Bending of plates, or plate bending, refers to the deflection of a plate perpendicular to the plane of the plate under the action of external

force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...

s and moments. The amount of deflection can be determined by solving the differential equations of an appropriate plate theory. The stresses in the plate can be calculated from these deflections. Once the stresses are known, failure theories can be used to determine whether a plate will fail under a given load.

Bending of Kirchhoff-Love plates

Definitions

For a thin rectangular plate of thickness

H

, Young's modulus

E

, and Poisson's ratio

\nu

, we can define parameters in terms of the plate deflection,

w

. The flexural rigidity is given by :

D = \frac

Moments

The bending moments per unit length are given by :

M_ = -D \left( \frac + \nu \frac \right)

M_ = -D \left( \nu \frac + \frac \right)

The twisting moment per unit length is given by :

M_ = -D \left( 1 - \nu \right) \frac

Forces

The

shear force In solid mechanics, shearing forces are unaligned forces acting on one part of a body in a specific direction, and another part of the body in the opposite direction. When the forces are collinear (aligned with each other), they are called t ...

s per unit length are given by :

Q_ = -D \frac \left( \frac + \frac \right)

Q_ = -D \frac \left( \frac + \frac \right)

Stresses

The bending stresses are given by :

\sigma_ = -\frac \left( \frac + \nu \frac \right)

\sigma_ = -\frac \left( \nu \frac + \frac \right)

The shear stress is given by :

\tau_ = -\frac \left(1-\nu\right) \frac

Strains

The bending strains for small-deflection theory are given by :

\epsilon_ = \frac = -z\frac

\epsilon_ = \frac = -z\frac

The shear strain for small-deflection theory is given by :

\gamma_ = \frac + \frac = -2z\frac

For large-deflection plate theory, we consider the inclusion of membrane strains :

\epsilon_ = \frac + \frac\left(\frac\right)^2

\epsilon_ = \frac + \frac\left(\frac\right)^2

\gamma_ = \frac + \frac + \frac \frac

Deflections

The deflections are given by :

u = -z\frac

v = -z\frac

Derivation

In the Kirchhoff–Love plate theory for plates the governing equations areReddy, J. N., 2007, Theory and analysis of elastic plates and shells, CRC Press, Taylor and Francis. :

N_ = 0

and :

M_ - q = 0

In expanded form, :

\cfrac + \cfrac = 0 ~;~~
      \cfrac + \cfrac = 0

and :

\cfrac + 2\cfrac +
      \cfrac = q

where

q(x)

is an applied transverse

load Load or LOAD may refer to: Aeronautics and transportation *Load factor (aeronautics), the ratio of the lift of an aircraft to its weight *Passenger load factor, the ratio of revenue passenger miles to available seat miles of a particular transpo ...

per unit area, the thickness of the plate is

H=2h

, the stresses are

\sigma_

, and :

N_ := \int_^h \sigma_~dx_3 ~;~~
   M_ := \int_^h x_3~\sigma_~dx_3~.

The quantity

N

has units of

per unit length. The quantity

M

has units of

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per unit length. For

isotropic Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...

homogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...

, plates with Young's modulus

E

and Poisson's ratio

\nu

these equations reduce toTimoshenko, S. and Woinowsky-Krieger, S., (1959), Theory of plates and shells, McGraw-Hill New York. :

\nabla^2\nabla^2 w = -\cfrac ~;~~ D := \cfrac = \cfrac

where

w(x_1,x_2)

is the deflection of the mid-surface of the plate.

Small deflection of thin rectangular plates

This is governed by the

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Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaFöppl–

von Kármán The term ''von'' () is used in German language surnames either as a nobiliary particle indicating a noble patrilineality, or as a simple preposition used by commoners that means ''of'' or ''from''. Nobility directories like the ''Almanach de Go ...

plate equations :

\cfrac +
   2\cfrac +
   \cfrac =
   E\left left(\cfrac\right)^2 -
   \cfrac
   \cfrac\right

\cfrac +
   2\cfrac +
   \cfrac =
   \cfrac + \cfrac\left(
   \cfrac\cfrac +
   \cfrac\cfrac -
   2\cfrac\cfrac
   \right)

where

F

is the stress function.

Circular Kirchhoff-Love plates

The bending of circular plates can be examined by solving the governing equation with appropriate boundary conditions. These solutions were first found by Poisson in 1829. Cylindrical coordinates are convenient for such problems. Here

z

is the distance of a point from the midplane of the plate. The governing equation in coordinate-free form is :

\nabla^2 \nabla^2 w = -\frac \,.

In cylindrical coordinates

(r, \theta, z)

, :

\nabla^2 w \equiv \frac\frac\left(r \frac\right) + 
      \frac\frac + \frac \,.

For symmetrically loaded circular plates,

w = w(r)

, and we have :

\nabla^2 w \equiv \frac\cfrac\left(r \cfrac\right) \,.

Therefore, the governing equation is :

= -\frac\,.

q

and

D

are constant, direct integration of the governing equation gives us

: $w(r) = -\frac + C_1\ln r + \cfrac + \cfrac(2\ln r - 1) + C_4$

where

C_i

are constants. The slope of the deflection surface is :

\phi(r) = \cfrac = -\frac + \frac + C_2 r + C_3 r \ln r \,.

For a circular plate, the requirement that the deflection and the slope of the deflection are finite at

r = 0

implies that

C_1 = 0

. However,

C_3

need not equal 0, as the limit of

r \ln r\,

exists as you approach

r = 0

from the right.

Clamped edges

For a circular plate with clamped edges, we have

w(a) = 0

and

\phi(a) = 0

at the edge of the plate (radius

a

). Using these boundary conditions we get

: $w(r) = -\frac (a^2 -r^2)^2 \quad \text \quad
\phi(r) = \frac(a^2-r^2) \,.$

The in-plane displacements in the plate are :

u_r(r) = -z\phi(r) \quad \text \quad u_\theta(r) = 0 \,.

The in-plane strains in the plate are :

\varepsilon_ = \cfrac = -\frac(a^2-3r^2) ~,~~
  \varepsilon_ = \frac = -\frac(a^2-r^2) ~,~~
  \varepsilon_ = 0 \,.

The in-plane stresses in the plate are :

~;~~ \sigma_ = 0 \,.

For a plate of thickness

2h

, the bending stiffness is

D = 2Eh^3/ (1-\nu^2) /math> and we
have

: $\ \sigma_ &= 0 \,. \end$

The moment resultants (bending moments) are :

M_ = -\frac\left 1+\nu)a^2-(3+\nu)r^2\right ~;~~
  M_ = -\frac\left 1+\nu)a^2-(1+3\nu)r^2\right ~;~~
  M_ = 0 \,.

The maximum radial stress is at

z = h

and

r = a

: :

\left.\sigma_\_ = \frac = \frac

where

H := 2h

. The bending moments at the boundary and the center of the plate are :

\left.M_\_ = \frac ~,~~
  \left.M_\_ = \frac ~,~~
  \left.M_\_ = \left.M_\_ = -\frac \,.

Rectangular Kirchhoff-Love plates

For rectangular plates, Navier in 1820 introduced a simple method for finding the displacement and stress when a plate is simply supported. The idea was to express the applied load in terms of Fourier components, find the solution for a sinusoidal load (a single Fourier component), and then superimpose the Fourier components to get the solution for an arbitrary load.

Sinusoidal load

Let us assume that the load is of the form :

q(x,y) = q_0 \sin\frac\sin\frac \,.

Here

q_0

is the amplitude,

a

is the width of the plate in the

x

-direction, and

b

is the width of the plate in the

y

-direction. Since the plate is simply supported, the displacement

w(x,y)

along the edges of the plate is zero, the bending moment

M_

is zero at

x=0

and

x=a

, and

M_

is zero at

y=0

and

y=b

. If we apply these boundary conditions and solve the plate equation, we get the solution :

w(x,y) = \frac\,\left(\frac+\frac\right)^\,\sin\frac\sin\frac \,.

Where D is the flexural rigidity :

D=\frac

Analogous to flexural stiffness EI. We can calculate the stresses and strains in the plate once we know the displacement. For a more general load of the form :

q(x,y) = q_0 \sin\frac\sin\frac

where

m

and

n

are integers, we get the solution

: $\text \qquad
w(x,y) = \frac\,\left(\frac+\frac\right)^\,\sin\frac\sin\frac \,.$

Navier solution

Double trigonometric series equation

We define a general load

q(x,y)

of the following form :

q(x,y) = \sum_^ \sum_^\infty a_\sin\frac\sin\frac

where

a_

is a Fourier coefficient given by :

a_ = \frac\int_0^b \int_0^a q(x,y)\sin\frac\sin\frac\,\textx\texty

. The classical rectangular plate equation for small deflections thus becomes: :

\cfrac + 2\cfrac + \cfrac =
   \cfrac \sum_^ \sum_^\infty a_\sin\frac\sin\frac

Simply-supported plate with general load

We assume a solution

w(x,y)

of the following form :

w(x,y) = \sum_^ \sum_^\infty w_\sin\frac\sin\frac

The partial differentials of this function are given by :

\cfrac = \sum_^ \sum_^\infty
   \left(\frac\right)^4 w_\sin\frac\sin\frac

\cfrac = \sum_^ \sum_^\infty
   \left(\frac\right)^2 \left(\frac\right)^2 w_\sin\frac\sin\frac

\cfrac = \sum_^ \sum_^\infty
   \left(\frac\right)^4 w_\sin\frac\sin\frac

Substituting these expressions in the plate equation, we have :

\sum_^ \sum_^\infty
   \left( \left(\frac\right)^2 + \left(\frac\right)^2 \right)^2
   w_\sin\frac\sin\frac =
   \sum_^ \sum_^\infty \cfrac \sin\frac\sin\frac

Equating the two expressions, we have :

\left( \left(\frac\right)^2 + \left(\frac\right)^2 \right)^2 w_ = \cfrac

which can be rearranged to give :

w_ = \frac\frac

The deflection of a simply-supported plate (of corner-origin) with general load is given by

: $w(x,y) = \frac \sum_^\infty \sum_^\infty
\frac
\sin\frac\sin\frac$

Simply-supported plate with uniformly-distributed load

: For a uniformly-distributed load, we have :

q(x,y) = q_0

The corresponding Fourier coefficient is thus given by :

a_ = \frac \int_0^a \int_0^b q_0\sin\frac\sin\frac\,\textx\texty

. Evaluating the double integral, we have :

a_ = \frac(1 - \cos m\pi)(1 - \cos n\pi)

, or alternatively in a piecewise format, we have :

a_ = \begin
            \cfrac & m~\text~n~\text \\
            0 & m~\text~n~\text
            \end

The deflection of a simply-supported plate (of corner-origin) with uniformly-distributed load is given by

: $w(x,y) = \frac \sum_^\infty \sum_^\infty
\frac
\sin\frac\sin\frac$

The bending moments per unit length in the plate are given by

: $M_ = \frac \sum_^\infty \sum_^\infty
\frac
\sin\frac\sin\frac$ : $M_ = \frac \sum_^\infty \sum_^\infty
\frac
\sin\frac\sin\frac$

Lévy solution

Another approach was proposed by LévyLévy, M., 1899, Comptes rendues, vol. 129, pp. 535-539 in 1899. In this case we start with an assumed form of the displacement and try to fit the parameters so that the governing equation and the boundary conditions are satisfied. The goal is to find

Y_m(y)

such that it satisfies the boundary conditions at

y = 0

and

y = b

and, of course, the governing equation

\nabla^2 \nabla^2 w = q/D

. Let us assume that :

w(x,y) = \sum_^\infty Y_m(y) \sin \frac \,.

For a plate that is simply-supported along

x=0

and

x=a

, the boundary conditions are

w=0

and

M_=0

. Note that there is no variation in displacement along these edges meaning that

\partial w/\partial y = 0

and

\partial^2 w/\partial y^2 = 0

, thus reducing the moment boundary condition to an equivalent expression

\partial^2 w/\partial x^2 = 0

Moments along edges

Consider the case of pure moment loading. In that case

q = 0

and

w(x,y)

has to satisfy

\nabla^2 \nabla^2 w = 0

. Since we are working in rectangular Cartesian coordinates, the governing equation can be expanded as :

\frac + 2 \frac
   + \frac  = 0 \,.

Plugging the expression for

w(x,y)

in the governing equation gives us :

\sum_^\infty \left left(\frac\right)^4 Y_m \sin\frac
   - 2\left(\frac\right)^2 \cfrac \sin\frac
   + \frac \sin\frac\right = 0

or :

\frac  - 2 \frac \cfrac + \frac Y_m = 0 \,.

This is an ordinary differential equation which has the general solution :

Y_m = A_m \cosh\frac + B_m\frac \cosh\frac + 
  C_m \sinh\frac + D_m\frac \sinh\frac

where

A_m, B_m, C_m, D_m

are constants that can be determined from the boundary conditions. Therefore, the displacement solution has the form

: $\sin \frac \,.$

Let us choose the coordinate system such that the boundaries of the plate are at

x = 0

and

x = a

(same as before) and at

y = \pm b/2

(and not

y=0

and

y=b

). Then the moment boundary conditions at the

y = \pm b/2

boundaries are :

w = 0 \,, -D\frac\Bigr, _ = f_1(x) \,, 
  -D\frac\Bigr, _ = f_2(x)

where

f_1(x), f_2(x)

are known functions. The solution can be found by applying these boundary conditions. We can show that for the ''symmetrical'' case where :

M_\Bigr, _ = M_\Bigr, _

and :

f_1(x) = f_2(x) = \sum_^\infty E_m\sin\frac

we have

: $w(x,y) = \frac\sum_^\infty \frac\,
\sin\frac\, \left(\alpha_m \tanh\alpha_m \cosh\frac
- \frac\sinh\frac\right)$

where :

\alpha_m = \frac \,.

Similarly, for the ''antisymmetrical'' case where :

M_\Bigr, _ = -M_\Bigr, _

we have

: $w(x,y) = \frac\sum_^\infty \frac\,
\sin\frac\, \left(\alpha_m \coth\alpha_m \sinh\frac
- \frac\cosh\frac\right) \,.$

We can superpose the symmetric and antisymmetric solutions to get more general solutions.

Simply-supported plate with uniformly-distributed load

For a uniformly-distributed load, we have :

q(x,y) = q_0

The deflection of a simply-supported plate with centre

\left(\frac, 0\right)

with uniformly-distributed load is given by

: $\begin
&w(x,y) = \frac \sum_^\infty
\left( A_m\cosh\frac + B_m\frac\sinh\frac + G_m\right)
\sin\frac\\\\
&\begin
\text\quad
&A_m = -\frac\\
&B_m = \frac\\
&G_m = \frac\\\\
\text\quad
&\alpha _m = \frac
\end
\end$

The bending moments per unit length in the plate are given by

: $M_x = -q_0\pi^2 a^2\sum_^\infty m^2\left(
\left(\left(\nu -1\right)A_m + 2\nu B_m\right)\cosh\frac +
\left(\nu -1\right)B_m\frac\sinh\frac - G_m\right)
\sin\frac$ : $M_y = -q_0\pi^2 a^2\sum_^\infty m^2\left(
\left(\left(1-\nu\right)A_m + 2B_m\right)\cosh\frac +
\left(1-\nu\right)B_m\frac\sinh\frac - \nu G_m\right)
\sin\frac$

Uniform and symmetric moment load

For the special case where the loading is symmetric and the moment is uniform, we have at

y=\pm b/2

, :

M_ = f_1(x) = \frac\sum_^\infty \frac\,\sin\frac \,.

: The resulting displacement is

: $\end$

where :

\alpha_m = \frac \,.

The bending moments and shear forces corresponding to the displacement

w

are :

\\ Q_ & = \frac-\frac \\ & = \frac\sum_^\infty \frac\,\times \\ & ~ \cos\frac\cosh\frac\,. \end

The stresses are :

\sigma_ = \frac\,M_ \quad \text \quad 
  \sigma_ = \frac\,Q_\left(1 - \frac\right)\,.

Cylindrical plate bending

Cylindrical bending occurs when a rectangular plate that has dimensions

a \times b \times h

, where

a \ll b

and the thickness

h

is small, is subjected to a uniform distributed load perpendicular to the plane of the plate. Such a plate takes the shape of the surface of a cylinder.

Simply supported plate with axially fixed ends

For a simply supported plate under cylindrical bending with edges that are free to rotate but have a fixed

x_1

. Cylindrical bending solutions can be found using the Navier and Levy techniques.

Bending of thick Mindlin plates

For thick plates, we have to consider the effect of through-the-thickness shears on the orientation of the normal to the mid-surface after deformation. Raymond D. Mindlin's theory provides one approach for find the deformation and stresses in such plates. Solutions to Mindlin's theory can be derived from the equivalent Kirchhoff-Love solutions using canonical relations.Lim, G. T. and Reddy, J. N., 2003, ''On canonical bending relationships for plates'', International Journal of Solids and Structures, vol. 40, pp. 3039-3067.

Governing equations

The canonical governing equation for isotropic thick plates can be expressed as :

\begin
    & \nabla^2 \left(\mathcal - \frac\,q\right) = -q \\
    & \kappa G h\left(\nabla^2 w + \frac\right) =
      -\left(1 - \cfrac\right)q \\
    & \nabla^2 \left(\frac - \frac\right)
      = c^2\left(\frac - \frac\right)
  \end

where

q

is the applied transverse load,

G

is the shear modulus,

D = Eh^3/ 2(1-\nu^2) /math>
is the bending rigidity, h is the plate thickness, c^2 = 2\kappa G h/ (1-\nu) /math>, \kappa is the shear correction factor, E is the Young's modulus, \nu is the Poisson's 
ratio, and
: \mathcal  = D\left mathcal\left(\frac + \frac\right)
    - (1-\mathcal)\nabla^2 w\right + \frac\mathcal  \,. In Mindlin's theory, w is the transverse displacement of the mid-surface of the plate
and the quantities \varphi_1 and \varphi_2 are the rotations of the mid-surface normal
about the x_2 and x_1 -axes, respectively.  The canonical parameters for this theory
are \mathcal = 1 and \mathcal = 0 .  The shear correction factor \kappa usually has the
value 5/6 .

The solutions to the governing equations can be found if one knows the corresponding 
Kirchhoff-Love solutions by using the relations
: \begin
    w & = w^K + \frac\left(1 - \frac\right)
        - \Phi + \Psi \\
    \varphi_1 & = - \frac
     - \frac\left(1 - \frac - \frac\right)Q_1^K
     + \frac\left(\frac\nabla^2 \Phi + \Phi - \Psi\right)
     + \frac\frac \\
    \varphi_2 & = - \frac
     - \frac\left(1 - \frac - \frac\right)Q_2^K
     + \frac\left(\frac\nabla^2 \Phi + \Phi - \Psi\right)
     + \frac\frac
  \end where w^K is the displacement predicted for a Kirchhoff-Love plate, \Phi is a 
biharmonic function such that \nabla^2 \nabla^2 \Phi = 0, \Psi is a function that satisfies the 
Laplace equation, \nabla^2 \Psi = 0, and
: \begin
    \mathcal & = \mathcal^K + \frac\,q + D \nabla^2 \Phi ~;~~ \mathcal^K := -D\nabla^2 w^K \\
    Q_1^K & = -D\frac\left(\nabla^2 w^K\right) ~,~~
    Q_2^K = -D\frac\left(\nabla^2 w^K\right) \\
    \Omega & = \frac - \frac ~,~~ \nabla^2 \Omega = c^2\Omega \,.
  \end

Simply supported rectangular plates

For simply supported plates, the ''Marcus moment'' sum vanishes, i.e., :

\mathcal = \frac(M_+M_) = D\left(\frac+\frac\right) = 0 \,.

In that case the functions

\Phi

\Psi

\Omega

vanish, and the Mindlin solution is related to the corresponding Kirchhoff solution by :

w = w^K + \frac \,.

Bending of Reissner-Stein cantilever plates

Reissner-Stein theory for cantilever platesE. Reissner and M. Stein. Torsion and transverse bending of cantilever plates. Technical Note 2369, National Advisory Committee for Aeronautics,Washington, 1951. leads to the following coupled ordinary differential equations for a cantilever plate with concentrated end load

q_x(y)

x=a

. :

\begin
    & bD \frac  = 0 \\
    & \frac\,\frac - 2bD(1-\nu)\cfrac = 0
  \end

and the boundary conditions at

x=a

are :

\begin
  & bD\cfrac + q_ = 0 \quad,\quad
   \frac\cfrac -2bD(1-\nu)\cfrac + q_ = 0 \\
  & bD\cfrac = 0 \quad,\quad  \frac\cfrac = 0 \,.
  \end

Solution of this system of two ODEs gives :

right)\right">nu_b(x-a).html" ;"title=" - \frac\, \left(\frac + \tanh[\nu_b(x-a)"> - \frac\, \left(\frac + \tanh[\nu_b(x-a)right)\right \end

where

\nu_b = \sqrt/b

. The bending moments and shear forces corresponding to the displacement

w = w_x + y\theta_x

are :

- \cosh[\nu_b(3x-4a)">nu_b(3x-2a).html" ;"title="2 + \cosh[\nu_b(3x-2a)">2 + \cosh[\nu_b(3x-2a)- \cosh[\nu_b(3x-4a)right. \\ & \qquad \left. - 16\cosh[2\nu_b(x-a)] + 23\cosh[\nu_b(x-2a)] - 23\cosh(\nu_b x)\right]\,. \end

The stresses are :

\sigma_ = \frac\,M_ \quad \text \quad 
  \sigma_ = \frac\,Q_\left(1 - \frac\right)\,.

If the applied load at the edge is constant, we recover the solutions for a beam under a concentrated end load. If the applied load is a linear function of

y

, then :

q_ = \int_^q_0\left(\frac - \frac\right)\,\texty = \frac ~;~~
  q_ = \int_^yq_0\left(\frac - \frac\right)\,\texty = -\frac \,.

References

{{DEFAULTSORT:Bending Of Plates Continuum mechanics

Bending of Kirchhoff-Love plates

Definitions

Moments

Forces

Stresses

Strains

Deflections

Derivation

Small deflection of thin rectangular plates

Circular Kirchhoff-Love plates

Clamped edges

Rectangular Kirchhoff-Love plates

Sinusoidal load

Navier solution

Double trigonometric series equation

Simply-supported plate with general load

Simply-supported plate with uniformly-distributed load

Lévy solution

Moments along edges

Simply-supported plate with uniformly-distributed load

Uniform and symmetric moment load

Cylindrical plate bending

Simply supported plate with axially fixed ends

Bending of thick Mindlin plates

Governing equations

Simply supported rectangular plates

Bending of Reissner-Stein cantilever plates

See also

References