MODELLING OF MECHANICAL SYSTEM VOLUME 2 Episode 9 pps

5.3.6.4 Plate loaded by a concentrated in-plane force: spatial attenuation of the local response A rectangular plate Lx, Ly, h is loaded by an in-plane force concentrated at P x0, y0, se

In the particular case of a square plate, they are immediately obtained as:in-phase modes:

Plates: in-plane motion 299

element method The computed natural frequencies are found to be less thanthe Rayleigh–Ritz values: f (1, 1) = 789 Hz, f (1, 2) out-of phase = 1048 Hz,

f (1, 2) in-phase= 1180 Hz, f (2, 2) = 1578 Hz, as expected from the Rayleighminimum principle The unsatisfactory result for the mode (2,2) concerning boththe value of the natural frequency and the mode shape, which is clearly marked

by a coupling between the x and the y directions, could be corrected by selectingmore complicated trial functions such as:

ψ2,2(X)= α2,2sin 2π x

Lx

sin 2π y

Ly

+ α1,3sin

Lx

sin 3π y

Ly

+ β1,3sin 3π x

Lx

sin

Ly



which would produce non-vanishing coupling terms between α2,2and β1,3as well

as between β2,2and α1,3

5.3.6.4 Plate loaded by a concentrated in-plane force:

spatial attenuation of the local response

A rectangular plate (Lx, Ly, h) is loaded by an in-plane force concentrated at

P (x0, y0), see Figure 5.18 The four edges are on sliding supports We are interested

in studying the field of the normal longitudinal stress Nxx(x, y) Colour plate 8illustrates the results of a finite element computation, referring to a longitudinalload applied to the plate centre It is worth noticing that magnitude of displacementand stress fields are sharply peaked in the close vicinity of the loaded point, inagreement with Saint-Venant’s principle It is also noted that the support reactionsare essentially normal to the edges and not distributed uniformly As expected,longitudinal reactions are negative along the edge x = L, and positive along theedge x = 0 Their magnitude is maximum at y = L/2 in agreement with thelongitudinal stress field Nxx(x, y) Lateral reactions are found to be antisymmetricabout the middle lateral axis, in agreement with the lateral stress field Nyy(x, y).Such behaviour is clearly related to the Poisson effect

Figure 5.18 Rectangular plate on sliding support loaded by a longitudinal force at point P

Here the selected boundary conditions allow us to expand the solution as amodal series by using [5.67] Furthermore, by making the distinction between thecontribution of the rectilinear modes and that of the membrane modes (n, m= 0) itbecomes possible to separate the global and the local responses The displacementfields are expanded as:

Ly



[5.75]

The problem is further analysed by assuming a longitudinal axial load Fxapplied

at a point P (x0, Ly/2) located on the middle longitudinal axis

1 Rectilinear mode contribution and global response

A simple calculation gives:

qn,0= 2(1− ν

2)FxEhLxLy

Lx

[5.78]

It could be shown that, provided x0 differs from zero, or Lx, the modalexpansion [5.78] is the Fourier series on the interval 0 ≤ x ≤ Lx of thefollowing step function:

Plates: in-plane motion 301

Figure 5.19 Distribution of the normal stress N(m=0)

xx along a longitudinal line of the plate

This is illustrated in Figure 5.19 which refers to a plate L = 10 m;

ℓ= 2 m loaded by a longitudinal force of 1 kN applied at midwidth y0= ℓ/2and three quarter length x0 = 0.75 L The series [5.78] is truncated to

N = 400 in order to minimize the Gibbs oscillations near the ity As expected, the step magnitude is Fx/Ly, the plate is compressed inthe domain [x0, Lx] and stretched in the domain [0, x0], the external load

discontinu-is exactly balanced by the longitudinal component of the support reactions:

Ly(Nxx(m=0)(Lx)− Nxx(m=0)(0)) = Fx On the other hand, the series [5.78]vanishes if the loading is applied at the sliding edges The shear stresses

Nxy(m=0) are null and the lateral normal stress field is given by the relation

Nyy(m=0)= νNxx(m=0)

The modal displacements are given by:

The membrane stresses related to the membrane modes are:

Lx

cos

Lx

cos

Lx

sin

to the singularity of the stress distribution at P Along the lateral direction the

Nxx(n,m)profile comprises a Dirac component δ(y− y0)and in the longitudinaldirection it comprises a Dirac dipole δ′(x− x0) It is thus appropriate to calcu-late [5.82] over a rectangular grid of elementary size x, y Then the seriesare truncated in such a way that the smallest wavelengths of the modes areroughly a few tenths of x, y This is illustrated in Figure 5.20 which refers

to a square plate L= ℓ = 2 m loaded by a longitudinal force of 1 kN applied

at y0= 1 m and x0 = 1.5 m The elementary lengths are x = y = 7 cmand the series are truncated up to the smallest wavelengths λx = λy = 5 cm

As expected, the larger the distance from P , the smaller is the response and theless is the number of modes which are necessary to compute the series [5.82]

Figure 5.20 Distribution of the local component of N over the plate

Plates: in-plane motion 303

Figure 5.21 Distribution of the local component of Nxxover the plate

In the same way, the larger the distance of the loaded point from one of the lateraledges, the more uniform is the stress distribution along this edge This is illustrated

in Figures 5.21 where the lateral distribution of LNxx(n,m)is plotted along the lateraledges x = 0 , x = L On such plots, the singular values at y0are replaced bythe nearest value, which is convenient to focus on the regular part of the localstress field The local normal stress is very important at the edge nearest to theloading and remains significant at the other edge, though of much less magnitude.One observes a stress peak centred about y0, which is rather broad and of thesame sign as the ‘global’ field Nxx(m=0), at about y = L/4, the sign is reversed.Such a sign reversal is a very necessary feature of the local field Nxx(n,m), since itmust vanish when integrated over the lateral edge Similar calculations carried out

on rectangular plates show that the local response vanishes with a characteristiclength of the order of a fraction of the width of the plate Such a result indicatesthat Saint-Venant’s principle can be applied to plate in-plane problems

As a final remark it is worth mentioning that the use of the finite element method

to study the local effects induced by a concentrated load gives rise to the same kind ofdifficulties as the semi-analytical method described here In both cases, the singularcomponent of the stress field is smoothed out by the discretization procedure and

no very reliable values of the stress can be obtained at the loaded point and evenalong the y0line, see Colour Plate 8

5.4 Curvilinear coordinates

If the plates are limited by curved edges, a mathematical difficulty arises asthe boundary conditions of the problem cannot be expressed in a tractable way byusing Cartesian coordinates Fortunately, the use of curvilinear coordinates is found

appropriate, to deal with rather simple geometries at least, for instance circular andelliptical plates, and more generally when an orthogonal curvilinear coordinatesystem can be fitted to the edge geometry.

5.4.1 Linear strain tensor

Let us define the position of a point lying on the midplane of a plate by curvilinearcoordinates denoted α and β as shown in Figure 5.22 The curves (Cα)defined by

α = constant are orthogonal to the curves (Cβ)defined by β = constant Theunit vectors tangent to these curves are denoted
tα,
tβ Transformation to Cartesiancoordinates is defined as:

The length of any segment drawn in the midplane is independent of the

coordinate system, then for any infinitesimal segment of length ds, the following

relationship holds:

ds2= dx2+ dy2= gα2dα2+ gβ2dβ2 [5.84]where:

gα, gβare termed the Lamé parameters of the plane surface

On the other hand, according to [5.84], in the α, β orthonormal system, the area

of the elementary surface must be defined as:

Figure 5.22 Curvilinear orthogonal coordinates

Plates: in-plane motion 305

The linear membrane strain tensor is

5.4.2 Equilibrium equations and boundary conditions

The kinetic energy has the form:

Ek =12

 α2 α1

Using [5.87] and integrating once by parts to express all the variations in terms

of δXα and δXβ, we arrive at the following expressions, written in a suitable form

to apply Hamilton’s principle, as detailed below:

Nααg

α)

∂α +δ Xg β∂g∂βα



gαgβdα dβ

 β 2

β1 [gβNααδXα]α2α1dβ+

 α2 α1

−

 α2

α1

 β2 β1

Plates: in-plane motion 307

The associated elastic boundary conditions are,

Nαα− KααXα = 0; Nββ− KββXβ = 0

Nαβ− KαβXβ = 0; Nβα− KβαXα = 0 [5.91]where Kαα, Kββ, Kαβ are the stiffness coefficients of the supports, acting in thenormal and tangential directions to the boundary lines (Cα)and (Cβ)

5.4.3 Elastic law in curvilinear coordinates

The invariance of the strain energy with respect to any coordinate transformationimplies that the strain–stress relationship [5.33] is not changed and is written as:



5.4.4 Circular cylinder loaded by a radial pressure

As an interesting application of plate in-plane equations in curvilinear ates, let us consider the problem sketched in Figure 5.23, which deals with a circularcylinder loaded by an external pressure Peand an internal pressure Pi Both Peand

coordin-Pi are assumed to be uniform and Pe= Pi A priori, the reader could be surprised

to find here a shell instead of a plate problem The shell is conveniently described

by using the cylindrical coordinate system r, θ , z Nevertheless, as here the pressure

is assumed to be independent of z, the dimension of the problem can be reduced

to two dimensions described by the polar coordinates r and θ By doing so, thecylinder is reduced to an annular plate of unit thickness The Lamé parameters arefound to be:

ds2= dr2+ (rdθ)2 ⇒ gr = 1; gθ = r [5.93]

Figure 5.23 Circular cylinder loaded by a uniform radial pressure

The radial and tangential equilibrium equations are:

∂V

∂θ +Ur + ν∂U∂r



[5.95]Obviously the solution is independent of θ and the equations [5.94] reduce to:

1r

∂(rNrr)

∂r −Nrθ θ = 0; r12∂(r

2Nrθ)

as r is strictly positive, the second equation implies r2Nrθ = constant The

bound-ary conditions imply Nrθ(R1) = Nrθ(R2) = 0, then Nrθ ≡ 0 for any r in theinterval R1≤ r ≤ R2 However, from [5.95] it is also found that:

by an angular sector dθ The condition of radial equilibrium is readily found to be:

dF = Nrr(r+ dr){(r + dr) dθ} − Nrr(r){r dθ}

≃Nrr(r)+∂Nrr

∂r dr{(r + dr) dθ} − Nrr(r){r dθ}

≃Nrr(r)+∂Nrr{dr dθ} = ∂rNrr{dr dθ} = F {dr dθ}

Plates: in-plane motion 309

Figure 5.24 Plate sector equilibrium

The resultant of the tangential forces is also radial−dF = −Nθ θ{drdθ} Assuitable, this force balance is consistent with the first equation [5.96] It must beemphasized that the curvature of the sector induces a coupling between the radialand tangential stresses; this is a very important feature which holds for any curvedstructures (beams, plates, shells) as will be stressed several times in Chapters 7and 8 To alleviate to some extent the algebra, without changing the physics of theproblem, Piis assumed hereafter to be negligible in comparison with Pe The radialequation is thus associated with the boundary conditions:

aand b are determined by the boundary conditions and the solution is

U= −PEe R

2 2

R22− R2

1

r(1− ν) + (1+ ν)R

2 1

r= R1 The sum of the stresses (radial and tangential) is constant; which is written

in terms of local stresses as:

If the cylinder thickness is small R1∼= R2∼= R; R2= R + e, with e/R ≪ 1, then

a first order approximation gives

Chapter 6

Plates: out-of-plane motion

Out-of-plane, or transverse, motions of plates is of paramount importancebecause, as in the case of straight beams and for the same reasons, plates aremuch less rigid when solicited in the out-of-plane than in the in-plane direction.Furthermore, the transverse response of a plate is sensitive to the presence ofin-plane stresses, just as the bending of a straight beam is sensitive to the pres-ence of a longitudinal stress Modelling of the out-of-plane motions of plates

is based on the so called Kirchhoff–Love hypotheses which extends the fying assumptions used to establish the Bernoulli–Euler model of straight beambending to the two-dimensional case Accordingly, the transverse motion can bedescribed in terms of a single displacement field, the so called transverse dis-placement denoted Z As Z depends on two coordinates (x, y in the case of arectangular plate), new interesting features arise in plate bending and torsion withrespect to straight beam bending, both from the mathematical and physical view-points The Kirchhoff–Love model is found appropriate to deal with thin plateswhere the thickness is small compared with either the other dimensions of the plate

simpli-or to the modal wavelengths of the highest modal frequency of interest Thoughthis book is strictly restricted to the case of thin plates and thin shells, it may beworth mentioning that the Kirchhoff–Love model can be improved by accountingfor rotatory inertia and out-of-plane shear deformation, in a similar manner as theBernoulli–Euler model can be improved, to give rise to the Rayleigh–Timoshenkomodel

6.1 Kirchhoff–Love hypotheses

6.1.1 Local displacements

As already stated in Chapter 5 section 5.2, the local displacement field

of a material point M, located at the distance z from the plate midplane is (seeFigure 6.1):

where

X= X
i + Y
j+ Z
k;
r = z
k; ψ
= ψx
i + ψyj
[6.2]Then the Cartesian components of the local displacements are:

ξx = X + zψy; ξy= Y − zψx; ξz= Z [6.3]According to the second hypothesis of the Kirchhoff–Love model, the rotationsare related to the transverse displacements through the derivatives:

Plates: out-of-plane motion 313

6.1.2 Local and global strains

6.1.2.2 Global flexure and torsional strains

As in the case of beams, the local strains [6.6] can be viewed as the sum oftwo distinct components, namely the membrane strains independent of z, and theflexure and torsional strains, proportional to z Thus, the tensor form of [6.6] isconveniently written as:

6.1.3 Local and global stresses: bending and torsion

The global stresses are obtained by integrating the local stresses through theplate thickness If the material is isotropic (which is the limitation adopted in thisbook), the Kirchhoff–Love model restricts the elastic stresses to the following threecomponents, σxx, σxy, σyywhich depend linearly on the coordinate z Using thematrix form [5.32] and [6.9], in Cartesian coordinates we obtain:



= [C]([
η] + z[
χ])

[6.10]Starting from [6.10], integration through the thickness of the bending terms z[
χ]gives zero, whereas integration of the bending stress moments z2[
χ] gives:[
M]T

to the Oz axis.

In a similar manner to that for beams, the equilibrium of plate elements loaded

by external transverse forces requires the presence of internal transverse forces,

Plates: out-of-plane motion 315

Figure 6.2a Bending moments

Figure 6.2b Torsion moments

as shown in Figure 6.3 They result from the shear local stresses, according tothe formulas:

concep-Figure 6.3 Transverse shear forces Qxz, Qyz

consider that the plate is made of an orthotropic material, which is characterized by

a finite Young modulus E and Poisson’s ratio ν in the in-plane directions, whereas

Eis assumed to be infinitely large and ν is zero in the out-of-plane direction

6.2 Bending equations

In the absence of in-plane loading, the small transverse displacements of a platedepend on the bending and torsion terms only The present section is restricted tothis case Study of the effect of in-plane stressed plates is postponed to section 6.4

6.2.1 Formulation in terms of stresses

As in the case of beams, the local equilibrium equations could be derived byusing again the Newtonian approach However difficulties would arise in writingdown the proper boundary conditions, as further discussed in subsection 6.2.2.Hence, in the present problem it is found more appropriate to use a variationalprinciple such as Hamilton’s principle Because the detailed calculation is rathercumbersome and tedious and because some boundary conditions give rise to a fewinteresting subtleties, we will proceed step by step in the analysis At first, a platewithout external loading of any kind will be considered Then the homogeneousand inhomogeneous boundary conditions will be discussed Finally surface andconcentrated loads applied on lines or points on the midplane will be included inthe analysis

6.2.1.1 Variation of the inertia terms

If the rotatory inertia of the transverse plate fibres is neglected, the kinetic energydensity is:

Plates: out-of-plane motion 317

The variation of the kinetic energy is:

6.2.1.2 Variation of the strain energy

Using a tensor formulation, the variation of the strain energy density is written as:

δ[es] = M : δ[χ] = [
M]Tδ[
χ] [6.16]Using [6.8] and [6.12] we get: