When a beam having an arbitrary cross section is subjected to a transverse loads the beam will bend. In addition to bending the other effects such as twisting and buckling may occur, and to investigate a problem that includes all the combined effects of bending, twisting and buckling could become a complicated one.

**Stresses in beams**

**Preamble:**

When a beam having an arbitrary
cross section is subjected to a transverse loads the beam will bend. In
addition to bending the other effects such as twisting and buckling may occur,
and to investigate a problem that includes all the combined effects of bending,
twisting and buckling could become a complicated one. Thus we are interested to
investigate the bending effects alone, in order to do so, we have to put
certain constraints on the geometry of the beam and the manner of loading.

**Assumptions:**

The
constraints put on the geometry would form the **assumptions:**

1. Beam is
initially **straight** , and has a **constant cross-section.**

2.
Beam is made of **homogeneous material** and
the beam has a **longitudinal plane of** **symmetry. **

3. Resultant
of the applied loads lies in the plane of symmetry.

4. The
geometry of the overall member is such that bending not buckling is the primary
cause of failure.

5. Elastic
limit is nowhere exceeded and **'E'** is same
in tension and compression.

6. Plane
cross - sections remains plane before and after bending.

Let us consider a beam initially
unstressed as shown in fig 1(a). Now the beam is subjected to a constant
bending moment (i.e. „Zero Shearing Force') along its length as would be
obtained by

applying equal couples at each end. The beam will bend to the
radius R as shown in Fig 1(b)

As a result of this bending, the
top fibers of the beam will be subjected to tension and the bottom to
compression it is reasonable to suppose, therefore, **that some where between
the two there** **are points at which the stress is zero. The locus of all
such points is known as neutral axis **.** **The radius of curvature R is
then measured to this axis. For symmetrical sections the N. A. is the axis of
symmetry but what ever the section N. A. will always pass through the centre of
the area or centroid.

As we are aware of the fact
internal reactions developed on any cross-section of a beam may consists of a
resultant normal force, a resultant shear force and a resultant couple. In
order to ensure that the bending effects alone are investigated, we shall put a
constraint on the loading such that the resultant normal and the resultant
shear forces are zero on any cross-section perpendicular to the longitudinal
axis of the member, That means F = 0 since or M = constant.

Thus, the zero shear force means
that the bending moment is constant or the bending is same at every
cross-section of the beam. Such a situation may be visualized or envisaged when
the beam

or some portion of the beam, as
been loaded only by pure couples at its ends. It must be recalled that the
couples are assumed to be loaded in the plane of symmetry.

When a member is loaded in such a
fashion it is said to be in **pure bending.** The examples of pure bending
have been indicated in EX 1and EX 2 as shown below :

When a beam is subjected to pure bending are loaded by the
couples at the ends, certain cross- section gets deformed and we shall have to
make out the conclusion that,

1. Plane
sections originally perpendicular to longitudinal axis of the beam remain plane
and perpendicular to the longitudinal axis even after bending , i.e. the
cross-section A'E', B'F' ( refer Fig 1(a) ) do not get warped or curved.

2. In the
deformed section, the planes of this cross-section have a common intersection
i.e. any time originally parallel to the longitudinal axis of the beam becomes
an arc of circle.

We know that when a beam is under
bending the fibres at the top will be lengthened while at the bottom will be
shortened provided the bending moment M acts at the ends. In between these
there are some fibres which remain unchanged in length that is they are not
strained, that is they do not carry any stress. The plane containing such fibres
is called neutral surface.

The line of intersection between
the neutral surface and the transverse exploratory section is called the
neutral axisNeutral axis **(N A)** .

**Bending Stresses in Beams or Derivation of Elastic
Flexural formula :**

In order to compute the value of
bending stresses developed in a loaded beam, let us consider the two
cross-sections of a beam **HE** and **GF** , originally parallel as shown
in fig 1(a).when the beam

is to bend it is assumed that
these sections remain parallel i.e. **H'E'** and **G'F'** , the final
position of the sections, are still straight lines, they then subtend some
angle q.

Consider now fiber AB in the
material, at adistance y from the N.A, when the beam bends this will stretch to
A'B'

Since CD and C'D' are on the
neutral axis and it is assumed that the Stress on the neutral axis zero.
Therefore, there won't be any strain on the neutral axis

Consider any arbitrary a
cross-section of beam, as shown above now the strain on a fibre at a distance
„y' from the N.A, is given by the expression

Now the termis the property of
the material and is called as a second moment of area of the cross-section and
is denoted by a symbol I.

Therefore M/I = sigma/y
= E/R

**This equation is known as the
Bending Theory Equation.**The above proof has involved the** **assumption
of pure bending without any shear force being present. Therefore this termed as
the pure bending equation. This equation gives distribution of stresses which
are normal to cross-section i.e. in x-direction.

**Stress
variation along the length and in the beam section**

**Bending Stress and Deflection Equation**

In this section, we consider the case of pure bending; i.e.,
where only bending stresses exist as a

result of applied bending
moments. To develop the theory, we will take the phenomenological approach to develop
what is called the '**Euler-Bernoulli theory of beam bending**.'
Geometry:

Consider a long slender straight
beam of length L and cross-sectional area A. We assume the beam is prismatic or
nearly so. The length dimension is large compared to the dimensions of the
cross-section. While the cross-section may be any shape, we will assume that it
is symmetric about the y axis

Loading: For our purposes, we
will consider shear forces or distributed loads that are applied in the y
direction only (on the surface of the beam) and moments about the z-axis. We
have consider examples of such loading in ENGR 211 previously and some examples
are shown below:

Kinematic Observations: In order to obtain a
'feel' for the kinematics (deformation) of a beam

subjected to pure bending loads,
it is informative to conduct an experiment. Consider a rectangular lines
have been scribed on the beam's surface, which are parallel to the top and

bottom surfaces (and thus
parallel to a centroidally placed x-axis along the length of the beam). Lines
are also scribed around the circumference of the beam so that they are
perpendicular to the longitudinals (these circumferential lines form flat
planes as shown). The longitudinal and circumferential lines form a square grid
on the surface. The beam is now bent by moments at each end as shown in the
lower photograph. After loading, we note that the top line has stretched and the
bottom line has shortened (implies that there is strain *exx*). If
measured carefully, we see that the longitudinal line at the center has not
changed length (implies that *exx* = 0 at *y* = 0). The longitudinal
lines now appear to form concentric circular lines.

We also note that the vertical
lines originally perpendicular to the longitudinal lines remain straight

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