MODELLING OF MECHANICAL SYSTEM VOLUME 2 Episode 3 ppt

How-ever, the interesting point here is that by solving the same problem along the Oy and Oz axes instead of Ox, we find much lower stresses which are, Thus, the idea which arises natura

hold, the dilatational (P ) waves are reflected without mode conversion Again, it

is stressed that sliding supports would be difficult to manufacture Incidentally, ifthe solid is replaced by a non-viscous fluid, i.e a medium in which the Poissoneffect and the shear stresses do not exist, the present calculation takes on its full

practical interest In a fluid, the dilatation waves are called acoustic waves and the natural modes defined by the formulas [1.136], [1.137] are the acoustic modes of

a rectangular enclosure provided with rigid and motionless walls

1.4.7.3 Shear plane modes of vibration

Starting from the plane shear waves [1.86], it is immediately verifiable thatshear stresses σxy = GdY /dx and σxz = GdZ/dx are the only nonzero stresscomponents They both must vanish at the faces x = 0 and x = Lx which areassumed to be free Whence the boundary conditions:

dYdx

#

#

0= dYdx

#

#

Lx

= 0; dZdx

#

#

0= dZdx

the corresponding natural modes of vibration can be split into two families whichare identical to each other, except that they vibrate into two orthogonal directions.The common modal quantities are:

where the vector (anj
+ bn
k) specifies the direction of vibration, that is the ization of the wave Clearly, any transverse direction is possible and equivalent

polar-to the others It can be also noted that modes at zero frequency exist, which areimmediately recognized as a rigid displacement of the whole body in an arbitrarytransverse direction

1.5 From solids to structural elements

1.5.1 Saint-Venant’s principle

The difficulty in obtaining the analytical solutions of elastodynamic problems isalso encountered when attempting to solve static problems, as for instance that ofdetermining the stresses in a three-dimensional body induced by forces applied onsmall portions of it As an example, let us consider a cylindrical rod inserted in atensile test machine, see Figure 1.23 If a numerical simulation of the experiment isperformed, by using the finite element method (regarding the finite element methodsee Chapter 3, section 3.4), quite enlightening results are obtained concerning thedisplacement field and the stress distribution, especially sufficiently near the grips.Major information can be suitably summarized as follows:

1 Near the loaded parts of the rod, in a portion extending no more than a fraction

of the rod radius in each direction, the displacement field, and even moreconspicuously the stress field, are found to vary in all directions, in connectionwith the detailed space distribution of the contact forces exerted by the grips

Such fields characterize the so called local response of the system rod plus

grips

Figure 1.23 Cylindrical rod set in a tensile test machine

2 In the current portion of the rod, sufficiently far from the grips, the stress field

is found to reduce practically to only one non-vanishing component, namelythe uniform axial stress σzz(r, θ , z) = T /S, where cylindrical coordinates r,

θ, z are used Here, T stands for the resultant of the tensile force system

exerted by the grips and S denotes the cross-sectional area of the rod Thedisplacement field is marked by an axial component denoted by W and aradial component denoted by U , which are independent of the polar angle

θ W is observed to increase in proportion to z, starting from the fixed grip,and U varies in proportion to r All these fields are independent of the dis-

tribution of the loading by the grips and characterize the global response of

the rod

As can be expected from the simplicity of the response observed sufficientlyfar from the grips, the analytical solution of the problem becomes straightforward,provided it is formulated in a suitable way A first idea is to adopt the model of acircular cylinder loaded at its bases by a uniform axial force density whose resultant

is T The problem can be thus solved by using a 2D model, since the θ dependency

is removed Further, if the interest is restricted to determining the axial ment and the axial stress, a 1D model becomes sufficient Finally, to determine

displace-σzz only, it becomes unnecessary to model the rod as an elastic body because

a single DOF system suffices This elementary example emphasizes, if necessary,the importance of modelling mechanical systems in close relation to the nature of theinformation desired In continuous systems, a major simplification occurs when the

study of the local and of the global responses can be split into two distinct

prob-lems This is often the case in solid mechanics and this kind of simplification hasbeen stated as a principle by Barré de Saint-Venant (1885), which is enunciated asfollows:

The elastic response induced by a local force system, whose resultant force and torque are both zero, become negligible far enough from the small loaded por- tion of the body In other words, if sufficiently far from the loaded domain, the response depends solely upon the resultant force and torque of the actual loading system.

Here ‘local’ means a part which is much smaller than the size of the whole body,

or of the boundary in the case of contact forces, and ‘far enough’ means distancessubstantially greater than the length scales of the loaded part

As detailed further in the next chapters, the Saint-Venant principle may be sidered rightly as the cornerstone for modelling solids as structural elements, even

con-if it is not explicitly invoked in most cases Nevertheless, care has to be taken whenusing it, as its validity is not universal In particular, an important exception isencountered in the case of thin shells of revolution loaded by concentrated forces.This can be easily realized by observing the deformation of a flexible pail filled ofwater, or sand, and held by its handle Indeed, it can be observed that the reactions at

the handle fasteners induce an ovalization of the pail cross-sections which extendsfar beyond the vicinity of the fasteners The colour Plates 1 and 2 illustrate suchresults which are further discussed in Chapter 8, see also Plates 13 and 14.

1.5.2 Shape criterion to reduce the dimension of a problem

In most engineering applications, use is made of structural elements which arecharacterized by a few generic particularities in their geometry Such peculiaritiescan be advantageously used to simplify their mathematical modelling In partic-ular, most structural elements can be characterized by the fact that one or twodimension(s) is/are much smaller than the other(s) Then, the corresponding straincomponents can be neglected and finally the dimensions connected to the smallscales can be suppressed, as detailed in the rest of the book for several types ofstructures A few simple ‘order of magnitude’ calculations can be used to supportthis simplifying statement

1.5.2.1 Compression of a solid body shaped as a slender parallelepiped

An elastic bar shaped as a slender parallelepiped is constrained on the face

x = 0 and loaded on the opposite face x = Lx by a compressive force
T(e) =

T0
i parallel to 0x, see Figure 1.24 The three-dimensional problem is not simple

if the local response is needed Indeed, it depends upon the stress distribution

on the loaded face, and also on the boundary conditions in the vicinity of theopposite face As already indicated in the example of the rod, the same problem isdrastically simplified when restricted to the global response According to Saint-Venant’s principle, the global solution may be expected to be valid in the largestportion of the bar, if Lxis much larger than Lyand Lz The actual loading exerted

on the face x = Lx is described by the force density tx(e)(y, z) However, fordetermining the global response, an equivalent load can be defined as the resultant

T0 = ''

(S)tx(e)(y, z) dy dz applied to the centre of the free end of the bar Formathematical convenience, the face x = 0 is provided with a sliding supportcondition, i.e (X(0, y, z)= 0) The other faces are free With these hypotheses, the

Figure 1.24 Bar shaped as a slender parallelepiped under compressive load

equilibrium is obtained from an axial state of stress whose nonzero component is:

σxx =LT0

Of course, this result fully agrees with the rod example of subsection 1.5.1

How-ever, the interesting point here is that by solving the same problem along the Oy and Oz axes instead of Ox, we find much lower stresses which are,

Thus, the idea which arises naturally is to neglect the transverse strains εyy, εzz

which are found to be much smaller than the axial strain In other words, ifthe body extends much more in the axial direction than in the transverse direc-tions, it will be modelled as an equivalent body which is completely rigid in thetransverse directions The corresponding stresses needed to obtain the equilibrium

of the solid are given by the Lagrange multipliers associated with the constraintconditions:

1.5.2.2 Shearing of a slender parallelepiped

What was said just above in the case of a normal loading is also true in thecase of a tangential loading For instance, let us consider the two similar problemssketched in Figure 1.25 In the case (a), a load T0
i is applied on the face z = Lz,the opposite face being fixed in any direction From the equilibrium conditions it isnot difficult to show that normal stresses are identically zero Hence div
X= 0 andthe displacement field is necessarily in the longitudinal direction (Y = Z = 0).The shear stress and strain fields are:

σzx= LT0

xLy; εzx= 12∂X∂z [1.148]

In the case (b), the load T0
k is applied on x = Lx Of course, the same reasoning

as in case (a) applies The shear stress and strain fields are:

σxz= LT0L ; εxz = 12∂Z∂x [1.149]

Figure 1.25 Shear deformation of a slender parallelepiped

As the stresses are again increasing functions of the strains, it follows that:

(σzx)(a)(σxz)(b) = LLz

x ≪ 1 ⇒ ∂X∂z ≪∂Z∂x [1.150]From similar calculations related to the other faces of the parallelepiped, it can also

corollary, the strain components in the planes parallel to Oyz can be discarded In other words, the cross-sections of the parallelepiped parallel to Oyz can be modelled

as rigid bodies

1.5.2.3 Validity of the simplification for a dynamic loading

The relations [1.136] and [1.137] defining the natural frequencies and modeshapes of a cubical solid are convenient to discuss the validity of the previoussimplifications in the dynamic domain Since we assume here that Lx ≫ Ly, Lz,

the first natural frequencies associated to the plane modes – indexed by the numbers(n, 0, 0) – are much lower than that of the other modes Therefore, if the frequencyrange of the excitation spectrum is much smaller than the lowest frequency (f1,1,0or

f1,0,1) of the first non-plane mode, then the dynamic response can be obtained withsufficient accuracy by taking into account the plane modes only (cf for instance[AXI 04], Chapters 7 and 9) In other words, the natural modes of vibration which

are related to transversal strains (Oy, Oz directions) can be safely neglected if their

frequency is much higher than the frequency range of the excitation It may bealso noted that, according to such an approximation, the frequency spectrum ofthe guided waves is restricted to the plane wave branches, which are the lowestbranches of the complete 3D frequency spectrum

1.5.2.4 Structural elements in engineering

From the previous considerations, it is concluded that to make tractable theanalysis of engineering structures, the first simplifying assumption is to replace thereal three-dimensional continuous medium by an equivalent continuous medium

of smaller dimension In this way, structural elements can be defined, as cables or beams which are one-dimensional structures and as plates and shells, which are

two-dimensional structures, see Figure 1.26 It is also worth emphasizing that if adimension is changed, all the formulation of the equilibrium equations is deeplymodified too, like the boundary conditions and space distribution of the loading Onthe other hand, the advantages gained by using such simplified models in structuralanalysis are very important The following chapters are devoted to work out suchsimplified models and to studying their properties both in the static and in thedynamic domains However, the present book is not exhaustive by far and has beenrestricted, on purpose, to the structural elements which are the most commonlyencountered in engineering For mathematical convenience, we start by studying

Figure 1.26 From solids to structural elements

the straight beams, then the plates and finally, the curved elements such as the archesand the shells The present order of presentation is appropriate to take into accountprogressively the increasing mathematical difficulties encountered in modellingsuch structures However, from a logical point of view, an exactly inverse order ofpresentation should be more concise, as, starting from a 3D solid body, it is possible

to deduce all the other models as particular cases, as indicated in Figure 1.26

Straight beam models: Newtonian approach

Beams are slender structural elements which are employed to support transverse

as well as axial loads They are of very common use as columns, masts, lintels, joistsetc, or as constitutive parts of supporting frames of buildings, cars, airplanes etc.They may be considered as the simplest structural element because of the relativesimplicity of the equilibrium equations arising from a suitable condensation ofthe space variables into a single one, namely the abscissa along the beam length.However, the approximations used to represent in an appropriate manner the 3Dsolid as an equivalent 1D solid involve several subtleties, and several degrees ofapproximation can be proposed to refine the beam model when necessary Theobject of this first of four chapters devoted to beams is to introduce the basic ideas

of the elementary theory of beams For the sake of simplicity at least, it is foundconvenient to adopt here a Newtonian instead of a variational approach The latterwill be adopted in the next chapters to establish more refined models than thosepresented here

2.1 Simplified representation of a 3D continuous medium by

fall within the class of ‘beams’ The ratio L/D is called the slenderness ratio.

As indicated in Chapter 1, subsection 1.5.2, the purpose of the present chapter is

to model a beam structure as a one-dimensional equivalent medium Having thispurpose in mind, a first step is to simplify the 3D expression of the strain anddisplacement fields in a suitable way

2.1.2 Global and local displacements

Let us cut a beam by a nearly transversal plane The centre of the area of this

section – which is supposed homogeneous and isotropic – is called the centroid denoted C The line of centroids, or central axis is defined as the line that passes

through the centroids along the beam The cross-sections are then defined as thebeam sections which are perpendicular to the central axis For the sake of simplicity,

Figure 2.1 Geometric representation of a beam

the material is assumed in what follows to be isotropic and homogeneous in thesame cross-section, but can vary along the beam central axis The result obtained

in subsection 1.5.2 of Chapter 1 in the particular case of a right parallelepiped isgeneralized as follows: if the slenderness ratio is high enough, the longitudinalstrains are much larger than the transverse ones, for a given external loading.Therefore, it is assumed that:

The cross-sections remain rigid during any possible motion Then, to describe the beam movement, six variables of displacement are needed which comprise three components of translation: the linear displacements of the centroid and three angular components: the small rotations of the cross-section about the centroid These functions are dependent upon one space variable only, which is defined as the curvilinear coordinate along the line of centroids.

As shown in Figure 2.1, two Cartesian coordinate systems are used for writingthe equilibrium equations The global frame Ox′y′z′serves to describe the structure

as a whole, and the local frame Cxyz serves to write the equilibrium equations of

a beam element of infinitesimal length between the abscissa s and s+ ds C(s) isthe centroid of the cross-section located at s, Cx is tangent to the line of centroids,

Cy and Cz are two principal axes of inertia of the cross-section However, inthis and in the next three chapters, consideration is restricted to straight beams,i.e beams which have a straight central axis in the non deformed state and in whichall the cross-sections have the same principal axes of inertia Then the global frameOxyzis defined by the direction Ox merged with the central axis Accordingly,the curvilinear abscissa s is replaced by the Cartesian coordinate x On the otherhand, the two transverse axes Oy and Oz are parallel to an orthogonal pair ofprincipal axes of inertia of the cross-sections As a further simplification, the cross-sections will be assumed to be symmetric with respect to the Cy and Cz axes, as

is the case for instance in Figure 2.2 A lack of central symmetry of the sections induces some specific complications which will be discussed separately,

cross-in subsection 2.2.6

The displacement of the centroid is defined by the translation vector
X ofCartesian components X(x), Y (x), Z(x) The rotation of the cross-section isdefined by the vector
ψ In accordance with the hypothesis of ‘small’ motions,the Cartesian components of
ψ, hereafter denoted ψx(x), ψy(x), ψz(x), are thesmall angles of rotation with respect to the axes of the local coordinate system

(Figures 2.1 and 2.2) These six quantities are the components of the global placement field of the straight beam They depend on the x abscissa and on the

dis-time t if the problem is dynamic

The local displacement field
ξ (
r) of a particle P , whose position is defined bythe vector
r (coordinates y, z in the local frame) is written as:

ξ(x, y, z) =
X(x) +
ψ(x) ×
r [2.1]

Figure 2.2 Particular case of straight beams

Figure 2.3 Local displacement of the current point P of the cross-section

This 3D field has the components:

2.1.3 Local and global strains

The components of the local strains are obtained by substituting [2.2] into the expression [1.22] or [1.25] of the local strain tensor The result is:

2 ηxy, ηxzare the transverse shear strains, in the planes Oxy or Oxz respectively.

3 χxxis the torsion (or twist) strain.

4 χyy and χzzare the bending (or flexure) strains, with respect to the Oy and

Ozaxes respectively

In a straight beam, all these modes of global deformation are uncoupled fromeach other, since each of them is expressed in terms of only one component ofthe global displacement field, which differs from one mode of deformation to theother In contrast to the global strains and angular displacements, the local strainsare expressed as a superposition of distinct components of the global strains, as

Figure 2.4 Global deformations of a beam element

Figure 2.5 Beam seen as a continuous set of fibres parallel to the beam axis

evidenced in the relations [2.3] To visualize the local strains, it is often foundconvenient to look at the beam as if it were made of a continuous set of material

lines, or fibres, parallel to the Ox axis, see Figure 2.5 The change in length of a

fibre passing through the point (x, y, z) depends on the bending and longitudinalglobal strains, see Figures 2.6a and 2.6b Taking into account bending deformationonly, the relation [2.2] reduces to ξx= zψy− yψzso a layer of neutral lines whoselength remains unchanged exists If the beam is homogeneous, the neutral layer isdefined by the condition y= 0, or z = 0, depending on the bending plane So the

central axis is often called the neutral fibre.

Figure 2.6a Local strains: superposition of bending and axial deformations

Figure 2.6b Local strains: superposition of bending and shear deformations

2.1.4 Local and global stresses

The local stress tensor due to the strains acting on a cross-section has threecomponents, which are associated with the stress vector:

t1= σxx
i + σxyj
+ σxz
k [2.6]

In agreement with the definition given in subsection 1.2.2,
t1 is the force perunit cross-sectional area exerted by the right-hand part of the beam on the left-hand

part, see Figure 2.7 Six variables of global stresses (also called resultant stresses)

are associated with the global displacements and strains They define a force vector

Figure 2.7 Local stress vector on the beam cross-section

Figure 2.8 Global stress components on a beam cross-section

The components of
T and
M are shown in Figure 2.8 and the relations [2.8],derived from [2.7], specify the correspondence between the components of globaldisplacements and global stresses Global displacements and global stresses are

thus found to be related to each other as pairs of conjugate quantities On the other

hand, the sign of the bending moments is controlled by the usual convention ofsigns applied to the rotations in a direct Cartesian frame

X→ N (x) ='

(S)σxxdS: normal force in the x direction

(axial, or string force)

cross-section is so small that resistance to a transverse load is provided almostexclusively by the tensile force N (x) A typical example is the strings of musicalinstruments Study of the transverse equilibrium of beams subjected to tensile

or compressive loads is postponed to Chapter 3, subsection 3.2.3 and Chapter 4,subsection 4.2.4

2.1.5 Elastic stresses

The three-dimensional Hooke’s law [1.37] is adapted here, taking into accountthe simplifications described just above As cross-sections remain non deformed,the local stresses acting in the transverse directions are not related to the transversestrains but to the Lagrange multipliers related to the rigidity conditions Accord-ingly, εyy, εzz, εyzand the elastic stresses σyy, σzz, σyzare assumed to be zero From[1.38] we get:

εxx = 1+ νE −Eν



σxx =σExx; εxy= 1+ νE σxy; εxz= 1+ νE σxz [2.10]or,

The area polar moment of inertia is:

2.1.6 Equilibrium in terms of generalized stresses

The equations of dynamical equilibrium are obtained by writing down the forceand moment balance, including all the strain, inertia, and external terms, whichact at the boundaries and within a beam element of infinitesimal length, lim-ited by the cross-sections S(x) and S(x+ dL) The actual loads exerted withinand at the boundary of the 3D body are averaged in a similar way to the localstress vector in relation [2.7], to produce an equivalent one-dimensional loadingcomprising:

1 A force density (force per unit length)
F(e)(x, t) (Newton/meter in S.I units)applied to the centroid

2 A moment density (moment per unit length) about the centroid
M(e)(x, t)(Newton)

Equation [1.32] still holds and can be expressed either by using a vector or asimplified indicial notation, according to which the first index of the stress term

is replaced by ‘x’, because (S) is normal to Ox In terms of local quantities itreduces to:

Using the relations [2.1], [2.6] and [2.8], the following one-dimensional equationsare obtained:

2.1.6.2 Equilibrium of the moments

The balance of moments is calculated with respect to the centre-of-mass of thebeam element, located at abscissa x It is written as: