indicates the product of two quantities. Forinstance, this is the case of the internal work done by a spring force and(the variation of)33.8 its associated elongation. This interpretation isalso valid for a point load and its displacement.

• The dot may imply a sum as well as a product, for instance over anumber of springs or point loads and their displacements.

• The internal virtual work in bars and beams are formed as integralsover the length of the body, so here the dot implies integration.

• In a plate or a shell the internal virtual work is given as a surfaceintegral over the body meaning that here, the dot signifies integration.The external virtual work in these structures may be the result of“body forces”33.9 working with the displacements over the body and,again it is a surface integral. Along the boundaries and at line loadsthe dot implies line integrals.

• For three-dimensional bodies the internal virtual work is given by avolume integral, and some external work may be associated with bodyforces whose virtual work is also computed as a volume integral. Forother contributions to the external virtual work in this case you mayquite easily apply some of the above definitions.

Before we can derive or apply the principle of virtual displacements weneed to know the expression giving δε in terms of u and δu. In the spiritof Section 2.4 let the total displacement field be utot, where

utot = u+ ǫδu (33.15)

and the total strain field εtot

εtot = ε+ ǫδε+O(δε2) where |ǫ| ≪ 1 (33.16)

Here, |ǫ| ≪ 1 is the amplitude of the variations whose shapes are givenas δu and δε.

Then, (33.3)–(33.9) make it possible to write the following expressionfor the total strain provided that it is composed by a linear and a quadraticterm in the displacements, which certainly is the case for the LagrangeStrain, the strains of the kinematically moderately nonlinear beam theory

33.7 For cases which are not in the list you should be able to extend it yourself.33.8 In the interest of brevity the following I omit “the variation of.”33.9 For the meaning of that term see Section 9.1.4.


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developed in Sections 7.3 and 8.2 and of the plate theory of Chapter 9. Wemay get

εtot = l1(utot) +12 l2(utot)

= l1(u+ ǫδu) + 12 l2(u+ ǫδu)

= l1(u) +12 l2(u) + ǫ(l1(δu) + l11(u, δu)) + ǫ2

(12 l2(δu)

)

= ε+ ǫ(l1(δu) + l11(u, δu)) +O(ǫ2)

(33.17)

with the consequence that

Strain variation δεδε = l1(δu) + l11(u, δu) (33.18)

which means that we may express the principle of virtual displacements(33.14) in the alternative way

σ ·

(l1(δu) + l11(u, δu)

)= T · δu (33.19)

At this point we may discuss the implications and use of the principleof virtual displacements.

• We may try to establish it separately for each kind of structure theway we did it for the three-dimensional body in Section 2.4, whichlead to (2.97) which also appears above as (33.13). To do this we needto be able to derive the equilibrium equations for the structure weconsider and this is not always an easy task.

• The other way is to postulate that the principle, given by (33.14)or (33.19), is valid and derive the static equations, the equilibriumequations, under this assumption. This is how we exploit the principlein Part II.

In the present context the most important implication of the principle of Stresses and strainsmust be generalized

= Work conjugatevirtual work is that we insist that all strain and stress quantities of theoriesvalid for specialized continua, such as beams and plates, are Generalized inthat they are work conjugate in the sense of (33.14). Therefore, in almostall cases, we shall choose the second of the above possibilities.

33.3 Variation of a PotentialBefore we proceed with the more continuum mechanics oriented materialwe may take a break and realize that we have talked about variations anumber of times, but admit that each time it has been in an ad hoc fashion.In order that the following derivations do not get cluttered in the same waywe interject this section.

Let us consider the functional Π(α), which—as indicated—depends on Potential Π(α) and

its variation δΠ(α)the field33.10 α and define the variation δΠ(α) ofΠ(α) by aGateaux Deriva-

33.10 You may think of α as a displacement field, if you like, but it is probably better notto do so.


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tive, see e.g. (Budiansky 1974)

δΠ(α) by

Gateaux

derivative

δΠ(α) = limǫ→0

Π(α+ ǫδα)−Π(α)

ǫ(33.20)

This may also be written

δΠ(α) δΠ(α) =∂Π(α+ ǫδα)

∂ǫ

∣∣∣∣ǫ=0

(33.21)

These are the two most common and simple notations, or definitions, ofthe variation of a potential. There exist a number of other, more sophisti-cated and general definitions, see e.g. (Budiansky 1974), but the ones givenhere should suffice for our purposes.

33.4 Potential Energy for Linear ElasticityThe potential energy, see Section 2.7, may also be expressed in a convenientform by use of the operator and dot notation. This is especially true whenthe constitutive model is linear, i.e. expresses linear (hyper)elasticity, see(2.101)

Linear constitutive

modelW (γij) =

12Eijklγijγkl (33.22)

and (2.102)

Linear constitutive

modeltij = Eijklγkl (33.23)

When we introduce the compact notationLinear constitutive

operator Hσ = H(ε) (33.24)

where H is a linear constitutive operator, then the potential energy (2.106)

Potential energy

ΠP(ui)ΠP(ui) =

∫

V 0

W (γij)dV0 −

∫

V 0

qiuidV0 −

∫

S0T

τiuidS0 (33.25)

may be writtenPotential energy

ΠP(u)ΠP(u) =

12H(ε) · ε− T · u (33.26)

Note that ε depends on u, which justifies the notation ΠP(u).

The following reciprocity relation is a consequence of the nature of theconstitutive “law,” namely that it is linearly elastic and, therefore, thestresses depend directly on the strains and not on the loading history

Reciprocity

relationH(εa)εb = H(εb)εa (33.27)

where subscripts a and b indicate two different fields. Note that the state-ment is valid without a dot, i.e. it is valid for all points of the structure,individually.


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A result of this is thatIntegrated

reciprocity

relation

H(εa) · εb = H(εb) · εa (33.28)

It is worth noticing that in (33.27) and (33.28) one of the fields may bethe variation δε.

33.4.1 Stationarity of the Potential Energy for LinearElasticity

By application of (33.28) and of the operator rules given above in Sec-tion 33.3 the variation of ΠP is

Variation of

Potential EnergyδΠP(u) = H(ε) · δε− T · δu (33.29)

where the conditions on the variations are given below.

By comparison of (33.29) and (33.14) and using (33.24) we may realizethat the first variation δΠP of the potential energy ΠP vanishes for theexact solution in that

Principle of

Stationary

Potential Energy

δΠP(u) = 0 (33.30)

because (33.30) in connection with (33.29) shows that the requirementδΠP = 0 is equivalent to the principle of virtual displacements (33.14),but with the stresses expressed in terms of the constitutive relation (33.24).Realizing this, we may conclude that the variations must be kinematically Variations must be

kinematically

admissible

admissible. Thus, the variations must be “sufficiently” smooth and satisfythe homogeneous kinematic boundary conditions.

We shall refer to (33.30) as an expression of the Principle of StationaryPotential Energy.

33.4.2 Minimum of the Potential Energy for LinearElasticity and Kinematic Linearity

For linear elasticity and kinematic linearity the potential energy is not onlystationary at the exact solution, it is also a minimum, which we prove below.Let the exact solution be denoted u and an approximate solution (whichsatisfies the kinematic conditions) be u = u+ ǫδu, where ǫ is the amplitudeof the difference between u and u. Note that δu satisfies the homogeneousboundary conditions.

Assuming kinematic linearity

Kinematically

linear strainε = l1(u) and ε = l1(u) + ǫl1(δu) (33.31)

the potential energy of (33.26) for the exact field u becomes

ΠP(u) for

kinematic linearityΠP(u) =

12H(l1(u)) · l1(u)− T · u (33.32)

and for the approximate field u the expression for the potential energy looks


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just as simple

ΠP(u) for

kinematic linearityΠP(u) =

12H(l1(u)) · l1(u)− T · u (33.33)

but, when we note that l1 is a linear operator, see also (33.31b), we may seethat it is more complicated

ΠP(u) =12H(l1(u+ ǫδu)) · l1(u + ǫδu)− T · (u+ ǫδu)

= 12H(l1(u)) · l1(u)− T · u

+ ǫ(H(l1(u)) · l1(δu)− T · δu

)

+ 12ǫ

2H(l1(δu)) · l1(δu)

(33.34)

The term of order ǫ is the Principle of Virtual Displacements with σ =H(l1(u)) and therefore vanishes. For ǫδu 6≡ 0 the term of order ǫ2 is greaterthan zero because the strain energy is positive definite and is quadratic inl1(δu). Since the term of order ǫ0 is the potential energy for the exact fieldwe may conclude that

ΠP is minimum

for the exact

solution

ΠP(u) > ΠP(u) , u 6≡ u (33.35)

which is a much stronger statement than that of stationarity, but we canget an even more useful result, see the next section.

33.4.3 Minimization of the Potential Energy for Lin-ear Elasticity and Kinematic Linearity Resultsin Too Stiff Structures

Below, as in Section 33.4.2, we assume kinematic and constitutive linearity.In most cases, the fact that the potential energy for an assumed displace-ment field is greater than for the exact one does not provide us with veryΠP(u) ≥ ΠP(u) is

useful, but we want

more

valuable information, except that it can be used to choose the best of aset of approximate solutions. Rather, we would like to know the quality ofthe displacements or the stresses, and in particular their maximum values.As regards the displacements we show that use of an assumed displacementfield, which is different from the exact one, in the principle of stationarity ofthe potential energy results in a behavior which is too stiff.33.11 The stresses,however, present more severe difficulties, and we shall not address the ques-tion of their properties except to mention that the stress field usually followsfrom the displacement field by a process which entails differentiation, and,therefore, the stresses usually are of poorer quality than the displacements.

From (33.35) it follows that

ΠP(u) =12H(ε) · ε− T · u < ΠP(u) =

12H(ε) · ε− T · u (33.36)

where the meaning of ε should be obvious. According to (33.30), we know

33.11 As you will see, the proof is quite simple, but—much to my surprise–many bookson finite element theory do not touch it.


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that for the exact displacement

δΠP(u) = 0 ⇒ H(ε) · δε− T · δu = 0 (33.37)

and similarly we demand that

δΠP(u) = 0 ⇒ H(ε) · δε− T · δu = 0 (33.38)

Provided that all prescribed displacements are zero, i.e. provided that allkinematic boundary conditions are homogeneous, we may introduce δu = u

in (33.37) and δu = u in (33.38) to get

H(ε) · ε = T · u and H(ε) · ε = T · u (33.39)

respectively. Note that in (33.39a) the field u and in (33.39b) the field u

are the ones determined by (33.37) and (33.38), respectively. Thus, bothfields are fixed in the present context and it would therefore not make senseto vary with respect to these fields in (33.39a) and (33.39b).33.12 Insert(33.39a) and (33.39b) in (33.36)

min(ΠP(u)) for

(u 6= u) ⇒ too

stiff structure

− 12T · u < − 1

2T · u ⇒ T · u > T · u (33.40)

which means that the work done by the loading T is greater for the exactsolution. This statement is very useful because it tells us that among a setof approximate solutions we should always choose the one that entails thelargest work of the loading.

33.4.3.1 Single Point Force

When the loading consists of only a single point force, say P at point P wecan derive a very strong statement regarding its work conjugate displace-ment uP . In this case, (33.40) yields

P uP > P uP (33.41)

and, since P is the same in both cases, we may conclude thatDisplacement of

load is largest for

exact solution

uP > uP (33.42)

implying that the value of the “characteristic” displacement uP is larger forthe exact solution than for any other. Therefore, an assumed displacementfield u 6= u, which is used in the principle of stationarity of the potentialenergy results in predicting too stiff a structural behavior. The result (33.42)is even more convenient than (33.40) since, for the case of a single point forceP , among approximate solutions we should choose the solution for whichthe displacement uP of the force is the largest.33.13

33.12 We may also see this from the fact that, otherwise, we would get results that disagreewith (33.37) and (33.38).33.13 In the above derivations it is tacitly assumed that the characteristic displace-ment is finite. This is not the case for a plate loaded in-plane with a point force, seee.g. (Muskhelishvili 1963).


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33.5 Complementary Energy for Linear Elas-ticity

As in the case of the potential energy it proves convenient to formulatethe complementary energy in terms of the Budiansky-Hutchinson Notation,see Section 33.4. Below, as in Section 33.4.2, we assume kinematic andconstitutive linearity. Noting (4.134) we may easily see that the equivalentof (33.26) is

Complementary

energy ΠCΠC(σ) =

12C(σ) · σ − T · u (33.43)

where I emphasize that ΠC(σ) is valid only for kinematic linearity, whereu denotes the prescribed displacements, T are the reactions to u, and thelinear constitutive operatorC is the inverse ofH and signifiesHooke’s “Law,”and, formally, we may write

Linear constitutive

relation C = H−1 C = H−1 ⇒ ε = C(σ) (33.44)

see (4.129).

Variation of ΠC(σ) providesVariation of

complementary

energy ΠC

δΠC(σ) = C(σ) · δσ − δT · u (33.45)

and requiring that δΠC(σ) vanishes for all statically admissible stress vari-ations, i.e. stress variations satisfying the equilibrium equations, results in

C(σ) · δσ = δT · u (33.46)

The Principle of Virtual Forces, see (4.124), is expressed as

Principle of

Virtual Forcesε · δσ = δT · u (33.47)

and when the constitutive relation (33.44) is introduced, we recover (33.46).This is comforting because it means that requiring δΠP(σ) = 0 is equivalentto to the principle of virtual forces with the strains expressed in terms ofthe stresses.

33.5.1 Minimum Complementary Energy

Let σ denote the exact solution to some problem and let σ = σ + ǫδσ,where, as usual |ǫ ≪ 1|, be another field, an approximate solution, whichsatisfies the equilibrium equations. Then, the complementary energy of theapproximate solution is

Complementary

energy ΠC(σ) is

minimum for the

exact solution

ΠC(σ) =12C(σ + ǫδσ) · (σ + ǫδσ)− (T + δT ) · u

= 12C(σ) · σ + ǫC(σ) · δσ + 1

2ǫ2C(δσ) · δσ

−T · u− ǫδT · u

= ΠC(σ) +12ǫ

2C(δσ) · δσ

≥ ΠC(σ)

(33.48)


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The terms of order ε1 vanish because of (33.46), and thus the Comple-mentary Energy attains a minimum for the correct solution.

33.5.2 Minimization of the Complementary Energy forLinear Elasticity and Kinematic Linearity Re-sults in Too Flexible Structures

As a consequence of (33.48)

ΠC(σ) =12C(σ) · σ − T · u ≤ ΠC(σ) =

12C(σ) · σ − T · u (33.49)

Here, σ and T denote fields that are associated with an approximationwhich satisfies all equilibrium conditions. Variation of the complementaryenergy for the exact and for the approximate field provides

δΠC(σ) = 0 ⇒ C(σ) · δσ − δT · u = 0 (33.50)

and

δΠC(σ) = 0 ⇒ C(σ) · δσ − δT · u = 0 (33.51)

respectively. Since the stress fields (σ,T ) and (σ, T ) satisfy the same con-

ditions as (δσ, δT ) and (δσ, δT ), respectively, we may get

C(σ) · σ = T · u and C(σ) · σ = T · u (33.52)

and thus

ΠC(σ) = − 12T · u and ΠC(σ) = − 1

2 T · u (33.53)

with the comment that in (33.53a) and (33.53b) variations are not permit-ted because the stress fields occurring here are those given by (33.50) and(33.51), and therefore are fixed fields.

Because of the result (33.48) we get33.14

min(ΠC(σ)) for

(σ 6= σ) ⇒ too

flexible structure

− 12 T · u ≥ − 1

2T · u ⇒ T · u ≤ T · u (33.54)

This result is in itself helpful because it says that the work done bythe reactions of the approximate field is less than (or equal to, if we haveguessed right) that of the exact field, i.e. the structure is predicted to bemore flexible than the real one.

33.14 I hope that you can see that this derivation looks very much like the one which weperformed in Section 33.4.3. This is one of the many examples of the duality betweenstatics and kinematics.


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33.5.2.1 Single Point Force

Now, the question is whether the structure responds by a too flexible behav-ior when it is subjected to a load instead of prescribed displacements. Letthe prescribed displacement be confined to a particular point and denote itu, then for the reactions, which we shall denote P and P , the relation is33.15

P ≤ P (33.55)

and thus the value of the reaction to a prescribed displacement is underes-timated if the approximate stress field differs from the exact one, meaningthat the structure is predicted to be too flexible. Based on this, it seemsa reasonable conjecture that the approximate solution, based on the mini-mum of the complementary energy, also for an applied load predicts a largerdisplacement than the correct one. The effect of distributed loads may befound by use of integration of point loads.

Ex 33-1 A Clamped-Clamped BeamAs an example of use of the Principle of Minimum Complementary

Energy consider the clamped-clamped beam shown in Fig. Ex. 33-1.1.The elastic properties, given by the bending stiffness EI , and the loadp are independent of x.

x, ξ

p = −q(1− ξ)

L

w

Fig. Ex. 33-1.1: A clamped-clamped Beam.

We shall only deal with the transverse displacements of the beam, and,therefore, the complementary energy is

Complementaryenergy ΠC(M)

ΠC(M) = 12

∫ L

0

1

EIM2dx (Ex. 33-1.1)

where the bending momentM must satisfy the auxiliary condition33.16

Load p = q(1− ξ) M ′′ = p = −q(1− ξ) , q = const. > 0 , ξ ≡ x/L (Ex. 33-1.2)

33.15 The remarks in the footnote p. 589 regarding point forces are still valid but must beunderstood in the sense that here the reaction to a prescribed displacement in a plate,which is subjected to in-plane loading, is zero.33.16 Note that because q is negative the load actually points downwards.


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We may get an admissible moment field M by use of the field forthe equivalent simply supported beam, see later, and add an arbitrarylinear field, but it is even easier instead of using the beam result justto integrate twice in (Ex. 33-1.2) to get

Assumed bendingmoment M(ξ)

M(ξ) = r(ξ) + vβ(1) + vβ(2)ξ (Ex. 33-1.3)

where vβ(j) are stress field parameters and

Load term r(ξ)r(ξ) = − 16qL2ξ2(3− ξ) (Ex. 33-1.4)

By this choice the only pertinent equilibrium equation, namely (Ex. 33-1.2) is satisfied—and there are no other possible moment fields whichsatisfy this equation. Changing the notation slightly (Ex. 33-1.3) maybe written

Assumed bendingmoment M(ξ)

M(ξ) = [Nβ ]{vβ}+ r (Ex. 33-1.5)

where

Stress matrix [Nβ ]Stress fieldparameters {vβ}

[Nβ ] = [1 ; ξ]

{vβ}T = [vβ(1) ; vβ(2)](Ex. 33-1.6)

Introduce the constitutive matrix [C], which is the material flexibilitymatrix, by

Constitutivematrix [C]

[C] =

[1

EI

](Ex. 33-1.7)

Rewrite (Ex. 33-1.1)


ΠC(M) = 12L

∫ 1

0

([Nβ ]{vβ}+ r

)T[C]([Nβ ]{vβ}+ r

)dξ (Ex. 33-1.8)

or


ΠC(M) = 12{vβ}T [kββ ]{vβ}+ {vβ}T {rβ}+ const. (Ex. 33-1.9)

where the constant term is independent of {vβ} and, therefore, con-tributes nothing to the variation of ΠC, and

Coefficient matrix[kββ]Right-hand sidevector {rβ}

[kββ] ≡ L

∫ 1

0

[Nβ ]T [C][Nβ ]dξ

{rβ} ≡ L

∫ 1

0

[Nβ ]T [C]rdξ

(Ex. 33-1.10)

where [kββ] is a flexibility matrix and {rβ} is the right-hand side vectorof the ensuing matrix equation.


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Variation of ΠC provides

Variation ofΠC(M)

δΠC(M) = δ{vβ}T([kββ ]{vβ}+ {rβ}

)(Ex. 33-1.11)

When we require that the variation of ΠC vanishesSystem ofequations

[kββ]{vβ} = −{rβ} (Ex. 33-1.12)

With {vβ} in hand we may determine the bending moments from(Ex. 33-1.5).

In the present caseCoefficient matrix

[kββ]Right-hand side

vector {rβ}[kββ] =

L

EI

[1 1

212

13

]and {rβ} = − qL3

8EI

[11115

](Ex. 33-1.13)

with the solution

Solution vector{vβ} {vβ} = − qL

2

20

[1

−7

](Ex. 33-1.14)

and thusResulting bending

moment M(ξ)M = − 1

60qL2(3− 21ξ + 30ξ2 − 10ξ3

)(Ex. 33-1.15)

The support moments are

Supportingmoments

M(0) = − 120

and M(1) = − 130qL2 (Ex. 33-1.16)

where it makes sense that the absolute value of the support moment islarger at the end where the load intensity is the higher.

The maximum value Mmax of the bending moment is

Maximumbending moment

Mmax

Mmax =−3√30− 30

300qL2 = −0.0214389qL2

for ξ = 1−√30

10= 0.4522774

(Ex. 33-1.17)

where it is reasonable to expect the maximum bending moment tooccur closest to the highest load intensity.

Ex 33-1.1 Is Our Solution the Exact One?

You may check against the exact solution and see that indeed M =MWe may see that Mis the exact solution in this case. When you think about it, this is not surprising because we

started out with a field comprising the only possible components of theexact field, namely the first term of (Ex. 33-1.3), which is a particularsolution of the governing differential equation (Ex. 33-1.3), and theonly two terms, the second and third term of (Ex. 33-1.3), which to-gether constitute the full solution to the corresponding homogeneousdifferential equation. At a first glance it seems remarkable that wemay find a solution to a statically indeterminate problem without in-voking the kinematic boundary conditions. However, since (Ex. 33-1.1)does not contain any boundary terms, cf. (33.43), this means that theboundary displacements are all prescribed to be zero. But, becauseour solution is formulated in terms of the bending moment, which may


566


be expressed by the constitutive relation and the strain-displacementrelation, only the second derivative of the transverse displacement isgiven. Thus, the transverse displacement is determined to within a lin- We need to do

more to get thedisplacements

ear function, and in order to find the displacement we must integratetwice and enforce the kinematic boundary condition explicitly. Thiswill not be done here.

Ex 33-1.2 Another, Better(?) Assumption

The choice (Ex. 33-1.3) is, of course, not the only possible one. Wemay, for instance, choose

A choice whichmakesinterpretation of{vβ} transparent

M(ξ) = 16qL2ξ(1− ξ)(2− ξ) + [Nβ ]{vβ}

with [Nβ ] = [(1− ξ) ; ξ](Ex. 33-1.18)

The first term on the right-hand side of the expression for M is thesolution for the equivalent simply supported beam, and the secondterm has the convenient property that here we may then identify vβ(1)and vβ(2) as the bending moments at the supports.

33.6 Auxiliary ConditionsIt is worthwhile noticing that there are a number of conditions that mustbe satisfied by the fields in the principle of minimum of the potential energyand in the principle of minimum of the complementary energy. Here, weshall focus on the potential energy and, as far as the complementary energyis concerned, refer to Chapter 27 where the issue of auxiliary conditions onthe complementary energy is treated in some detail.

For the potential energy as given by (33.26) the most fundamental ofthese conditions, i.e. those which must always be satisfied, are kinematicconditions

1. The displacements u must be sufficiently smooth.2. The displacements u must satisfy the kinematic boundary conditions.3. The strains ε must be derived from the displacements u according to

(33.3).4. The displacement variations δu must be sufficiently smooth.5. The displacement variations δu must satisfy the homogeneous kine-

matic boundary conditions, i.e. δu must vanish on the kinematicboundary.

6. The strain variations δε must be derived from the displacements u

and displacement variations δu according to (33.18).33.17

In addition to these conditions, there may be others, which are specificto the particular problem at hand, e.g. inextensibility of a beam or a mem-brane; incompressibility of a solid; a condition, which expresses a couplingbetween a number of displacement components; etc.

33.17 Obviously, when we utilize the alternative formulation of the Principle of VirtualDisplacements (33.19) we satisfy (33.18) right from the outset.


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Such conditions are called Auxiliary Conditions or Side Conditions.Auxiliary conditions

= side conditions Sometimes it proves to be impossible to satisfy a number of these condi-tions, and then the method of Lagrange Multipliers, see Section 33.7, comesin handy.

33.7 Lagrange MultipliersIt is not always possible or feasible in advance to satisfy all auxiliary condi-Lagrange multiplier

tions that apply to the Principle of Virtual Displacements or the Principleof Stationarity Potential Energy, see Sections 33.2 and 33.4. If this is thecase, the method of Lagrange Multipliers, see e.g. (Arfken & Weber 1995),proves to be valuable. This is particularly true in connection with approx-imate solutions, but here we concentrate on the derivation of the methoditself.

33.7.1 Principle of Virtual Displacements

Suppose that we must fulfill an auxiliary condition on the Principle of Vir-tual Displacements, and that the condition may be written

Ψ(u) = 0 (33.56)

where it is indicated explicitly that the condition depends on the displace-ments.33.18 Then, we may augment the Principle of Virtual Displacements(33.14) with a term δ(Ψ(u) · η) and write

Lagrange

multiplier (field) ησ · δε− T · δu+ δ(Ψ(u) · η) = 0 (33.57)

where η denotes the Lagrange Multiplier (Field).33.19 The last term of(33.57) is, of course, the most interesting in the current connection. Writtenout (33.57) becomes

σ · δε− T · δu+ δΨ(u) · η +Ψ(u) · δη = 0 (33.58)

Now, because there are no conditions on δη, except maybe continuityconditions, (33.58) implies that

Ψ(u) = 0 (33.59)

as required.

33.18 You may think of this condition as one of incompressibility or some other kinematicconstraint.33.19 Since the term Lagrange Multiplier is used in honor of the French mathematicianJ.L. Lagrange, the symbol η is somewhat unnatural and, indeed, in most literaturesymbols such as λ or Λ are used. The reason why I have chosen η instead is that theGreek letter lambda, frequently λ, in literature associated with structural stability is themost commonly used symbol for load parameter. In the present context this justificationis, admittedly, not a very valid one, but all authors have their quirks, and I hope thatyou—and Lagrange—will forgive me.


568


Further, (33.59) means that

δΨ(u) = 0 (33.60)

and therefore (33.14) is recovered. Just from reading the above derivation Are Lagrange

Multipliers at all

useful?

the idea of applying Lagrange Multipliers may seem like an awkward way ofwriting the original equations, but in some cases—in particular in connectionwith approximate solutions—the method proves to be extremely efficient,see for instance Example Ex 32-5.2.

33.8 Interpretation of the Budiansky-Hutch-

inson Notation for Selected ExamplesThe previous derivations may be rather straightforward to follow, but my ex- Interpretation of

Budiansky-

Hutchinson

Notation

perience is that application of the formalism may be less obvious. Therefore,we interpret the Budiansky-Hutchinson Notation for some of the examplesgiven in Part II and in earlier sections of Part VI and begin with the latter.

33.8.1 Interpretations Related to Example Ex 32-2

For the one-degree “structure” One-degree

“structure”u ∼ v , ε ∼ v , H(ε) ∼ cv

σ · δε = T · δu ∼ Fδv = P δv(33.61)

where F = cv is the force in the spring.


For the “structure” with two degrees of freedom Two-degree

“structure”

u ∼ {v} =

[v1

v2

], ε ∼ {v} =

[v1

v2

](33.62)

H(ε) ∼ [c]{v} ∼[c1 0

0 c2

][v1

v2

](33.63)

σ · δε = T · δu ∼ c[F1 F2

][δv1

δv2

]= P δv (33.64)


For the structure with auxiliary conditions Structure with

auxiliary conditions,

the Euler columnu ∼ w(x) , ε ∼ κ(x) = ε , H(ε) ∼ EIκ(x)

Ψ(u) = 0 ∼ (κ− w′′) = 0 , η ∼ η

σ · δε = T · δu ∼∫ a

0

(Nδε+Mδκ)dx = −P δu(L)(33.65)


569


33.8.4 Interpretations Related to Sections 7.3 and 7.7

For the moderately kinematically nonlinear Bernoulli-Euler beams we citeModerately

kinematically

nonlinear

Bernoulli-Euler

beam

the following relations, which can also be found in Section 7.3

u ∼[u

w

], ε ∼

[ε

κ

]= l1(u) +

12 l2(u) (33.66)

l1(u) ∼[u′

w′′

], l2(u) ∼

[(w′)2

0

], l11(u

a,ub) ∼[w′

aw′b

0

](33.67)

σ ∼[N

M

], H(ε) ∼

[EA 0

0 EI

] [εκ

](33.68)

σ · δε = T · δu

∼∫ b

a

(Nδε+Mδκ)dx

=

∫ b

a

(puδu+ pwδw)dx

−Pu(a)δu(a)− Pw(a)δw(a) − C(a)δw′(a)+Pu(b)δu(b) + Pw(b)δw(b) + C(b)δw′(b)

(33.69)

33.8.5 Interpretations Related to Section 9.1

For the kinematically moderately nonlinear plane plates we cite the followingKinematically

moderately

nonlinear plane

plates

relations, which can also be found in Chapter 9

u ∼[uα

w

], ε ∼

[εαβ

καβ

]= l1(u) +

12 l2(u) (33.70)

l1(u) ∼[

12 (uα,β + uβ,α)

w,αβ

], l2(u) ∼

[w,αw,β

0

](33.71)

l11(ua,ub) ∼

[12

(wa

,αwb,β + wb

,αwa,β

)

0

](33.72)

H(ε) ∼[EA 0

0 EI

][ε

κ

]σ ∼

[Nαβ

Mαβ

](33.73)

σ · δε =

∫

A0

(Nαβδεαβ +Mαβδκαβ) dA0 (33.74)


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