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# Bending

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### Bending

Bending of an I-beam

In applied 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. For example, 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. Therefore, to make the usage of the term more precise, engineers refer to a specific object such as; the bending of rods,[2] the bending of beams,[1] the bending of plates,[3] the bending of shells[2] and so on.

## Contents

• Quasistatic bending of beams 1
• Euler–Bernoulli bending theory 1.1
• Extensions of Euler-Bernoulli beam bending theory 1.2
• Plastic bending 1.2.1
• Complex or asymmetrical bending 1.2.2
• Large bending deformation 1.2.3
• Timoshenko bending theory 1.3
• Dynamic bending of beams 2
• Euler-Bernoulli theory 2.1
• Free vibrations 2.1.1
• Timoshenko-Rayleigh theory 2.2
• Free vibrations 2.2.1
• Quasistatic bending of plates 3
• Kirchhoff-Love theory of plates 3.1
• Mindlin-Reissner theory of plates 3.2
• Dynamic bending of plates 4
• Dynamics of thin Kirchhoff plates 4.1
• References 6

## Quasistatic bending of beams

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 when some simplifying assumptions are used.[1]

### Euler–Bernoulli bending theory

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.

Consider beams where the following are true:

• The beam is originally straight and slender, and any taper is slight
• The material is isotropic (or orthotropic), linear elastic, and homogeneous across any cross section (but not necessarily along its length)
• Only small deflections are considered

In this case, the equation describing beam deflection (w) can be approximated as:

\cfrac{\mathrm{d}^2 w(x)}{\mathrm{d} x^2}=\frac{M(x)}{E(x) I(x)}

where the second derivative of its deflected shape with respect to x is interpreted as its curvature, E is the Young's modulus, I is the area moment of inertia of the cross-section, and M is the internal bending moment in the beam.

If, in addition, the beam is homogeneous along its length as well, and not tapered (i.e. constant cross section), and deflects under an applied transverse load q(x), it can be shown that:[1]

EI~\cfrac{\mathrm{d}^4 w(x)}{\mathrm{d} x^4} = q(x)

This is the Euler-Bernoulli equation for beam bending.

After a solution for the displacement of the beam has been obtained, the bending moment (M) and shear force (Q) in the beam can be calculated using the relations

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 (or orthotropic) 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]

{\sigma}= \frac{M y}{I_x}

where

• {\sigma} is the bending stress
• M - the moment about the neutral axis
• y - the perpendicular distance to the neutral axis
• Ix - the second moment of area about the neutral axis x.

### Extensions of Euler-Bernoulli beam bending theory

#### Plastic bending

The equation \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

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

{\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 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, M_y and M_z are the bending moments about the y and z centroid axes, I_y and I_z are the second moments of area (distinct from moments of inertia) about the y and z axes, and 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 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

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 \rho is smaller than ten section heights h:

\rho < 10 h.

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

\sigma = \frac {F} {A} + \frac {M} {\rho A} + {\frac {M} y{\frac {\rho}{\rho +y}}

where

F is the normal force
A is the section area
M is the bending moment
\rho is the local bending radius (the radius of bending at the current section)

and the bending equation can be written as

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

\hat{w} = A_1\cosh(\beta x) + A_2\sinh(\beta x) + A_3\cos(\beta x) + A_4\sin(\beta x)

where A_1,A_2,A_3,A_4 are constants and

\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

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]

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 J = \tfrac{mI}{A} is the polar moment of inertia of the cross-section, m = \rho A is the mass per unit length of the beam, \rho is the density of the beam, A is the cross-sectional area, G is the shear modulus, and k is a shear correction factor. For materials with Poisson's ratios (\nu) close to 0.3, the shear correction factor are approximately

\begin{array}{rcl} 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{array}

#### Free vibrations

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 w must have the same form to cancel out and hence as solution of the form e^{kx} may be expected. This observation leads to the characteristic equation

\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

k_1 = +\sqrt{z_+} ~,~~ k_2 = -\sqrt{z_+} ~,~~ k_3 = +\sqrt{z_-} ~,~~ k_4 = -\sqrt{z_-}

where

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

\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

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

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

\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 \mathbf{u} is the displacement of a point in the plate and w^0 is the displacement of the mid-surface.

The strain-displacement relations are

\begin{align} \varepsilon_{\alpha\beta} & = - x_3~w^0_{,\alpha\beta} \\ \varepsilon_{\alpha 3} & = 0 \\ \varepsilon_{33} & = 0 \end{align}

The equilibrium equations are

M_{\alpha\beta,\alpha\beta} + q(x) = 0 ~;~~ M_{\alpha\beta} := \int_{-h}^h x_3~\sigma_{\alpha\beta}~dx_3

where 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

w^0_{,1111} + 2~w^0_{,1212} + w^0_{,2222} = 0

In direct tensor notation,

\nabla^2\nabla^2 w = 0

### Mindlin-Reissner theory of plates

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

\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 \varphi_\alpha are the rotations of the normal.

The strain-displacement relations that result from these assumptions are

\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 \kappa is a shear correction factor.

The equilibrium equations are

\begin{align} & M_{\alpha\beta,\beta}-Q_\alpha = 0 \\ & Q_{\alpha,\alpha}+q = 0 \end{align}

where

Q_\alpha := \kappa~\int_{-h}^h \sigma_{\alpha 3}~dx_3

## Dynamic bending of plates

### Dynamics of thin Kirchhoff plates

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

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 \rho = \rho(x),

J_1 := \int_{-h}^h \rho~dx_3 ~;~~ J_3 := \int_{-h}^h x_3^2~\rho~dx_3

and

\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.

## References

1. ^ a b c d Boresi, A. P. and Schmidt, R. J. and Sidebottom, O. M., 1993, Advanced mechanics of materials, John Wiley and Sons, New York.
2. ^ a b Libai, A. and Simmonds, J. G., 1998, The nonlinear theory of elastic shells, Cambridge University Press.
3. ^ Timoshenko, S. and Woinowsky-Krieger, S., 1959, Theory of plates and shells, McGraw-Hill.
4. ^ Shigley J, "Mechanical Engineering Design", p44, International Edition, pub McGraw Hill, 1986, ISBN 0-07-100292-8
5. ^ Gere, J. M. and Timoshenko, S.P., 1997, Mechanics of Materials, PWS Publishing Company.
6. ^ Cook and Young, 1995, Advanced Mechanics of Materials, Macmillan Publishing Company: New York
7. ^ a b c Thomson, W. T., 1981, Theory of Vibration with Applications
8. ^ Han, S. M, Benaroya, H. and Wei, T., 1999, "Dynamics of transversely vibrating beams using four engineering theories," Journal of Sound and Vibration, vol. 226, no. 5, pp. 935-988.
9. ^ Rosinger, H. E. and Ritchie, I. G., 1977, On Timoshenko's correction for shear in vibrating isotropic beams, J. Phys. D: Appl. Phys., vol. 10, pp. 1461-1466.