## There edges, there weight which were attached to the

There are various aims and objectives
which need to be completed within this project. By the end of this project
these aims and objectives shall be completed:

A.
To confirm that a beam of even cross section
bends to a circular arc when exposed to a constant bending moment

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B.
To
demonstrate the radius of curvature, R, changes along the beam when the bending
moment, M, changes

C.
To find the relationship between M and R

D.
To determine Young’s Modulus of elasticity, E,
for the beam material utilizing different beam bending experiments

Introduction

When a beam undergoes bending it doesn’t decrease or increase
in general length. But, the surface of the beam which is on the outside of the
curve should increase in length and the surface on the inside should decrease. To
attain this the longitudinal fibers of the beam should be subjected compression
or tension.

Figure 1 https://skyciv.com/media/tutorials/stress/bending_stress_beam_compression_tension.png

Bending beams leads to compression and tension along the
span of the beam. If an edge on is in tension then it shall have positive
stress (+ve), alternatively if an edge is under compression then it shall have
negative stress (-ve), wherever the distribution of stress is there should be a
point in which the stress is zero. This point is named the neutral area and is
the area with the center line in the above diagram. Engineers remain intrigued
by the deformation of loaded beams. An example of this is simply supported
beams. In this experiment, the primary features of the effect of different
types of bending moments on a beams deflection are investigated.

Theory

When it comes to beading beams there is stress and it is
important that the beam is able to distribute the stress along its length.

Usually, the beam shall bend into a complex shape however, if a small amount of
length is regarded. The bent form may be thought of as an arc of a circle of
radius (R), measured to the neutral axis or neutral plane. In this experiment a
beam was put two simple support knife edges, there weight which were attached
to the sides of the beam and the deflection was measured. It imperative to
understand that a bending beam is in tension on the outside of the curve
(hogging) and is in compression on the inside of the curve (sagging). The
longitudinal planes in the beam stay parallel to the radius (R) throughout
bending.

Figure 2

Figure 3

To find out the position of the neutral axis, the cross
section of the beam should be known. Figure 4 indicates the linear stress
distribution across a beam. The formula for bending moment is:

In this equation the M is the bending moment, this is the
sum of all the moments given by the stresses on all such foundations:

The amount,

, is called second moment of area, which is about the
neutral plane and is given as I, so the second moment of area formula is:

thus making bending moment formula:

The E is the young’s modulus; it is the amount a material
may withstand changes in length under compression or tension. The R is the
radius of curvature of the beam when it undergoes compression and tension, the
radius is infinite when the beam has no bending moments applied and gradually
gets a radius after a moment is applied. There are three different types of
supports, simple support which is a type of support that can only handle
vertical forces and not horizontal, it also can’t handle moments. Another type
of support is pin support, this kind of support handles moments and vertical
forces however it can’t handle horizontal forces, the final type of support is
fixed and it can handle moments, vertical and horizontal forces.

Figure 4

Equipment used

In this lab various apparatus was used beam which was steel,
its width was 31.75mm and it had a height of 6.35mm. In order to calculate the
breadth of the beam three measurement s were taken and an average was taken and
it equated to 31.73mm, the same was done for the depth and it equaled 6.37mm.

Figure 5

Another bit of equipment utilized in this lab was knife edge
supports, these helped support the beam on two knife edges, these simple
supports are able to support vertical forces however can not support horizontal
forces, the beam rests onto the simple supports were the force of the bending
is transferred to.

Figure 6

Picture source: https://community.ptc.com/legacyfs/online/87678_beam.png

A marker pen and a ruler were used to mark out the points in
which the simple supports would be held on the knife edge, the track or marking
for the bridge gauge was done by the ruler and pen, it was an important bit of
apparatus as it was used for all the marking out, without it, it would be
difficult to know were to move the bridge gauge or were to put the supports for
the beam.

Figure 7

Picture source: https://images-na.ssl-images-amazon.com/images/I/51FgCX1NPvL._SY355_.jpg

The most essential piece of apparatus was the bridge gauge;
the bridge gauge was used to measure the deflection. Deflection in structures equates
to the movement of a node or beam from its initial place because of the loads
and forces which are applied to the beam. Deflection may happen from externally
applied forces like weights or from the beams on weight due to gravity.   There are
two types of bridge gauges, the analogue version and the digital, they are both
high in accuracy.