MODELLING OF MECHANICAL SYSTEM VOLUME 2 Episode 9 pps

298 Structural elements
In the particular case of a square plate, they are immediately obtained as:
in-phase modes:
f
n,m
=
c
2L

n
2
+
m
2
2
+
8n
2
m
2
π
2
(n
2
− m
2
)
2
; α
n,m
= β

n,m
out-of-phase modes:
f
n,m
=
c
2L

n
2
+
m
2
2
−
8n
2
m
2
π
2
(n
2
− m
2
)
2
; α
n,m
=−β

n,m
Figure 5.17 shows the mode shapes (1, 1), (1, 2) and (2, 2) of a steel plate
(though assuming ν = 0) L
x
= 4.04 m, L
y
= 4 m, as computed by the finite
f(1,1) = 783 Hz f(1,2) out-of- phase = 1037 Hz
f(1,2) in-phase = 1172 Hz f(2,2) = 1444 Hz
Figure 5.17. First modes of a plate fixed at the four edges
Plates: in-plane motion 299
element method. The computed natural frequencies are found to be less than
the 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 Rayleigh
minimum principle. The unsatisfactory result for the mode (2,2) concerning both
the 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 selecting
more complicated trial functions such as:
ψ
(X)
2,2
= α
2,2
sin

2πx
L
x

sin


2πy
L
y

+ α
1,3
sin

πx
L
x

sin

3πy
L
y

ψ
(Y )
2,2
= β
2,2
sin

2πx
L
x


sin

2πy
L
y

+ β
1,3
sin

3πx
L
x

sin

πy
L
y

which would produce non-vanishing coupling terms between α
2,2
and β
1,3
as well
as between β
2,2
and α
1,3
.

5.3.6.4 Plate loaded by a concentrated in-plane force:
spatial attenuation of the local response
A rectangular plate (L
x
, L
y
, h) is loaded by an in-plane force concentrated at
P(x
0
, y
0
), see Figure 5.18. The four edges are on sliding supports. We are interested
in studying the field of the normal longitudinal stress N
xx
(x,y). Colour plate 8
illustrates the results of a finite element computation, referring to a longitudinal
load applied to the plate centre. It is worth noticing that magnitude of displacement
and stress fields are sharply peaked in the close vicinity of the loaded point, in
agreement with Saint-Venant’s principle. It is also noted that the support reactions
are essentially normal to the edges and not distributed uniformly. As expected,
longitudinal reactions are negative along the edge x = L, and positive along the
edge x = 0. Their magnitude is maximum at y = L/2 in agreement with the
longitudinal stress field N
xx
(x,y). Lateral reactions are found to be antisymmetric
about the middle lateral axis, in agreement with the lateral stress field N
yy
(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

300 Structural elements
Here the selected boundary conditions allow us to expand the solution as a
modal series by using [5.67]. Furthermore, by making the distinction between the
contribution of the rectilinear modes and that of the membrane modes (n, m = 0) it
becomes possible to separate the global and the local responses. The displacement
fields are expanded as:
X(x, y) =
∞

m=0
∞

n=1
q
n,m
sin

nπx
L
x

cos

mπy
L
y

Y(x, y) =
∞


m=1
∞

n=0
p
n,m
cos

nπx
L
x

sin

mπy
L
y

[5.75]
The problem is further analysed by assuming a longitudinal axial load F
x
applied
at a point P(x
0
, L
y
/2) located on the middle longitudinal axis.
1. Rectilinear mode contribution and global response
A simple calculation gives:
q

n,0
=
2(1 − ν
2
)F
x
EhL
x
L
y

L
x
nπ

2
sin

nπx
0
L
x

; p
n,0
= 0 [5.76]
It follows the displacements in terms of rectilinear modes,
X
(m=0)
(x) = F

x
2(1 − ν
2
)
EhL
x
L
y
∞

n=1

L
x
nπ

2
sin

nπx
0
L
x

sin

nπx
L
x


;
Y
(m=0)
= 0 [5.77]
This field is independent of the lateral distribution of the load and characterises
the global response ofthe plate. The longitudinal normal stress field is givenby:
N
(m=0)
xx
(x) =
Eh
1 − ν
2
∂X
(m=0)
∂x
=

F
x
L
y

2
π

∞

n=1
1

n
sin

nπx
0
L
x

cos

nπx
L
x

[5.78]
It could be shown that, provided x
0
differs from zero, or L
x
, the modal
expansion [5.78] is the Fourier series on the interval 0 ≤ x ≤ L
x
of the
following step function:
+
F
x
L
y


L
x
− x
0
L
x

;if0≤ x ≤ x
0
−
F
x
L
y

x
0
L
x

;ifx
0
≤ x ≤ L
x
[5.79]
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 y
0
= ℓ/2
and three quarter length x
0
= 0.75 L. The series [5.78] is truncated to
N = 400 in order to minimize the Gibbs oscillations near the discontinu-
ity. As expected, the step magnitude is F
x
/L
y
, the plate is compressed in
the domain [x
0
, L
x
] and stretched in the domain [0, x
0
], the external load
is exactly balanced by the longitudinal component of the support reactions:
L
y
(N
(m=0)
xx
(L
x
) − N

(m=0)
xx
(0)) = F
x
. On the other hand, the series [5.78]
vanishes if the loading is applied at the sliding edges. The shear stresses
N
(m=0)
xy
are null and the lateral normal stress field is given by the relation
N
(m=0)
yy
= νN
(m=0)
xx
.
2. Contribution of the membrane modes (n, m = 0) and local response
The modal displacements are given by:
q
n,m
=
8(1 + ν)F
x
EhL
x
L
y



nπ
L
x

2
+

1 − ν
2

mπ
L
y

2

sin

nπx
0
L
x

cos

mπy
0
L
y




nπ
L
x

2
+

mπ
L
y

2

2
[5.80]
p
n,m
=−
8(1 + ν)F
x
EhL
x
L
y

1 + ν
2


nπ
L
x

mπ
L
y

sin

nπx
0
L
x

cos

mπy
0
L
y



nπ
L
x

2
+


mπ
L
y

2

2
[5.81]
302 Structural elements
The membrane stresses related to the membrane modes are:
N
(n,m)
xx
=
Eh
1 − ν
2
∞

n=1
∞

m=1

q
n,m
nπ
L
x

+ νp
n,m
mπ
L
y

cos

nπx
L
x

cos

mπy
L
y

N
(n,m)
yy
=
Eh
1 − ν
2
∞

n=1
∞


m=1

νq
n,m
nπ
L
x
+ p
n,m
mπ
L
y

cos

nπx
L
x

cos

mπy
L
y

N
(n,m)
xy
=
Eh

1 − ν
2
∞

n=1
∞

m=1

q
n,m
mπ
L
y
+ p
n,m
nπ
L
x

sin

nπx
L
x

sin

mπy
L

y

[5.82]
These describe the local response of the plate in the vicinity of the loaded
point P . In numerical evaluation of the series some difficulties arise related
to the singularity of the stress distribution at P . Along the lateral direction the
N
(n,m)
xx
profile comprises a Dirac component δ(y −y
0
) and in the longitudinal
direction it comprises a Dirac dipole δ
′
(x −x
0
). It is thus appropriate to calcu-
late [5.82] over a rectangular grid of elementary size x, y. Then the series
are truncated in such a way that the smallest wavelengths of the modes are
roughly 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 y
0
= 1 m and x
0
= 1.5 m. The elementary lengths are x = y = 7cm
and the series are truncated up to the smallest wavelengths λ
x
= λ
y

= 5cm.
As expected, the larger the distance from P , the smaller is the response and the
less is the number of modes which are necessary to compute the series [5.82].
Figure 5.20. Distribution of the local component of N
xx
over the plate
Plates: in-plane motion 303
Figure 5.21. Distribution of the local component of N
xx
over the plate
In the same way, the larger the distance of the loaded point from one of the lateral
edges, the more uniform is the stress distribution along this edge. This is illustrated
in Figures 5.21 where the lateral distribution of LN
(n,m)
xx
is plotted along the lateral
edges x = 0,x = L. On such plots, the singular values at y
0
are replaced by
the nearest value, which is convenient to focus on the regular part of the local
stress field. The local normal stress is very important at the edge nearest to the
loading and remains significant at the other edge, though of much less magnitude.
One observes a stress peak centred about y
0
, which is rather broad and of the
same sign as the ‘global’ field N
(m=0)
xx
, at about y = L/4, the sign is reversed.
Such a sign reversal is a very necessary feature of the local field N

(n,m)
xx
, since it
must vanish when integrated over the lateral edge. Similar calculations carried out
on rectangular plates show that the local response vanishes with a characteristic
length of the order of a fraction of the width of the plate. Such a result indicates
that 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 studythe local effectsinduced by aconcentrated load gives rise tothe same kindof
difficulties as the semi-analytical method described here. In both cases, the singular
component 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 even
along the y
0
line, see Colour Plate 8.
5.4. Curvilinear coordinates
If the plates are limited by curved edges, a mathematical difficulty arises as
the boundary conditions of the problem cannot be expressed in a tractable way by
using Cartesian coordinates. Fortunately, the use of curvilinear coordinates is found
304 Structural elements
appropriate, to deal with rather simple geometries at least, for instance circular and
elliptical plates, and more generally when an orthogonal curvilinear coordinate
system 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 curvilinear
coordinates denoted α and β as shown in Figure 5.22. The curves (C
α
) defined by
α = constant are orthogonal to the curves (C
β

) defined by β = constant. The
unit vectors tangent to these curves are denoted

t
α
,

t
β
. Transformation to Cartesian
coordinates is defined as:
x = x(α, β); y = y(α, β) [5.83]
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:
ds
2
= dx
2
+ dy
2
= g
2
α
dα
2
+ g
2
β
dβ

2
[5.84]
where:
g
2
α
=

∂x
∂α

2
+

∂y
∂α

2
; g
2
β
=

∂x
∂β

2
+

∂y

∂β

2
g
α
, g
β
are 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:
g
α
g
β
dαdβ [5.85]
Figure 5.22. Curvilinear orthogonal coordinates
Plates: in-plane motion 305
The linear membrane strain tensor is
[η]=
1
2


∇(X
α

t
α
+ X
β


t
β
)

+

∇

X
α

t
α
+ X
β

t
β

T

[5.86]
As detailed in Appendix 3 (formula [A.3.13]), it can be written as:
[η]=




1

g
α

∂X
α
∂α
+
X
β
g
β
∂g
α
∂β

1
2

g
α
g
β
∂
∂β

X
α
g
α


+
g
β
g
α
∂
∂α

X
β
g
β

1
2

g
α
g
β
∂
∂β

X
α
g
α

+
g

β
g
α
∂
∂α

X
β
g
β

1
g
β

∂X
β
∂β
+
X
α
g
α
∂g
β
∂α






[5.87]
The components of [η] can also be used to define a strain vector [η]
T
=
[η
αα
2η
αβ
η
ββ
].
5.4.2 Equilibrium equations and boundary conditions
The kinetic energy has the form:
E
k
=
1
2

α2
α1

β2
β1
ρh

˙
X
2

α
+
˙
X
2
β

g
α
g
β
dα dβ [5.88]
The strain energy is also invariant in any coordinate transformation, so its
variation is written as:
δ[E
s
]=

α2
α1

β2
β1
[

N ]
T
δ[[η]]g
α
g

β
dα dβ
=

α2
α1

β2
β1
{N
αα
δ[η
αα
]+N
αβ
δ[η
αβ
]+N
ββ
δ[η
ββ
]}g
α
g
β
dα dβ [5.89]
where the factor g
α
g
β

comes from the definition [5.85] of the elementary area in
curvilinear coordinates.
Using [5.87] and integrating once by parts to express all the variations in terms
of δX
α
and δX
β
, we arrive at the following expressions, written in a suitable form
to apply Hamilton’s principle, as detailed below:

α2
α1

β2
β1
{N
αα
δ[η
αα
]}g
α
g
β
dα dβ
=

α2
α1

β2

β1

N
αα
g
α

∂(δ X
α
)
∂α
+
δX
β
g
β
∂g
α
∂β

g
α
g
β
dα dβ
306 Structural elements
=

β2
β1

[g
β
N
αα
δX
α
]
α2
α1
dβ
+

α2
α1

β2
β1

−
∂(g
β
N
αα
)
∂α
δX
α
+
∂g
α

∂β
δX
β

dα dβ
and,

α2
α1

β2
β1
{N
ββ
δ[η
ββ
]}g
α
g
β
dα dβ
=

α2
α1

β2
β1

N

ββ
g
β

∂(δ X
β
)
∂β
+
δX
α
g
α
∂g
β
∂α

g
α
g
β
dα dβ
=

α2
α1
[g
α
N
ββ

δX
β
]
β2
β1
dα
+

α2
α1

β2
β1

−
∂(g
α
N
ββ
)
∂β
δX
β
+
∂g
β
∂α
δX
α


dα dβ
2

α2
α1

β2
β1
{N
αβ
δ[η
αβ
]}g
α
g
β
dα dβ
=

α2
α1

β2
β1

N
αβ

g
α

gβ
∂
∂β

∂X
α
g
α

+
g
β
g
α
∂
∂α

δX
β
g
β

g
α
g
β
dα dβ
=

α2

α1
[g
β
N
αβ
δX
β
]
β2
β1
dα +

β2
β1
[g
α
N
αβ
δX
α
]
α2
α1
dβ
−

α2
α1

β2

β1

1
g
α
∂(g
2
α
)N
αβ
)
∂β
δX
α
+
1
g
β
∂(g
2
β
)N
αβ
)
∂α
δX
β

dα dβ
So, the two following equations of motion are found:

ρh
¨
X
α
−
1
g
α
g
β

∂(g
β
N
αα
)
∂α
− N
ββ
∂g
β
∂α
+
1
g
α
∂(g
2
α
N

βα
)
∂β

= f
(e)
α
ρh
¨
X
β
−
1
g
α
g
β

∂(g
α
N
ββ
)
∂β
− N
αα
∂g
α
∂β
+

1
g
β
∂(g
2
β
N
αβ
)
∂α

= f
(e)
β
[5.90]
where f
(e)
α
and f
(e)
β
are the external forces per unit length applied along the

t
α
and

t
β
directions, respectively.

Plates: in-plane motion 307
The associated elastic boundary conditions are,
N
αα
− K
αα
X
α
= 0; N
ββ
− K
ββ
X
β
= 0
N
αβ
− K
αβ
X
β
= 0; N
βα
− K
βα
X
α
= 0
[5.91]
where K

αα
, K
ββ
, K
αβ
are the stiffness coefficients of the supports, acting in the
normal and tangential directions to the boundary lines (C
α
) and (C
β
).
5.4.3 Elastic law in curvilinear coordinates
The invariance of the strain energy with respect to any coordinate transformation
implies that the strain–stress relationship [5.33] is not changed and is written as:


N
αα
N
ββ
N
αβ


=
Eh
1 − ν
2




1 ν 0
ν 10
00
1 − ν
2





η
αα
η
ββ
2η
αβ


[5.92]
5.4.4 Circular cylinder loaded by a radial pressure
As an interesting application of plate in-plane equations in curvilinear coordin-
ates, let us consider the problem sketched in Figure 5.23, which deals with a circular
cylinder loaded by an external pressure P
e
and an internal pressure P
i
. Both P
e
and

P
i
are assumed to be uniform and P
e
= P
i
. 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, the
cylinder is reduced to an annular plate of unit thickness. The Lamé parameters are
found to be:
ds
2
= dr
2
+ (rdθ )
2
⇒ g
r
= 1; g
θ
= r [5.93]
Figure 5.23. Circular cylinder loaded by a uniform radial pressure
308 Structural elements
The radial and tangential equilibrium equations are:
1
r
∂(rN

rr
)
∂r
+
∂(N
rθ
)
r∂θ
−
N
θθ
r
= 0;
∂(N
θθ
)
r∂θ
+
1
r
2
∂(r
2
N
θr
)
∂r
= 0 [5.94]
U denoting the radial and V the tangential displacements, [5.87] and [5.92] give:
η

rr
=
∂U
∂r
; η
rθ
=
1
2

1
r
∂U
∂θ
+ r
∂
∂r

V
r

; η
θθ
=
1
r
∂V
∂θ
+
U

r
N
rr
=
Eh
1 − ν
2

∂U
∂r
+ ν

1
r
∂V
∂θ
+
U
r

; N
rθ
= Gh

1
r
∂U
∂θ
+ r
∂

∂r

V
r

N
θθ
=
Eh
1 − ν
2

1
r
∂V
∂θ
+
U
r
+ ν
∂U
∂r

[5.95]
Obviously the solution is independent of θ and the equations [5.94] reduce to:
1
r
∂(rN
rr
)

∂r
−
N
θθ
r
= 0;
1
r
2
∂(r
2
N
rθ
)
∂r
= 0 [5.96]
as r is strictly positive, the second equation implies r
2
N
rθ
= constant. The bound-
ary conditions imply N
rθ
(R
1
) = N
rθ
(R
2
) = 0, then N

rθ
≡ 0 for any r in the
interval R
1
≤ r ≤ R
2
. However, from [5.95] it is also found that:
N
rθ
=
Eh
1 − ν
2
(1 − ν)η
rθ
= Ghr
∂
∂r

V
r

[5.97]
which means
∂
∂r

V
r


= 0 ⇒ V = αr.
The constant α is arbitrary; as the physical meaning of V is a rotation of the
plate (considered as a rigid solid) around the Oz axis, α may be chosen equal to
zero. This result confirms the intuition of a pure radial displacement of the cylinder
which is a contraction if the pressure differential P
e
− P
i
is positive. Figure 5.24
visualizes the equilibrium of a plate sector, delimited by the radii r and r +dr and
by an angular sector dθ. The condition of radial equilibrium is readily found to be:
dF = N
rr
(r + dr){(r + dr) dθ}−N
rr
(r){rdθ}
≃

N
rr
(r) +
∂N
rr
∂r
dr

{(r + dr) dθ}−N
rr
(r){rdθ}
≃


N
rr
(r) +
∂N
rr
∂r

{dr dθ}=
∂rN
rr
∂r
{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 =−N
θθ
{drdθ}.As
suitable, this force balance is consistent with the first equation [5.96]. It must be
emphasized that the curvature of the sector induces a coupling between the radial
and tangential stresses; this is a very important feature which holds for any curved
structures (beams, plates, shells) as will be stressed several times in Chapters 7
and 8. To alleviate to some extent the algebra, without changing the physics of the
problem, P
i
is assumed hereafter to be negligible in comparison with P
e
. The radial
equation is thus associated with the boundary conditions:
N

rr
(R
1
) = 0; N
rr
(R
2
) =−P
e
h [5.98]
Strains and stresses reduce to
η
rr
=
dU
dr
; η
θθ
=
U
r
N
rr
=
Eh
1 − ν
2

dU
dr

+ ν

U
r

; N
θθ
=
Eh
1 − ν
2

U
r
+ ν
dU
dr

[5.99]
and using [5.96], we arrive at the differential equation:
r
d
2
U
dr
2
+
dU
dr
−

U
r
= 0 [5.100]
The solutions have the general form r
m
with the values m = 1 and m =−1, so
U = ar + b
1
r
[5.101]
310 Structural elements
a and b are determined by the boundary conditions and the solution is
U =−
P
e
E
R
2
2
R
2
2
− R
2
1

r(1 − ν) +
(1 + ν)R
2
1

r

; N
θr
= 0
N
rr
=
−P
e
h
1 −
(
R
1
/R
2
)
2

1 −

R
1
r

2

; N
θθ

=
−P
e
h
1 −
(
R
1
/R
2
)
2

1 +

R
1
r

2

[5.102]
The tangential stress is always greater than the pressure load; it is maximum for
r = R
1
. The sum of the stresses (radial and tangential) is constant; which is written
in terms of local stresses as:
σ
rr
+ σ

θθ
=−2P
e

1 −

R
1
R
2

2

[5.103]
If the cylinder thickness is small R
1
∼
=
R
2
∼
=
R; R
2
= R +e, with e/R ≪ 1, then
a first order approximation gives
U
∼
=
−

P
e
R
2
Ee
; σ
rr
∼
=
−P
e
(r − R)
e
; σ
θθ
∼
=
−P
e
R
e
[5.104]
The tangential stress is much greater than the radial stress:
σ
θθ
∼
=
−P
e
R

e
≫ σ
rr
|
max
∼
=
−P
e
[5.105]
Because of the curvature the radial loading can be equilibrated essentially by the
tangential stresses.
Chapter 6
Plates: out-of-plane motion
Out-of-plane, or transverse, motions of plates is of paramount importance
because, as in the case of straight beams and for the same reasons, plates are
much 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 of
in-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 simpli-
fying assumptions used to establish the Bernoulli–Euler model of straight beam
bending to the two-dimensional case. Accordingly, the transverse motion can be
described 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 a
rectangular plate), new interesting features arise in plate bending and torsion with
respect to straight beam bending, both from the mathematical and physical view-
points. The Kirchhoff–Love model is found appropriate to deal with thin plates
where the thickness is small compared with either the other dimensions of the plate
or to the modal wavelengths of the highest modal frequency of interest. Though

this book is strictly restricted to the case of thin plates and thin shells, it may be
worth mentioning that the Kirchhoff–Love model can be improved by accounting
for rotatory inertia and out-of-plane shear deformation, in a similar manner as the
Bernoulli–Euler model can be improved, to give rise to the Rayleigh–Timoshenko
model.
312 Structural elements
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 (see
Figure 6.1):

ξ(x, y,z; t) =

X +

ψ ×r [6.1]
where

X = X

i + Y

j +Z

k; r = z

k;

ψ = ψ

x

i + ψ
y

j [6.2]
Then the Cartesian components of the local displacements are:
ξ
x
= X +zψ
y
; ξ
y
= Y − zψ
x
; ξ
z
= Z [6.3]
According to the second hypothesis of the Kirchhoff–Love model, the rotations
are related to the transverse displacements through the derivatives:
ψ
x
=+
∂Z
∂y
; ψ
y
=−
∂Z
∂x

[6.4]
The signs of these derivatives are consistent with a direct frame of reference. Using
[6.4], [6.3] becomes:
ξ
x
= X −z
∂Z
∂x
; ξ
y
= Y − z
∂Z
∂y
; ξ
z
= Z [6.5]
z
x
y
PЈ
P
M
x

PЈ
MЈ
Figure 6.1. Local (M point) and global (P point) displacements
Plates: out-of-plane motion 313
6.1.2 Local and global strains
6.1.2.1 Local strains

Starting from [6.5], the local strains are found to be:
ε
xx
=
∂ξ
x
∂x
=
∂X
∂x
− z
∂
2
Z
∂x
2
; ε
yy
=
∂ξ
y
∂y
=
∂Y
∂y
− z
∂
2
Z
∂y

2
ε
xy
= ε
yx
=
1
2

∂ξ
x
∂y
+
∂ξ
y
∂x

=
1
2

∂X
∂y
+
∂Y
∂x
− 2z
∂
2
Z

∂x∂y

[6.6]
As a statement of the Kirchhoff–Love model, the other components (in particular
the shear strains ε
xz
and ε
yz
) are nil, leading thus to a flexure model without shear.
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 of
two distinct components, namely the membrane strains independent of z, and the
flexure and torsional strains, proportional to z. Thus, the tensor form of [6.6] is
conveniently written as:
ε = η + zχ [6.7]
where η stands for the membrane strain tensor, already introduced in the preceding
chapter, and χ stands for the flexure stress tensor, written in the matrix form as:
[χ]=

χ
xx
χ
xy
χ
yx
χ
yy

=






−
∂
2
Z
∂x
2
−
∂
2
Z
∂y∂x
−
∂
2
Z
∂x∂y
−
∂
2
Z
∂y
2






[6.8]
The coefficients of [χ]are interpreted as small curvatures of the deformed mid-
plane; the diagonal terms are the flexure curvatures (similar to the beam bending
curvatures) whereas the non diagonal terms represent a torsion of the plate which
induces shear of the cross-sections. It is also worth mentioning that the tensor form
[6.7] can be replaced by the vector form:
[ε]=[η]+z[χ] [6.9]
314 Structural elements
where,
[η]
T
=

η
xx
η
yy
2η
xy

=

∂X
∂x
∂Y
∂y
∂X
∂y
+

∂Y
∂x

[χ]
T
=

χ
xx
χ
yy
2χ
xy

=

−
∂
2
Z
∂x
2
−
∂
2
Z
∂y
2
−2
∂

2
Z
∂y∂x

6.1.3 Local and global stresses: bending and torsion
The global stresses are obtained by integrating the local stresses through the
plate thickness. If the material is isotropic (which is the limitation adopted in this
book), the Kirchhoff–Love model restricts the elastic stresses to the following three
components, σ
xx
, σ
xy
, σ
yy
which depend linearly on the coordinate z. Using the
matrix form [5.32] and [6.9], in Cartesian coordinates we obtain:
[σ ]=


σ
xx
σ
yy
σ
xy


=[C][ε]=
E
1 −ν

2


1 ν 0
ν 10
00(1 −ν)/2




ε
xx
ε
yy
2ε
xy


=[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 z
2
[χ] gives:
[

M]
T
=


M
xx
M
yy
M
xy

=

h/2
−h/2
zσ
T
dz=

h/2
−h/2
z
2
[χ
T
]dz=
h
3
12
[C][χ
T
]
[6.11]
The moments M

xx
and M
yy
are termed bending moments, and M
xy
= M
yx
is a
torsion moment. Actually, these quantities are dimensioned as a moment, or torque,
per unit length, i.e. as a force. Of course [6.11] can also be written as a tensor
M
written in matrix form as:
[M]=

M
xx
M
xy
M
yx
M
yy

[6.12]
In Figures 6.2a and 6.2b the components of
M acting on a rectangular plate
element are sketched in relation to the local stresses, which helps to make clear
the appropriate sign. As will be shown in subsection 6.2.1, it is also useful to
visualize these moments by equivalent torques whose force components are parallel
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 to
the formulas:
Q
xz
=

+h/2
−h/2
σ
xz
dz; Q
yz
=

+h/2
−h/2
σ
yz
dz [6.13]
However, in the same way as in the Bernoulli–Euler model for beams, the trans-
verse shear stresses cannot be introduced through a material law, since according
to the Kirchhoff–Love model ε
xz
= ε
yz

= 0. So they must be introduced concep-
tually via the Lagrange multipliers related to the conditions ε
xz
= ε
yz
= 0 and
derived in practice by balancing the transverse forces applied to a plate element.
Another way to give a physical interpretation of the Kirchoff–Love model is to
316 Structural elements
Figure 6.3. Transverse shear forces Q
xz
, Q
yz
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
E is 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 plate
depend on the bending and torsion terms only. The present section is restricted to
this 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 by
using again the Newtonian approach. However difficulties would arise in writing
down 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 variational
principle such as Hamilton’s principle. Because the detailed calculation is rather
cumbersome and tedious and because some boundary conditions give rise to a few
interesting subtleties, we will proceed step by step in the analysis. At first, a plate
without external loading of any kind will be considered. Then the homogeneous
and inhomogeneous boundary conditions will be discussed. Finally surface and

concentrated loads applied on lines or points on the midplane will be included in
the analysis.
6.2.1.1 Variation of the inertia terms
If the rotatory inertia of the transverse plate fibres is neglected, the kinetic energy
density is:
de
κ
=
1
2
ρh
˙
Z
2
[6.14]
Plates: out-of-plane motion 317
The variation of the kinetic energy is:
ρh

t
2
t
1


L
x
0

L

y
0
˙
Zδ[
˙
Z]dx dy

dt
=−

t
2
t
1


L
x
0

L
y
0
ρh
¨
ZδZ dx dy

dt [6.15]
6.2.1.2 Variation of the strain energy
Using a tensor formulation, the variationof the strain energy density is written as:

δ[e
s
]=
M : δ[χ]=[

M]
T
δ[χ ] [6.16]
Using [6.8] and [6.12] we get:
−δ[e
s
]=M
xx
∂
2
δZ
∂x
2
+ (M
xy
+ M
yx
)
∂
2
δZ
∂x∂y
+ M
yy
∂

2
δZ
∂y
2
[6.17]
Despite being equal to each other, the terms M
xy
and M
yx
are still considered
separately because they do not act in the same direction. As the time dependent
variations are not considered, the calculus of variations can be restricted to the
space domain:

L
x
0

L
y
0
M
xx
∂
2
δZ
∂x
2
dx dy =


L
x
0

L
y
0
∂
2
M
xx
∂x
2
δZ dx dy
+

L
y
0

M
xx
∂δZ
∂x
−
∂M
xx
∂x
δZ


L
x
0
dy [6.18]

L
y
0

L
x
0
M
yy
∂
2
δZ
∂y
2
dx dy =

L
x
0

L
y
0
∂
2

M
yy
∂y
2
δZ dx dy
+

L
x
0

M
yy
∂δZ
∂y
−
∂M
yy
∂y
δZ

L
y
0
dx [6.19]
and for the torsion terms,

L
y
0


L
x
0
M
xy
∂
2
δZ
∂x∂y
dx dy =

L
x
0

L
y
0
∂
2
M
xy
∂x∂y
δZ dx dy
+

L
x
0


M
xy
∂δZ
∂x

L
y
0
dx −

L
y
0

∂M
xy
∂y
δZ

L
x
0
dy
[6.20]
318 Structural elements

L
x
0


L
y
0
M
yx
∂
2
δZ
∂x∂y
dx dy =

L
x
0

L
y
0
∂
2
M
yx
∂x∂y
δZ dx dy
+

L
y
0


M
yx
∂δZ
∂y

L
x
0
dy −

L
x
0

∂M
yx
∂x
δZ

L
y
0
dx
[6.21]
6.2.1.3 Local equilibrium without external loads
Putting together all the surface integrals, we obtain:
−δ[E
s
]=


L
x
0

L
y
0

∂
2
M
xx
∂x
2
+ 2
∂
2
M
xy
∂x∂y
+
∂
2
M
yy
∂y
2

δZ


dx dy [6.22]
Then, in the absence of any external loading, Hamilton’s principle is found to
reduce to:

t
2
t
1

L
x
0

L
y
0

−ρh
¨
Z +

∂
2
M
xx
∂x
2
+ 2
∂

2
M
xy
∂x∂y
+
∂
2
M
yy
∂y
2

δZ dx dy dt = 0
[6.23]
which gives the local equilibrium equation at a current point of the midplane of the
plate:
ρh
¨
Z −

∂
2
M
xx
∂x
2
+ 2
∂
2
M

xy
∂x∂y
+
∂
2
M
yy
∂y
2

= 0 [6.24]

k denoting the unit vector normal to the plate midplane, [6.24] is written in intrinsic
form as:
ρh

¨
X ·

k −div(div
=
M
) = 0 [6.25]
note. – Shear loads and gradient of the moments
Equation [6.25] represents the equilibrium of the internal forces. Accordingly,
it may be written as:
ρh

¨
X ·n −div


Q = 0 where

Q = div
=
M
[6.26]
Plates: out-of-plane motion 319
Figure 6.4. Shear loads related to the moment derivatives

Q is the shear force vector which equilibrates the inertia forces. Its components are
given by:
Q
xz
=
∂M
xx
∂x
+
∂M
yx
∂y
; Q
yz
=
∂M
yy
∂y
+
∂M

xy
∂x
[6.27]
These components are represented in Figure 6.4 in the same way as in
Figure 2.21. Their orientation is consistent with the orientation of the local stresses
acting on the facets of a cuboid, or of the global stresses acting on a straight beam.
As M
xy
= M
yx
, [6.24] can also be expressed as:
ρh
¨
Z −

∂Q
xz
∂x
+
∂Q
yz
∂y

= 0 [6.28]
6.2.2 Boundary conditions
6.2.2.1 Kirchhoff effective shear forces and corner forces
In [6.17] to [6.21], cancellation of the integral variations on the edges of
the plate gives the boundary conditions. A correct formulation of these condi-
tions is not simple and concerning it one might be reminded of an historic story.
At the beginning of the nineteen century, the physicist and acoustician E. Chladni

(1756–1829) displayed many complex and beautiful geometrical figures when,
using a bow, he excited a plate covered with a thin sand layer. This experiment was
presented to the emperor Napoléon Bonaparte who was very puzzled and decided
in 1809 to transfer 3000 francs to the French Academy of Sciences to be given as
a prize for anybody who could be able to explain the vibrations of the plate. After
several attempts, Miss S. Germain won the prize in 1816; she produced a correct
320 Structural elements
differential equation but derived erroneous boundary conditions. Though Lagrange,
Cauchy and Poisson participated in her research work, it was not until about
1850 that the consistent boundary conditions were established by G.R. Kirchhoff
(1824–1887).
1. Components of the moment densities. Let us consider a plate edge parallel to
Oy. The terms involving the moments in formulas [6.18] to [6.21] are:

L
y
0

M
xx
δ

∂Z
∂x

+ M
yx
δ

∂Z

∂y

L
x
0
dy [6.29]
Before Kirchhoff, the rotations were considered as two independent variables.
Accordingly, at a free edge the following boundary conditions were assumed to
occur: M
xx
= M
yx
= 0. Such a result cannot be true because an additional
boundary condition has still to be complied with, which concerns the shear forces,
as further discussed in the next subsection. Hence, it would turn out that three
conditions ought to be fulfilled at a plate edge, producing thus an oversized sys-
tem of equations since shear forces and moments are already related through
equation [6.28]. The difficulty was overcome by Kirchhoff who correctlyaccounted
for the fact that rotations and displacements are dependent variables by integrating
the torsion term to express the variation in terms of δZ solely. So, in place of [6.29]
he obtained:

L
y
0

M
xx
δ


∂Z
∂x

+ M
yx
δ

∂Z
∂y

L
x
0
dy
=

L
y
0

M
xx
δ

∂Z
∂x

−
∂M
yx

∂y
δZ

L
x
0
dy +

[M
yx
δZ]
L
x
0

L
y
0
[6.30]
Then, in [6.30] the sole condition to be fulfilled concerning the moment is:

L
y
0

M
xx
δ

∂Z

∂x

L
x
0
dy = 0 [6.31]
Similarly along an edge parallel to the Ox axis we get:

L
x
0

M
yy
δ

∂Z
∂y

+ M
xy
δ

∂Z
∂x

L
y
0
dx

=

L
x
0

M
yy
δ

∂Z
∂x

−
∂M
xy
∂x
δZ

L
y
0
dx +

[M
xy
δZ]
L
x
0


L
y
0
[6.32]
Plates: out-of-plane motion 321

L
x
0

M
yy
δ

∂Z
∂y

L
y
0
dx = 0 [6.33]
The other terms are to be included in the boundary condition concerning the
shear forces, as detailed below.
2. Components of the shear force densities. By gathering the pertinent terms of the
relations [6.18] to [6.21] together with the Kirchhoff term in [6.30] along an edge
parallel to Oy, we obtain:

L
y

0

−

∂M
xx
∂x
+
∂M
yx
∂y
+
∂M
xy
∂y

L
x
0
δZ dy = 0 [6.34]
and for an edge parallel to Ox:

L
x
0

−

∂M
yy

∂y
+
∂M
xy
∂x
+
∂M
yx
∂x

L
y
0
δZ dx = 0 [6.35]
The terms between brackets are interpreted as shear force densities which are
known as effective Kirchhoff shear forces per unit length, expressed as:
V
xz
=
∂M
xx
∂x
+
∂M
yx
∂y
+
∂M
xy
∂y

= Q
xz
+
∂M
yx
∂y
=
∂M
xx
∂x
+ 2
∂M
xy
∂y
V
yz
=
∂M
yy
∂y
+
∂M
xy
∂x
+
∂M
yx
∂x
= Q
yz

+
∂M
xy
∂x
=
∂M
yy
∂y
+ 2
∂M
yx
∂x
[6.36]
where use is made of the equality M
xy
= M
yx
.
3. Corner forces. Corner forces originate from the Kirchhoff relations [6.30] and
[6.32] which imply the boundary condition at a corner:

[(M
xy
+ M
yx
)δZ]
L
x
0


L
y
0
= 0 [6.37]
If the corner is supported in such a way that δZ =0, a corner force
M
xy
+ M
yx
= 0 arises in the transverse direction, as further discussed in
subsection 6.2.2.3.
322 Structural elements
6.2.2.2 Elastic boundary conditions
When applied to elastic stresses, the preceding relations lead to the following
homogeneous boundary conditions:
1. Clamped edge

parallel to Ox : Z(x, y = 0orL
y
) = 0; ∂Z/∂y|
y=0orL
y
= 0
parallel to Ox : Z(x = 0orL
x
, y) = 0; ∂Z/∂x|
x=0orL
x
= 0
[6.38]

2. Sliding edge

parallel to Ox : Z(x, y = 0, or x, L
y
) = 0; V
yz
|
y=0orL
y
= 0
parallel to Oy : Z(x = 0, or L
x
, y) = 0; V
xz
|
x=0orL
x
= 0
[6.39]
3. Hinged edge

parallel to Ox : Z(x, y = 0orx, L
y
) = 0; M
yy
|
y=0orL
y
= 0
parallel to Oy : Z(x = 0orL

x
, y) = 0; M
xx
|
x=0orL
x
= 0
[6.40]
4. Free edge

parallel to Ox : V
yz
|
x,o or L
y
= 0; M
yy
|
y=0orL
y
= 0
parallel to Oy : V
xz
|
x=0orL
x
,y
= 0; M
xx
|

x=0orL
x
= 0
[6.41]
5. Free corner
(M
yx
= M
xy
)|
0,0 or 0,L
y
or L
x
,0 or L
x
,L
y
= 0 [6.42]
6.2.2.3 External loading of the edges and inhomogeneous
boundary conditions
The external loading is comprised of forces or/and moments per unit length
applied to the edges of the plate, the latter defining the plate contour denoted (C):

Q
(e)
(x, y; t) =

Q
(e)

z

k;

M
(e)
(x, y; t) =

M
(e)
x

i + M
(e)
y

j

x, y ∈ (C) [6.43]
The analysis is restrained to the simplest configuration which supposes the moment

M
(e)
parallel to the edge; then the virtual work is:

(C)

Q
(e)
z

δZ + M
(e)
x
∂δZ
∂y
− M
(e)
y
∂δZ
∂x

ds [6.44]