# Dictionary Definition

bending adj : not remaining rigid or straight;
"tried to support his weight on a bending cane"

### Noun

1 movement that causes the formation of a curve
[syn: bend]

2 the property of being bent or deflected [syn:
deflection, deflexion]

3 the act of bending something

# User Contributed Dictionary

## English

### Verb

bending- present participle of bend

#### Declension

# Extensive Definition

- This article is about structural behavior. For other meanings see Bending (disambiguation).

In engineering
mechanics, bending (also known as flexure) characterizes the
behavior of a structural element
subjected to an external load applied perpendicular to the
axis of the element. A structural element subjected to bending is
known as a beam. A
closet rod sagging under the weight of
clothes on clothes
hangers is an example of a beam experiencing bending.

Bending produces reactive forces inside a beam as the beam
attempts to accommodate the flexural load; the material at the top
of the beam is being compressed while the material at the bottom is
being stretched. There are three notable internal forces caused by
lateral loads: shear
parallel to the lateral loading, compression
along the top of the beam, and tension
along the bottom 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 produces the sagging deformation characteristic of
compression
members experiencing bending.

This stress distribution is dependent on a number
of assumptions. First, that 'plane sections remain plane'. In
otherwords, 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.

The compressive and tensile 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.

## Simple or symmetrical bending

Beam bending is analyzed with the Euler-Bernoulli beam equation. The classic formula for determining the bending stress in a member is:- = \frac

simplified for a beam of rectangular
cross-section to:

- = \frac

- is the bending stress
- M - the moment at the neutral axis
- y - the perpendicular distance to the neutral axis
- Ix - the area moment of inertia about the neutral axis x
- b - the width of the section being analyzed
- h - the depth of the section being analyzed

This equation is valid only when the stress at
the extreme fiber (i.e. the portion of the beam furthest from the
neutral axis) is below the yield
stress of the material it is constructed from. 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 unsymmetrical bending

The equation above is, also, only valid if the cross-section is symmetrical. For unsymmetrical sections, the full form of the equation must be used (presented below):- = -\frac x - \frac y.

### Complex bending of homogeneous beams

The complex bending stress equation for elastic,
homogeneous beams is given as where Mx and My are the bending
moments about the x and y centroid axes, respectively. Ix
and Iy are the second moments of area (also known as moments of
inertia) about the x and y axes, respectively, and Ixy is the
product of inertia. Using this equation it would be possible to
calculate the bending stress at any point on the beam cross section
regardless of moment orientation or cross-sectional shape. Note
that Mx, My, Ix, Iy, and Ixy are all unique for a given section
along the length of the beam. In other words, they will not change
from one point to another on the cross section. However, the x and
y variables shown in the equation correspond to the coordinates of
a point on the cross section at which the stress is to be
determined.

## Stress in 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:- Assumption of flat sections - before and after deformation the considered section of body remains flat (i.e. is not swirled).
- 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

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

\sigma = \frac + \frac + 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)
- is the area moment of inertia along the x axis, at the y place (see Steiner's theorem)
- y is the position along y axis on the section area in which the stress \sigma is calculated

When bending radius \rho approaches infinity and
y is zero, the original formula is back:

- \sigma = \pm \frac .

## References

## External links

bending in Bulgarian: Огъване

bending in German: Biegung (Mechanik)

bending in Modern Greek (1453-): Κάμψη

bending in Spanish: Flexión (ingeniería)

bending in French: Flexion (matériau)

bending in Korean: 휨

bending in Italian: Flessione retta

bending in Hebrew: כפיפה

bending in Polish: Zginanie