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Continuum mechanics
Laws Conservation of mass Conservation of momentum Conservation of energy Entropy inequality
Solid mechanics Solids Stress · Deformation Compatibility Finite strain · Infinitesimal strain Elasticity (linear) · Plasticity Bending · Hooke's law Failure theory Fracture mechanics Frictionless/Frictional Contact mechanics
Fluid mechanics Fluids Fluid statics · Fluid dynamics Surface tension Navier–Stokes equations Viscosity: Newtonian, Non-Newtonian
Rheology Viscoelasticity Smart fluids: Magnetorheological Electrorheological Ferrofluids Rheometry · Rheometer
Scientists Bernoulli · Cauchy · Hooke Navier · Newton · Stokes
v t e

Bending of an I-beam

In engineering mechanics, bending (also known as flexure) characterizes the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element. The structural element is assumed to be such that at least one of its dimensions is a small fraction, typically 1/10 or less, of the other two.^[1] When the length is considerably longer than the width and the thickness, the element is called a beam. A closet rod sagging under the weight of clothes on clothes hangers is an example of a beam experiencing bending. On the other hand, a shell is a structure of any geometric form where the length and the width are of the same order of magnitude but the thickness of the structure (known as the 'wall') is considerably smaller. A large diameter, but thin-walled, short tube supported at its ends and loaded laterally is an example of a shell experiencing bending.

In the absence of a qualifier, the term bending is ambiguous because bending can occur locally in all objects. To make the usage of the term more precise, engineers refer to the bending of rods,^[2] the bending of beams,^[1] the bending of plates,^[3] the bending of shells^[2] and so on.

Quasistatic bending of beams[link]

A beam deforms and stresses develop inside it when a transverse load is applied on it. In the quasistatic case, the amount of bending deflection and the stresses that develop are assumed not to change over time. In a horizontal beam supported at the ends and loaded downwards in the middle, the material at the over-side of the beam is compressed while the material at the underside is stretched. There are two forms of internal stresses caused by lateral loads:

• Shear stress parallel to the lateral loading plus complementary shear stress on planes perpendicular to the load direction;
• Direct compressive stress in the upper region of the beam, and direct tensile stress in the lower region of the beam.

These last two forces form a couple or moment as they are equal in magnitude and opposite in direction. This bending moment resists the sagging deformation characteristic of a beam experiencing bending. The stress distribution in a beam can be predicted quite accurately even when some simplifying assumptions are used.^[1]

Euler-Bernoulli bending theory[link]

Element of a bent beam: the fibers form concentric arcs, the top fibers are compressed and bottom fibers stretched.

Bending moments in a beam

In the Euler-Bernoulli theory of slender beams, a major assumption is that 'plane sections remain plane'. In other words, any deformation due to shear across the section is not accounted for (no shear deformation). Also, this linear distribution is only applicable if the maximum stress is less than the yield stress of the material. For stresses that exceed yield, refer to article plastic bending. At yield, the maximum stress experienced in the section (at the furthest points from the neutral axis of the beam) is defined as the flexural strength.

The Euler-Bernoulli equation for the quasistatic bending of slender, isotropic, homogeneous beams of constant cross-section under an applied transverse load Failed to parse (Missing texvc executable; please see math/README to configure.): q(x)

is[1]

Failed to parse (Missing texvc executable; please see math/README to configure.): EI~\cfrac{\mathrm{d}^4 w(x)}{\mathrm{d} x^4} = q(x)

where Failed to parse (Missing texvc executable; please see math/README to configure.): E

is the Young's modulus, Failed to parse (Missing texvc executable; please see math/README to configure.): I
is the area moment of inertia of the cross-section, and Failed to parse (Missing texvc executable; please see math/README to configure.): w(x)
is the deflection of the neutral axis of the beam.

After a solution for the displacement of the beam has been obtained, the bending moment (Failed to parse (Missing texvc executable; please see math/README to configure.): M ) and shear force (Failed to parse (Missing texvc executable; please see math/README to configure.): Q ) in the beam can be calculated using the relations

Failed to parse (Missing texvc executable; please see math/README to configure.): M(x) = -EI~\cfrac{\mathrm{d}^2 w}{\mathrm{d} x^2} ~;~~ Q(x) = \cfrac{\mathrm{d}M}{\mathrm{d}x}

Simple beam bending is often analyzed with the Euler-Bernoulli beam equation. The conditions for using simple bending theory are ^[4]:

1. The beam is subject to pure bending. This means that the shear force is zero, and that no torsional or axial loads are present.
2. The material is isotropic and homogeneous.
3. The material obeys Hooke's law (it is linearly elastic and will not deform plastically).
4. The beam is initially straight with a cross section that is constant throughout the beam length.
5. The beam has an axis of symmetry in the plane of bending.
6. The proportions of the beam are such that it would fail by bending rather than by crushing, wrinkling or sideways buckling.
7. Cross-sections of the beam remain plane during bending.

Deflection of a beam deflected symmetrically and principle of superposition

Compressive and tensile forces develop in the direction of the beam axis under bending loads. These forces induce stresses on the beam. The maximum compressive stress is found at the uppermost edge of the beam while the maximum tensile stress is located at the lower edge of the beam. Since the stresses between these two opposing maxima vary linearly, there therefore exists a point on the linear path between them where there is no bending stress. The locus of these points is the neutral axis. Because of this area with no stress and the adjacent areas with low stress, using uniform cross section beams in bending is not a particularly efficient means of supporting a load as it does not use the full capacity of the beam until it is on the brink of collapse. Wide-flange beams (I-beams) and truss girders effectively address this inefficiency as they minimize the amount of material in this under-stressed region.

The classic formula for determining the bending stress in a beam under simple bending is^[5]:

Failed to parse (Missing texvc executable; please see math/README to configure.): {\sigma}= \frac{M y}{I_x}

where

• Failed to parse (Missing texvc executable; please see math/README to configure.): {\sigma}

is the bending stress

• M - the moment about the neutral axis
• y - the perpendicular distance to the neutral axis
• I_x - the second moment of area about the neutral axis x

Extensions of Euler-Bernoulli beam bending theory[link]

Plastic bending[link]

The equation Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma = \tfrac{M y}{I_x}

is valid only when the stress at the extreme fiber (i.e. the portion of the beam farthest from the neutral axis) is below the yield stress of the material from which it is constructed. At higher loadings the stress distribution becomes non-linear, and ductile materials will eventually enter a plastic hinge state where the magnitude of the stress is equal to the yield stress everywhere in the beam, with a discontinuity at the neutral axis where the stress changes from tensile to compressive.  This plastic hinge state is typically used as a limit state in the design of steel structures.

Complex or asymmetrical bending[link]

The equation above is only valid if the cross-section is symmetrical. For homogeneous beams with asymmetrical sections, the axial stress in the beam is given by

Failed to parse (Missing texvc executable; please see math/README to configure.): {\sigma_x}(y,z) = -\frac {(M_z~I_y + M_y~I_{yz})} {I_y~I_z - I_{yz}^2}y + \frac {(M_y~I_z + M_z~I_{yz})} {I_y~I_z - I_{yz}^2}z

^[6]

where Failed to parse (Missing texvc executable; please see math/README to configure.): y,z

are the coordinates of a point on the cross section at which the stress is to be determined as shown to the right, Failed to parse (Missing texvc executable; please see math/README to configure.): M_y
and Failed to parse (Missing texvc executable; please see math/README to configure.): M_z
are the bending moments about the y and z centroid axes, Failed to parse (Missing texvc executable; please see math/README to configure.): I_y
and Failed to parse (Missing texvc executable; please see math/README to configure.): I_z
are the second moments of area (distinct from moments of inertia) about the y and z axes, and Failed to parse (Missing texvc executable; please see math/README to configure.): I_{yz}
is the product of moments of area. Using this equation it is possible to calculate the bending stress at any point on the beam cross section regardless of moment orientation or cross-sectional shape. Note that Failed to parse (Missing texvc executable; please see math/README to configure.): M_y, M_z, I_y, I_z, I_{yz}
do not change from one point to another on the cross section.

Large bending deformation[link]

For large deformations of the body, the stress in the cross-section is calculated using an extended version of this formula. First the following assumptions must be made:

1. Assumption of flat sections - before and after deformation the considered section of body remains flat (i.e. is not swirled).
2. Shear and normal stresses in this section that are perpendicular to the normal vector of cross section have no influence on normal stresses that are parallel to this section.

Large bending considerations should be implemented when the bending radius Failed to parse (Missing texvc executable; please see math/README to configure.): \rho

is smaller than ten section heights h:

Failed to parse (Missing texvc executable; please see math/README to configure.): \rho < 10 h

With those assumptions the stress in large bending is calculated as:

Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma = \frac {F} {A} + \frac {M} {\rho A} + {\frac {M} {{I_x}'}}y{\frac {\rho}{\rho +y}}

where

Failed to parse (Missing texvc executable; please see math/README to configure.): F

is the normal force

Failed to parse (Missing texvc executable; please see math/README to configure.): A

is the section area

Failed to parse (Missing texvc executable; please see math/README to configure.): M

is the bending moment

Failed to parse (Missing texvc executable; please see math/README to configure.): \rho

is the local bending radius (the radius of bending at the current section)

Failed to parse (Missing texvc executable; please see math/README to configure.): {{I_x}'}

is the area moment of inertia along the x axis, at the Failed to parse (Missing texvc executable; please see math/README to configure.): y
place (see Steiner's theorem)

Failed to parse (Missing texvc executable; please see math/README to configure.): y

is the position along y axis on the section area in which the stress Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma
is calculated

When bending radius Failed to parse (Missing texvc executable; please see math/README to configure.): \rho

approaches infinity and y<<Failed to parse (Missing texvc executable; please see math/README to configure.): \rho

, the original formula is back:

Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma = {F \over A} \pm \frac {My}{I}

.

Timoshenko bending theory[link]

Deformation of a Timoshenko beam. The normal rotates by an amount Failed to parse (Missing texvc executable; please see math/README to configure.): \theta

which is not equal to Failed to parse (Missing texvc executable; please see math/README to configure.): dw/dx

.

In 1921, Timoshenko improved upon the Euler-Bernoulli theory of beams by adding the effect of shear into the beam equation. The kinematic assumptions of the Timoshenko theory are

• normals to the axis of the beam remain straight after deformation
• there is no change in beam thickness after deformation

However, normals to the axis are not required to remain perpendicular to the axis after deformation.

The equation for the quasistatic bending of a linear elastic, isotropic, homogeneous beam of constant cross-section beam under these assumptions is^[7]

Failed to parse (Missing texvc executable; please see math/README to configure.): EI~\cfrac{\mathrm{d}^4 w}{\mathrm{d} x^4} = q(x) - \cfrac{EI}{k A G}~\cfrac{\mathrm{d}^2 q}{\mathrm{d} x^2}

where Failed to parse (Missing texvc executable; please see math/README to configure.): I

is the area moment of inertia of the cross-section, Failed to parse (Missing texvc executable; please see math/README to configure.): A
is the cross-sectional area, Failed to parse (Missing texvc executable; please see math/README to configure.): G
is the shear modulus, and Failed to parse (Missing texvc executable; please see math/README to configure.): k
is a shear correction factor.  For materials with Poisson's ratios (Failed to parse (Missing texvc executable; please see math/README to configure.): \nu

) close to 0.3, the shear correction factor for a rectangular cross-section is approximately

Failed to parse (Missing texvc executable; please see math/README to configure.): k = \cfrac{5 + 5\nu}{6 + 5\nu}

The rotation (Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi(x) ) of the normal is described by the equation

Failed to parse (Missing texvc executable; please see math/README to configure.): \cfrac{\mathrm{d}\varphi}{\mathrm{d}x} = -\cfrac{\mathrm{d}^2w}{\mathrm{d}x^2} -\cfrac{q(x)}{kAG}

The bending moment (Failed to parse (Missing texvc executable; please see math/README to configure.): M ) and the shear force (Failed to parse (Missing texvc executable; please see math/README to configure.): Q ) are given by

Failed to parse (Missing texvc executable; please see math/README to configure.): M(x) = -EI~ \cfrac{\mathrm{d}\varphi}{\mathrm{d}x} ~;~~ Q(x) = kAG\left(\cfrac{\mathrm{d}w}{\mathrm{d}x}-\varphi\right) = -EI~\cfrac{\mathrm{d}^2\varphi}{\mathrm{d}x^2} = \cfrac{\mathrm{d}M}{\mathrm{d}x}

Dynamic bending of beams[link]

The dynamic bending of beams,^[8] also known as flexural vibrations of beams, was first investigated by Daniel Bernoulli in the late 18th century. Bernoulli's equation of motion of a vibrating beam tended to overestimate the natural frequencies of beams and was improved marginally by Rayleigh in 1877 by the addition of a mid-plane rotation. In 1921 Stephen Timoshenko improved the theory further by incorporating the effect of shear on the dynamic response of bending beams. This allowed the theory to be used for problems involving high frequencies of vibration where the dynamic Euler-Bernoulli theory is inadequate. The Euler-Bernoulli and Timoshenko theories for the dynamic bending of beams continue to be used widely by engineers.

Euler-Bernoulli theory[link]

The Euler-Bernoulli equation for the dynamic bending of slender, isotropic, homogeneous beams of constant cross-section under an applied transverse load Failed to parse (Missing texvc executable; please see math/README to configure.): q(x,t)

is[7]

Failed to parse (Missing texvc executable; please see math/README to configure.): EI~\cfrac{\partial^4 w}{\partial x^4} + m~\cfrac{\partial^2 w}{\partial t^2} = q(x,t)

where Failed to parse (Missing texvc executable; please see math/README to configure.): E

is the Young's modulus, Failed to parse (Missing texvc executable; please see math/README to configure.): I
is the area moment of inertia of the cross-section, Failed to parse (Missing texvc executable; please see math/README to configure.): w(x,t)
is the deflection of the neutral axis of the beam, and Failed to parse (Missing texvc executable; please see math/README to configure.): m
is mass per unit length of the beam.

Free vibrations[link]

For the situation where there is no transverse load on the beam, the bending equation takes the form

Failed to parse (Missing texvc executable; please see math/README to configure.): EI~\cfrac{\partial^4 w}{\partial x^4} + m~\cfrac{\partial^2 w}{\partial t^2} = 0

Free, harmonic vibrations of the beam can then be expressed as

Failed to parse (Missing texvc executable; please see math/README to configure.): w(x,t) = \text{Re}[\hat{w}(x)~e^{-i\omega t}] \quad \implies \quad \cfrac{\partial^2 w}{\partial t^2} = -\omega^2~w(x,t)

and the bending equation can be written as

Failed to parse (Missing texvc executable; please see math/README to configure.): EI~\cfrac{\mathrm{d}^4 \hat{w}}{\mathrm{d}x^4} - m\omega^2\hat{w} = 0

The general solution of the above equation is

Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{w} = A_1\cosh(\beta x) + A_2\sinh(\beta x) + A_3\cos(\beta x) + A_4\sin(\beta x)

where Failed to parse (Missing texvc executable; please see math/README to configure.): A_1,A_2,A_3,A_4

are constants and

Failed to parse (Missing texvc executable; please see math/README to configure.): \beta := \left(\cfrac{m}{EI}~\omega^2\right)^{1/4}

The mode shapes of a cantilevered I-beam
1st lateral bending	1st torsional	1st vertical bending
2nd lateral bending	2nd torsional	2nd vertical bending

Timoshenko-Rayleigh theory[link]

In 1877, Rayleigh proposed an improvement to the dynamic Euler-Bernoulli beam theory by including the effect of rotational inertia of the cross-section of the beam. Timoshenko improved upon that theory in 1922 by adding the effect of shear into the beam equation. Shear deformations of the normal to the mid-surface of the beam are allowed in the Timoshenko-Rayleigh theory.

The equation for the bending of a linear elastic, isotropic, homogeneous beam of constant cross-section beam under these assumptions is ^[7][9]

Failed to parse (Missing texvc executable; please see math/README to configure.): EI~\cfrac{\partial^4 w}{\partial x^4} + m~\cfrac{\partial^2 w}{\partial t^2} - \left(J + \cfrac{E I m}{k A G}\right)\cfrac{\partial^4 w}{\partial x^2~\partial t^2} + \cfrac{J m}{k A G}~\cfrac{\partial^4 w}{\partial t^4} = q(x,t) + \cfrac{J}{k A G}~\cfrac{\partial^2 q}{\partial t^2} - \cfrac{EI}{k A G}~\cfrac{\partial^2 q}{\partial x^2}

where Failed to parse (Missing texvc executable; please see math/README to configure.): J = \tfrac{mI}{A}

is the polar moment of inertia of the cross-section, Failed to parse (Missing texvc executable; please see math/README to configure.): m = \rho A
is the mass per unit length of the beam, Failed to parse (Missing texvc executable; please see math/README to configure.): \rho
is the density of the beam, Failed to parse (Missing texvc executable; please see math/README to configure.): A
is the cross-sectional area, Failed to parse (Missing texvc executable; please see math/README to configure.): G
is the shear modulus, and Failed to parse (Missing texvc executable; please see math/README to configure.): k
is a shear correction factor.  For materials with Poisson's ratios (Failed to parse (Missing texvc executable; please see math/README to configure.): \nu

) close to 0.3, the shear correction factor are approximately

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} k &= \tfrac{5 + 5\nu}{6 + 5\nu} \quad \text{rectangular cross-section}\\ &= \tfrac{6 + 12\nu + 6\nu^2}{7 + 12\nu + 4\nu^2} \quad \text{circular cross-section} \end{align}

Free vibrations[link]

For free, harmonic vibrations the Timoshenko-Rayleigh equations take the form

Failed to parse (Missing texvc executable; please see math/README to configure.): EI~\cfrac{\mathrm{d}^4 \hat{w}}{\mathrm{d} x^4} + m\omega^2\left(\cfrac{J}{m} + \cfrac{E I}{k A G}\right)\cfrac{\mathrm{d}^2 \hat{w}}{\mathrm{d} x^2} + m\omega^2\left(\cfrac{\omega^2 J}{k A G}-1\right)~\hat{w} = 0

This equation can be solved by noting that all the derivatives of Failed to parse (Missing texvc executable; please see math/README to configure.): w

must have the same form to cancel out and hence as solution of the form Failed to parse (Missing texvc executable; please see math/README to configure.): e^{kx}
may be expected.  This observation leads to the characteristic equation

Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha~k^4 + \beta~k^2 + \gamma = 0 ~;~~ \alpha := EI ~,~~ \beta := m\omega^2\left(\cfrac{J}{m} + \cfrac{E I}{k A G}\right) ~,~~ \gamma := m\omega^2\left(\cfrac{\omega^2 J}{k A G}-1\right)

The solutions of this quartic equation are

Failed to parse (Missing texvc executable; please see math/README to configure.): k_1 = +\sqrt{z_+} ~,~~ k_2 = -\sqrt{z_+} ~,~~ k_3 = +\sqrt{z_-} ~,~~ k_4 = -\sqrt{z_-}

where

Failed to parse (Missing texvc executable; please see math/README to configure.): z_+ := \cfrac{-\beta + \sqrt{\beta^2 - 4\alpha\gamma}}{2\alpha} ~,~~ z_-:= \cfrac{-\beta - \sqrt{\beta^2 - 4\alpha\gamma}}{2\alpha}

The general solution of the Timoshenko-Rayleigh beam equation for free vibrations can then be written as

Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{w} = A_1~e^{k_1 x} + A_2~e^{-k_1 x} + A_3~e^{k_3 x} + A_4~e^{-k_3 x}

Quasistatic bending of plates[link]

Deformation of a thin plate highlighting the displacement, the mid-surface (red) and the normal to the mid-surface (blue)

The defining feature of beams is that one of the dimensions is much larger than the other two. A structure is called a plate when it is flat and one of its dimensions is much smaller than the other two. There are several theories that attempt to describe the deformation and stress in a plate under applied loads two of which have been used widely. These are

• the Kirchhoff-Love theory of plates (also called classical plate theory)
• the Mindlin-Reissner plate theory (also called the first-order shear theory of plates)

Kirchhoff-Love theory of plates[link]

The assumptions of Kirchhoff-Love theory are

• straight lines normal to the mid-surface remain straight after deformation
• straight lines normal to the mid-surface remain normal to the mid-surface after deformation
• the thickness of the plate does not change during a deformation.

These assumptions imply that

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} u_\alpha(\mathbf{x}) & = - x_3~\frac{\partial w^0}{\partial x_\alpha} = - x_3~w^0_{,\alpha} ~;~~\alpha=1,2 \\ u_3(\mathbf{x}) & = w^0(x_1, x_2) \end{align}

where Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{u}

is the displacement of a point in the plate and Failed to parse (Missing texvc executable; please see math/README to configure.): w^0
is the displacement of the mid-surface.

The strain-displacement relations are

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \varepsilon_{\alpha\beta} & = - x_3~w^0_{,\alpha\beta} \\ \varepsilon_{\alpha 3} & = 0 \\ \varepsilon_{33} & = 0 \end{align}

The equilibrium equations are

Failed to parse (Missing texvc executable; please see math/README to configure.): M_{\alpha\beta,\alpha\beta} + q(x) = 0 ~;~~ M_{\alpha\beta} := \int_{-h}^h x_3~\sigma_{\alpha\beta}~dx_3

where Failed to parse (Missing texvc executable; please see math/README to configure.): q(x)

is an applied load normal to the surface of the plate.

In terms of displacements, the equilibrium equations for an isotropic, linear elastic plate in the absence of external load can be written as

Failed to parse (Missing texvc executable; please see math/README to configure.): w^0_{,1111} + 2~w^0_{,1212} + w^0_{,2222} = 0

In direct tensor notation,

Failed to parse (Missing texvc executable; please see math/README to configure.): \nabla^2\nabla^2 w = 0

Mindlin-Reissner theory of plates[link]

The special assumption of this theory is that normals to the mid-surface remain straight and inextensible but not necessarily normal to the mid-surface after deformation. The displacements of the plate are given by

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} u_\alpha(\mathbf{x}) & = - x_3~\varphi_\alpha ~;~~\alpha=1,2 \\ u_3(\mathbf{x}) & = w^0(x_1, x_2) \end{align}

where Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi_\alpha

are the rotations of the normal.

The strain-displacement relations that result from these assumptions are

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \varepsilon_{\alpha\beta} & = - x_3~\varphi_{\alpha,\beta} \\ \varepsilon_{\alpha 3} & = \cfrac{1}{2}~\kappa\left(w^0_{,\alpha}- \varphi_\alpha\right) \\ \varepsilon_{33} & = 0 \end{align}

where Failed to parse (Missing texvc executable; please see math/README to configure.): \kappa

is a shear correction factor.

The equilibrium equations are

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} & M_{\alpha\beta,\beta}-Q_\alpha = 0 \\ & Q_{\alpha,\alpha}+q = 0 \end{align}

where

Failed to parse (Missing texvc executable; please see math/README to configure.): Q_\alpha := \kappa~\int_{-h}^h \sigma_{\alpha 3}~dx_3

Dynamic bending of plates[link]

Dynamics of thin Kirchhoff plates[link]

The dynamic theory of plates determines the propagation of waves in the plates, and the study of standing waves and vibration modes. The equations that govern the dynamic bending of Kirchhoff plates are

Failed to parse (Missing texvc executable; please see math/README to configure.): M_{\alpha\beta,\alpha\beta} - q(x,t) = J_1~\ddot{w}^0 - J_3~\ddot{w}^0_{,\alpha\alpha}

where, for a plate with density Failed to parse (Missing texvc executable; please see math/README to configure.): \rho = \rho(x) ,

Failed to parse (Missing texvc executable; please see math/README to configure.): J_1 := \int_{-h}^h \rho~dx_3 ~;~~ J_3 := \int_{-h}^h x_3^2~\rho~dx_3

and

Failed to parse (Missing texvc executable; please see math/README to configure.): \ddot{w}^0 = \frac{\partial^2 w^0}{\partial t^2} ~;~~ \ddot{w}^0_{,\alpha\beta} = \frac{\partial^2 \ddot{w}^0}{\partial x_\alpha \partial x_\beta}

The figures below show some vibrational modes of a circular plate.

• 

  mode k = 0, p = 1

• 

  mode k = 0, p = 2

• 

  mode k = 1, p = 2

See also[link]

• Bending of plates
• Contraflexure
• Flexure bearing
• Shear strength
• Sandwich theory
• Timoshenko beam theory
• Vibration
• Vibration of plates
• List of area moments of inertia

References[link]

External links[link]

• Flexure formulae

Chris Powell

Personal information
Full name	Christopher George Robin Powell
Date of birth	(1969-09-08) 8 September 1969 (age 42)
Place of birth	Lambeth, England
Height	5 ft 10 in (1.78 m)
Playing position	Left back
Club information
Current club	Charlton Athletic (manager)
Senior career*
Years	Team	Apps†	(Gls)†
1987–1990	Crystal Palace	3	(0)
1990	→ Aldershot (loan)	11	(0)
1990–1996	Southend United	248	(3)
1996–1998	Derby County	91	(1)
1998–2004	Charlton Athletic	200	(1)
2004	→ West Ham United (loan)	13	(0)
2004–2005	West Ham United	23	(0)
2005–2006	Charlton Athletic	27	(0)
2006–2007	Watford	15	(0)
2007–2008	Charlton Athletic	17	(1)
2008–2010	Leicester City	19	(0)
Total		667	(6)
National team
2001–2002	England	5	(0)
Teams managed
2010	Leicester City (caretaker)
2011–	Charlton Athletic
* Senior club appearances and goals counted for the domestic league only. † Appearances (Goals).

Christopher George Robin "Chris" Powell^[1] (born 8 September 1969) is an English football manager and retired player, who had a long career as a left back, crowned by five caps for the English national team. He is currently manager of Charlton Athletic, where he had three successful spells as a player over eight seasons.^[2]

He served as chairman of the Professional Footballers' Association from November 2005 until he retired. He finished his playing career at Leicester City where he then became a first team coach. In October 2009 he was named as one of 50 ambassadors for England's 2018 World Cup bid.^[3] In April 2012, Charlton Athletic were promoted to The Championship as League One Champions in Powell's first full season as a manager.

Playing career[link]

Early career[link]

After leaving Raynes Park High School, Powell spent the 1985/86 season playing for the Epsom & Ewell youth team alongside a young Matt Elliott. He began his professional career in December 1987 as a trainee at Crystal Palace with Ian Wright but failed to become a regular with the first team. He spent time on loan at Aldershot, where he made 11 appearances, before moving on a free transfer in August 1990 to Essex club Southend United. In six years at the club Powell made 288 first team appearances. In November 2004 a BBC poll of Southend United supporters voted Powell their all-time cult hero. One fan described him as "A very skilled, left-footed defender, who could quite possibly be considered one of the nicest men in football, a true gent."^[4]

Derby County[link]

In January 1996 Powell moved to Derby County for a fee of £750,000. He made 99 first team appearances before being transferred to Charlton Athletic in June 1998 for a fee of £825,000. Powell was a favourite of the Derby fans, and won the Supporters' Player of the Year award for season 1996/7. He scored twice during his spell at Derby with both goals coming in the 1997/98 season against Everton in the league^[5] and Southampton in the FA Cup.^[6]

Charlton Athletic[link]

While at Charlton, he made 206 first team appearances, and evidently caught the eye of Sven-Göran Eriksson, who picked him for the England team in 2001, at the unusually late age of 31. He was the oldest England debutant since Syd Owen in 1954. At Charlton he delighted the crowds after each home victory with his exuberant leaps of celebration. Powell scored twice during his first spell with the Addicks, with both goals coming against Tottenham Hotspur; once in the league^[7] and once in the FA Cup.^[8]

Shortly after the start of the 2004–05 season, Powell was allowed to move to West Ham United, where he played a major part in their successful promotion campaign. However, during the summer of 2005 he and West Ham were unable to agree terms for a further contract and he returned to Charlton on a one year contract, with a suggestion that he would develop his coaching skills.

Typically consistent performance belying his years throughout 2005–06 saw Powell regain his place as Charlton's first choice left back, rotated with loanee Jonathan Spector, due to Hermann Hreiðarsson's move to the centre of defence. On 1 July 2006 he signed a one year contract with Premiership newcomers Watford as Aidy Boothroyd looked to add some experience to his youthful squad. He was released by the club after one year.^[9] In July 2007, Powell returned to Charlton for a third spell, primarily as a coach.^[10] He made his last appearance for Charlton on 4 May 2008, coming on as substitute for the last five minutes in the final game of the season, against Coventry, and scoring only his third goal for the club to seal a 4–1 victory.^[11]

Leicester City[link]

Powell joined League Two side Bournemouth on a trial basis in July 2008.^[12] whilst the club looked to see if he could bring much needed experience to their defence.^[13] He joined Leicester City on a six-month contract on 23 August 2008.^[14] He made his debut against Fulham in Leicester's League Cup second round 3–2 defeat, playing the full 90 minutes.^[15]

On 6 December 2008 Powell started his 750th career appearance against former club Southend; Leicester won the game 3–0.^[16] He signed a contract extension that would see him stay at the club intil the end of the 2008–09 season on 5 January 2009.^[17]

In July 2009, Powell signed a contract to be a player-coach at Leicester City, becoming first team development coach, and also re-registering himself as a player.^[18] With teammates Bruno Berner out with a calf injury and Ryan McGivern struggling with form, Powell played his first league match on 26 January 2010, in a 1–0 defeat to Barnsley.^[19] He retired from playing at the end of the 2009–10 season; his final appearance came on 27 March 2010 against his former club Derby County.^[20]

Coaching career[link]

Having served as a player coach under Nigel Pearson, Powell was retained by incoming Leicester manager Paulo Sousa in the summer of 2010. In October, Powell was named caretaker manager with Mike Stowell of the Foxes after Sousa was sacked. Their first, and only, game in charge came against Scunthorpe United; Leicester won the game 3–1. After Leicester's third goal, Powell showed his excitement by running down to the corner flag to celebrate with the players.^{[citation needed]} He remained a coach at the club after Sousa's replacement, Sven-Göran Eriksson, was appointed. Eriksson—who selected Powell for his England caps—commented in December 2010 that Powell "will be a very good manager in the future".^[21]

Management career[link]

On 14 January 2011 Powell was appointed manager of Charlton Athletic,^[22] who were by now languishing in League One. He signed a three and a half year contract^[23] following the sacking of Phil Parkinson.

His first signing of his managerial career was Bradley Wright-Phillips, who went on to score on his debut on 1 February 2011, against Colchester United. Wright-Phillips then progressed to score in all of the next 3 games. Powell won his opening four games at Charlton, but then went 11 games without a win, falling out of the play-off places and leaving the Addicks in mid-table by the end of the season.

Powell signed 19 new players in summer 2011, including midfielders Danny Green, Danny Hollands, Dale Stephens, defenders Matt Taylor (footballer), Rhoys Wiggins and goalkeeper Ben Hamer. Bradley Pritchard was signed from non-league Hayes & Yeading United Football Club and has excelled in midfield, with Yann Kermorgant forming a formidable strike partnership alongside Wright-Phillips. Captain Johnnie Jackson has been a talisman all season despite numerous injuries, and Chris Solly, a product of the youth academy has made the right back berth his own. Clever loan signings were made throughout the season including Hogan Ephraim, Darel Russell, Dany N'Guessan and the return of Lee Cook. Charlton powered strongly into the second half of the 2011/12 season and sealed promotion back to the Championship as Champions of League One.

Managerial statistics[link]

Team	Nat	From	To	Record
				P	W	D	L	Win %
Leicester City (caretaker)		1 October 2010	3 October 2010	1	1	0	0	100
Charlton Athletic		17 January 2011	Present	76	39	18	19	51.32
Total				77	40	18	19	51.95

As of 5 May 2012

Career statistics[link]

Correct as of 17 May 2009 (2009 -05-17)^[update]

• NOTE teams in italics indicate loan period

Honours[link]

Derby County

• Football League First Division runners-up (1): 1995–96

Charlton Athletic

• Football League First Division Winner (1): 1999–2000

West Ham United

• Football Championship play-off winner: 2004–05

Leicester City

• Football League One Winner: 2008–09

Managerial Honours[link]

Charlton Athletic

• Football League One (1): 2011–12
• Football League One Manager of the Month (1): November 2011, January 2012, February 2012.
• League Managers Association League One Manager of the Year (1): 2011–12

References[link]

External links[link]

Wikimedia Commons has media related to: Chris Powell

• Chris Powell career stats at Soccerbase
• Chris Powell Official Website at officialplayerwebsites.com