MODELLING OF MECHANICAL SYSTEM VOLUME 2 Episode 8 potx

Plates are intended toresist various load conditions, broadly classified as in-plane and out-of-plane loads,that is forces either parallel or perpendicular to the plane of the plate, res

Figure 4.54 Collision of two beams, asymmetrical case

which start with equal and opposite velocities at t = 0 The contact time is equal

to the back and forth travelling time along an equivalent beam of length 2L Nowaves are excited and the velocities of the beams are simply interchanged by thecollision The last case corresponds to an asymmetrical system The upper plot ofFigure 4.54 shows a global view of the impact, which agrees satisfactorily with theconditions [4.100] and the lower plot focuses on the collision itself Contact timedepends on the beam properties and it may be noticed that waves are excited bythe impact, which are clearly visible in the trajectory of the second beam (dashedline in Figure 4.54)

Plates: in-plane motion

Plates are structural components used as walls, roofs, panels, windows, etc.They are modelled as two-dimensional structures characterized by a plane geometrybounded by a contour comprising straight and/or curved lines Plates are intended toresist various load conditions, broadly classified as in-plane and out-of-plane loads,that is forces either parallel or perpendicular to the plane of the plate, respectively

It is found convenient to study their response properties by separating first thein-plane and the out-of-plane motions In-plane motions of plates are generallymarked by the coupling between the two in-plane components of the displacementfield Such a coupling gives rise to new interesting features of in-plane motions withrespect to the longitudinal motions of straight beams Coupling between in-planeand out-of-plane motions occurs in the presence of in-plane preloads and will beconsidered in the next chapter On the other hand, for mathematical convenience

in using Cartesian coordinates, presentation will focus first on rectangular plates.Then use of curvilinear coordinates will be introduced to consider plates bounded

by curved contours

5.1 Introduction

5.1.1 Plate geometry

As shown in Figure 5.1, structural elements called plates are characterized by

the two following geometrical properties:

1 One dimension, termed the thickness, denoted h, is much smaller than theother two (length L1and width L2for a rectangular shape, diameter D for acircular one etc.) This allows one to model the mechanical properties of plates

by using a two-dimensional solid medium

2 The geometrical support of the 2D model is the midsurface, taken at h/2 if thematerial is homogeneous In contrast to the case of shells to be studied later, this

surface is flat and called the midplane of the plate The border lines of the tour of the midplane are called edges, which can be straight or/and curved lines.

con-A few other definitions concerning plate geometry are useful con-As athree-dimensional body, the plate is bounded by closed surfaces comprising the so

called faces and edge surfaces If the plate thickness is uniform, the faces are flat

and parallel to the midplane, whereas the edge surfaces can be cylindrical or plane,depending on the plate geometry Intersection of two edge surfaces defines a corneredge Finally, it is also found convenient to orientate the contour by defining a posit-ive unit vector normal to the midplane, which points from the lower face to the upperface of the plate The usual positive orientation is in the anticlockwise direction

5.1.2 Incidence of plate geometry on the mechanical response

As a preliminary, it is useful to outline a few generic features of the responseproperties of plates which serve as a guideline to organize the presentation of this

Figure 5.1 Plates: geometrical definitions

and the next chapters As will be verified ‘a posteriori’, they arise as a consequence

of the concept of midplane

Considering first linear motions about an equilibrium position with zero ornegligible stress level, it can be stated that:

1 In-plane loads, i.e parallel to the midplane, are balanced by in-plane stresses,which can be normal (compressive or tensile forces) and/or tangential (in-planeshear forces)

2 Out-of-plane loads, i.e perpendicular to the midplane, are balanced by bendingand torsional moments and by transverse shear forces

3 If the load is oblique, all the previous stresses participate with the librium However, in the absence of initial stresses, coupling between thein-plane and the out-of-plane motions of the plate can be discarded Sothe response to an oblique load is obtained by superposing the responses on thein-plane and to the out-of-plane components of the load, which can be studiedseparately

equi-Considering then linear motions about a prestressed equilibrium position, it can

be stated that:

1 In-plane initial stresses contribute to the out-of-plane equilibrium of a platethrough restoring, or alternatively, destabilizing forces, similar to those arising

in straight beams when prestressed axially

2 As a limit case, the forces that contribute to the out-of-plane equilibrium ofsuitably stretched skin structures originate almost completely from the in-planestresses which are prescribed initially Such skin structures are referred to asmembranes A membrane can be seen as an idealized two-dimensional mediumwhich has no flexural rigidity, in contrast with plates and shells

3 Like cables, or strings, membranes belong to the general class of the socalled tension structures which can support mechanical loads by tensilestresses only

In agreement with these preliminary remarks, it is found appropriate to igate first the motions of plates which involve in-plane components of force and

invest-displacement fields only Such in-plane fields are termed membrane components, or

fields, as their nature is basically the same as those induced when a skin is stretched

in its own plane On the other hand, it is also found convenient to study first thecase of rectangular plates as they can be described using Cartesian coordinates.Non rectangular plates have to be analysed by using oblique or curved coordin-ates, depending on the specificities of their shapes, which are more difficult tomanipulate mathematically

5.2 Kirchhoff–Love model

5.2.1 Love simplifications

Let us consider a rectangular plate of uniform thickness h The coordinates of

a material point are expressed in a direct Cartesian frame Oxyz, where Oxy is the midplane of the plate, the origin O is at a corner, the axes Ox and Oy are parallel

to the edges of lengths Lxand Lyrespectively The unit vectors of the Ox, Oy, Oz

axes are denoted
i,
j,
krespectively A main cross-section is obtained by cutting

mentally the plate with a plane which is perpendicular to the midplane and parallel

either to Ox or to Oy An ordinary cross-section is neither parallel to Ox nor to Oy,

P(x, y, 0) or P (x, y) designating a point of the midplane; the intersection of two

cross-sections passing through P defines a normal fibre (see Figure 5.2) As h is

very small with respect to Lx and to Ly, the following simplifying assumptionscan be made, which hold in the case of small deformations:

1 The normal fibres behave as a rigid body

2 As the midplane is deformed, the normal fibres remain perpendicular to it

3 The normal stresses acting on the planes parallel to the midplane are consistentwith the first hypothesis of rigidity of the normal fibres

These assumptions can be understood as a natural extension to the dimensional case of the basic beam model In particular, the second hypothesismeans that there is no shear between two neighbouring normal fibres as the platebends, in agreement with the Bernoulli–Euler model

two-5.2.2 Degrees of freedom and global displacements

According to the first assumption made above, the motion of a normal fibre can

be described by using five independent parameters, termed global displacements,

Figure 5.2 Cross-sections and normal fibre in a plate

Figure 5.3 Global displacements: translation and rotation variables

which are referred to a current point P (x, y) of the midplane They are shown inFigure 5.3 and defined as follows:

1 the longitudinal displacement in the direction Ox: X(x, y; t)
i

2 the lateral displacement in the direction Oy: Y (x, y; t)
j

3 the transverse displacement in the direction Oz: Z(x, y; t)
k

4 the rotation around the Ox axis: ψx(x, y; t)
i

5 the rotation around the Oy axis: ψy(x, y; t)
j

After deformation, the point P (x, y, 0) is transformed into P (x+ X, y + Y , Z)

5.2.3 Membrane displacements, strains and stresses

5.2.3.1 Global and local displacements

The components X, Y are termed membrane displacements, which are sufficient

to describe the in-plane motions of the plate In such motions, all the material pointslying on a same normal fibre have the same displacements So, global and localdisplacements are also the same, see Figure 5.4

ξ(x, y; t) =
X = X(x, y; t)
i + Y (x, y; t)
j [5.1]

5.2.3.2 Global and local strains

In agreement with [5.1], the local and the global strains are also the same: ε= η.Restricting the study to the case of small deformations, by substituting [5.1] into

Figure 5.4 Local and global displacements for in-plane motion

Figure 5.5 Membrane strains

the small strain tensor [1.25], we get:

ηxx = ∂X∂x; ηyy=∂Y∂y; ηxy= ηyx= 12 ∂y +∂Y∂x



[5.2]

The geometrical meaning of these quantities is illustrated in Figure 5.5 Theycomprise two normal and one shear component ηxx is the relative longitudinal

elongation (Ox direction), ηyyis the relative lateral elongation (Oy direction) and

ηxy = ηyx = γ /2 is the in-plane shear strain which is also expressed in terms ofthe shear angle γ Finally, the strain tensor [5.2] can be conveniently written in thefollowing matrix form:

Figure 5.6 Global stress components

As shown in Figure 5.6, they are consistent with the sign convention adoptedfor the three-dimensional stresses (see Figure 1.6) Nxx and Nyy are the normal

stresses, directed along the normal vectors of the edge surfaces parallel to Oy and

Ox respectively, and Nxy = Nyx are the in-plane shear stresses directed in the

tangential directions Oy and Ox respectively The global stress tensor is written in

matrix notation as:

[N ] =Nxx Nyx

Nxy Nyy



[5.5]

5.3 Membrane equilibrium of rectangular plates

5.3.1 Equilibrium in terms of generalized stresses

Because the geometry of rectangular plates is very simple, there is no difficulty

in deriving the equilibrium equations using the Newtonian approach i.e directbalancing of the forces acting on an infinitesimal rectangular element, as shown inthe next subsection However, it is also of interest to solve the problem by usingHamilton’s principle, which deals with scalar instead of vector quantities It is

applied first to the case of rectangular plates in subsection 5.3.1.2, and then to thecase of orthogonal curvilinear coordinates in section 5.4.

5.3.1.1 Local balance of forces

Let us consider a rectangular plate loaded by the external force field:

f(e)(x, y; t)= f(e)

x
i + f(e)

where
f(e)is a force density per unit area of the midplane surface

In Figure 5.7, the balances of the longitudinal and lateral forces acting on an

elementary rectangle dx, dy are sketched, which extend to the two-dimensional case

the longitudinal force balance of straight beams shown in Figure 2.9 Of course,

in the 2D case, it is appropriate to distinguish between a longitudinal and a lateralforce balance, giving rise thus to two distinct equilibrium equations Furthermore,the contribution of the tangential stresses due to the in-plane shear must be added

to the in-plane normal stresses As shear stresses are symmetric Nxy = Nyx, themoments are automatically balanced, not requiring any additional condition todescribe the equilibrium of the rectangle

The two equilibrium equations, in the Ox, Oy directions respectively, are

written as:

ρh ¨X− ∂xxx +∂N∂yyx



= f(e) x

ρh ¨Y − ∂yyy +∂N∂xxy



= f(e) y

[5.7]

Figure 5.7 In-plane forces acting on an infinitesimal element of the rectangular plate

In tensor notation, the system [5.7] is expressed as:

ρh
¨X·
ℓ −divN·
ℓ =
f(e)·
ℓ [5.8]

where
ℓdenotes a unit vector in the plane of the plate

It may be noticed that equation [5.8] is similar to the general 3D equation [1.32]and is independent of the coordinate system (intrinsic form), in contrast with [5.7]which holds in the case of Cartesian coordinates only

Hamilton’s principle is first written as:

The remaining task is to evaluate the terms of [5.11] in a suitable way, as detailedjust below

The kinetic energy density is found to be:

eκ= 12ρh( ˙X2+ ˙Y2) [5.12]

Its variation is integrated with respect to time, giving:

As long as the material law is not specified, it is not possible to express thestrain energy density analytically Nevertheless, its virtual variation is still given



dx dy

dt

conditions are readily derived The contour integral [5.18] is developed as:2

It may be checked that the signs of the different stress terms which appear

in the boundary conditions [5.20] are consistent with the general definition ofstresses given in Chapter 1, subsection 1.2.3 This is illustrated in Figure 5.8 inwhich positive edge forces are applied to the four edges of the plate For instance,according to [5.20] Nxxis found to be equal to the external edge force tx(e)at x= Lx

and to−tx(e)at x= 0 A slightly different way to check the signs consists of claimingthat a positive external force tx(e)applied along x= Lxmust induce tensile, hencepositive, stresses and the same force when applied along x = 0 must inducecompressive, hence negative stresses Of course, the signs of the stresses will alsoagree with those found in a 3D rectangular parallelepiped and in a straight beam

Figure 5.8 Stresses induced by a uniform edge loading

5.3.1.3 Homogeneous boundary conditions

If
t(e)≡ 0, the following standard homogeneous boundary conditions arise:

1 Fixed edge:

2 Sliding edge:

%parallel to O
x : Y = 0; Nyx = 0parallel to O
y : X= 0; Nxy= 0 [5.22]

3 Free edge:

%parallel to O
x : Nyy= 0; Nyx = 0parallel to O
y : Nxx= 0; Nxy= 0 [5.23]

5.3.1.4 Concentrated loads

As in the case of beams, it is found convenient to describe concentrated loads byusing singular distributions For example a force per unit length distributed alongthe line x= x0is written as:

f(e)=fx(e)(y; t)
i+ f(e)

y (y; t)
jδ(x− x0) [5.24]Then the equilibrium equations are written in terms of distributions as:

The integration in the domain[x0− ε, x0+ ε] where ε is arbitrarily small givesthe equations:

In [5.27] the symbol∩ means that the application point of the loading is located

at the intersection of the lines x = x0and y = y0 Accordingly, the equilibriumequations are written in terms of distributions as:

By integrating [5.28] over the intervals[x0− ε, x0+ ε; y0− η, y0+ η] where

εand η are arbitrarily small, the equations of motion can be written as:

manner, calculation performed on a plate of infinitesimal width 2ε would result inequation [5.26].

5.3.2 Elastic stresses

The three-dimensional law of elasticity as given by equations [1.36] to [1.38]

is particularized here to the case of plane stresses In agreement with theKirchhoff–Love assumptions, the out-of-plane components of the strain tensorand of the elastic stress tensor are set to zero Substituting such simplifications into[1.38], the following strain-stress relationships are derived:



where the global strain and stress components are used to form a strain vector and

a stress vector denoted[
η] and [
N] respectively

note –way of writing the stress–strain relationships

Because of the symmetry of the strain and stress tensors it follows that 2σxyεxy=

σxyεxy+ σyxεyx Hence, in [5.32] the notation 2εxy = γ in place of εxy allowsone to write the elastic energy density as the scalar product of a stress and a strainvectors:

note –3D elasticity versus plane stress and strain approximations

According to the 3D elasticity law, there is an incompatibility between a planestress model and a plane strain model, because if σzz is assumed to vanish, then

εzz= −ν(σxx+ σyy)/Ediffers from zero as a result of the Poisson effect theless, in the Kirchhoff–Love model the strains in the direction of plate thicknessare discarded and at the same time a plane stress model is assumed to hold Again,such a simplification is of the same nature as that made in beams by assuming thatthe cross-sections are rigid

Never-5.3.3 Equations and boundary conditions in terms of displacements

Substituting the strain–displacement relationships [5.3] into the stress–strainrelationships [5.33], we obtain the global membrane stresses:

On the left-hand side of [5.35] it is appropriate to make the distinction between

at least two stiffness operators The first one, proportional to E, describes the

stretching forces, and the second one, proportional to G, describes the in-planeshear forces Furthermore, both the stretching and the shear operators are found tocouple the longitudinal and the lateral motions together, since a displacement inthe Ox direction is found to induce a displacement in the Oy direction and viceversa The coupling terms appearing in the stretching operator is clearly due tothe Poisson effect The coupling terms appearing in the shear operator are inherent

in 2D shear effect They can be put in evidence even more clearly by writing theequations [5.35] in matrix form:

The homogeneous boundary conditions describing edge elastic supports can beinferred from the inhomogeneous conditions [5.20], by assimilating first the supportforces to an “external” load applied to the edges:

tx(e)(0, y)= −KxxX(0, y); tx(e)(Lx, y)= −KxxX(Lx, y)

ty(e)(0, y)= −KxyY (0, y); ty(e)(Lx, y)= −KxyY (Lx, y)

ty(e)(x, 0)= −KyyY (x, 0); ty(e)(x, Ly)= −KyyY (x, Ly)

tx(e)(x, 0)= −KyxY (x, 0); tx(e)(x, Ly)= −KyxY (x, Ly)

The supports are described by the stiffness coefficients Kxx, Kyy, Kxy, Kyx

defined as forces per unit area (Nm−2or Pa using the S.I units) If the edge librium conditions are considered as force balances between the ‘external’ loads

equi-and the internal stresses, then they are written as:

Of course, the standard boundary conditions [5.21] to [5.23] may be recovered

as limit cases of [5.37] where the appropriate stiffness coefficients tend either tozero or to infinity

5.3.4 Examples of application in elastostatics

5.3.4.1 Sliding plate subject to a uniform longitudinal load at the free edge

The rectangular plate (h, Lx, Ly) shown in Figure 5.9, is loaded along the edge

x= Lxby a uniform longitudinal force density tx(e)
i The edge x = 0 slides freely

Figure 5.9 Plate subject to a uniform longitudinal force density at one edge

in the lateral direction Oy The other edges are free In terms of distributions theequilibrium equations are:

As is rather obvious, any polynomial which is linear in x and y satisfiesthe homogeneous partial derivative equations So the general solution may bewritten as:

X= a1x+ b1y+ c1; Y = a2x+ b2y+ c2

where the coefficients a1, b1, etc are determined by the boundary conditions, asdetailed below.

3 Free edges y= ±Ly/2

Nyy= 0 ⇒ b2+ νa1= 0Hence, Nyy as well as Nxy, Nyx are found to vanish everywhere in the plate,including at the edges; so the stress field is uniaxial and uniform Nxx = tx(e) Thedisplacement field is found to be:

X= t

(e) x

Ehx; Y = −νt

(e) x

Ehx; Y =−νt

(e) x

The only difference between the plate solution [5.38] and the corresponding beamsolution lies in the lateral contraction proportional to Poisson’s ratio, which isrelated to the longitudinal stretching Finite element solution of the problem isshown in colour plate 4, the load being applied to the x= Lxand to the x= Lx/2line successively In the last case, only the part of the plate 0 ≤ x ≤ Lx/2 isdeformed The finite jump of Nxx across the loaded line is smoothed out by thefinite element discretization process

5.3.4.2 Fixed instead of sliding condition at the supported edge

The condition along the edge x= 0 becomes X(0, y) = 0; Y (0, y) = 0 A priori,

it could be surmised that the plate response does not differ much from the former