E. Byskov, Elementary Continuum Mechanics for Everyone,Solid Mechanics and Its Applications 194, DOI: 10.1007/978-94-007-5766-0_6,Ó Springer Science+Business Media Dordrecht 2013

Chapter 6

The Idea of SpecializedContinuaIntroductionIn this Part we develop theories for various types of structural elements,such as beams and plates. We may derive the theories in several differentways. The first two mentioned here are often used, while the third—whichI find much more safe and satisfactory—receives less attention.

• Always a New Topic

One possibility consists in regarding each type as a completely newtopic and establish the formulas without regard to general continuummechanics—and to some extent also knowledge about other kinds ofstructural elements. This is often done in courses on mechanics simplybecause the students have not been taught continuum mechanics andbecause of tradition.

• Simplified Three-Dimensional Body

Another way is to consider a structural element as a three-dimensionalbody subject to some constraints, be they of a kinematic or staticnature. This procedure works well in many cases, but it is oftendifficult to see whether the set of formulas derived in this way areinternally consistent, e.g. whether a principle of virtual work applies.

In Part III we shall use this method for a particular purpose, namelyto determine cross-sectional properties of beams, not to establish atheory for beams as such.

• Specialized Continuum

The one used here differs from the first ones in that here we insistthat general principles form the basis for the theories. In particular,we shall demand that the principle of virtual work,6.1 is valid for ourtheory.

First, we define the so-called generalized kinematic quantities, i.e. gen-

6.1 Either the principle of virtual forces or, almost always, the principle of virtualdisplacements.

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The Idea of Specialized Continua

eralized displacements, which e.g. may include rotations,6.2 and pos-tulate the relations between the generalized displacements and thegeneralized strains, which may also entail change in curvatures.6.3

How we select the relations between generalized displacements andgeneralized strains is the crux of the procedure.6.4 In many cases, thisdoes not cause problems, but under some circumstances, for instancefor beams with strong initial curvatures, the choice may not be thatobvious.

When we have set up these relations we stick them into the principleof virtual displacements and, after a number of manipulations, we mayget the static equations, i.e. the equilibrium equations that connectthe generalized stresses to the generalized loads.6.5

This means that the static quantities defined in this way are gener-alized in the sense that they are the work conjugate of the properkinematic quantities, i.e. that the applied generalized loads producevirtual work together with the appropriate generalized displacementsand that the generalized stresses and the generalized strains do thesame.

In this way we make sure that our theory is valid, for instance for thetheorems concerning extrema. Only when the static quantities are thework conjugate of the kinematic ones does the Principle of MinimumValue of the Potential Energy for elastic structures apply, and thesame is the case of the upper and lower bounds of the load-carryingcapacity of structures made of a material obeying perfect plasticity.Otherwise you may find that upper and lower bound solutions changeplace, which must never happen.

I shall try to explain my reason for saying that the third procedure isconcerned with specialized continua. While Euclidean geometry describesthe three-dimensional world, Riemannian geometry is concerned with othertypes of “spaces,” such as the two-dimensional surface of a sphere. In orderfor a theory to be valid for the description of such specialized spaces it mustinclude certain subjects, and among the most central ones are a measure oflength etc. through a metric. In much the same way a beam or plate may beconsidered to be a specialized continuum when we insist that it obeys somefundamental principles, most importantly the principle of virtual work.

6.2 In order to describe beam and plate bending some measure of rotation is needed.6.3 Again, bending of beams and plates necessitates such generalized strains.6.4 The choice of strain-displacement relations determines whether we get a good theory.6.5 In the case of kinematic nonlinearity the static equations may include displacements,

as we have seen in Chapter 2.

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