Analysis of deflection is of great importance in design, any
structure without prior testing and analysis when working in engineering could
have major ramifications. Excessive deflection has the possibility to result in
damage of moving parts, broken or jammed parts. This means that knowledge of Deflection
in materials is of paramount importance for many engineers. Therefore it is
necessary for students aspiring to become engineers to have a firm understanding
During first year study in mechanical engineering students
are exposed to many forms of stress and bending which results from this. These stresses
are as follows; shear, torsion and compression. The form of stress and
resulting deformation being considered in this study is shear stress which
results in bending.
These stresses are applicable in the real worlds with many
examples through experience students have seen these occur. However these
examples that students have experienced have limited aid in helping students
understand the underlying principals and concepts. Therefore For many students
it will be helpful in gaining a deeper understanding of bending theory through
experimentation and help students integrate theory into practical knowledge.
Laboratories allow student to appreciate and verify theories
taught and therefore my purpose is to review and improve the existing
laboratories using the latest teaching methods with the intent to improve the
students experience and learning while ensuring that students are adequately
prepared and have sufficient understanding of the theory demonstrated.
To ensure students have sufficient preparation to perform
the correct calculations and understanding relevant to the experiment thorough research
into teaching methods is necessary. This will be extremely useful in finding
the most useful techniques and methodologies which will both aid students in
the long term retention of information and prepare the students for the
experiments which will allow them to gain the most out of the experience.
2.0 Literature review
This projects aim is to improve existing laboratories and
therefore research and understanding into both teaching methods and Bending
theory are necessary as such this literature review will be divided up into 3
sections as follows;
Applying teaching methods to bending theory
Bending is a process that occurs when force is acted upon a
material which produces a V shape along an axis in ductile materials. When bending
occurs the inside surface is getting compressed with the outer surface is in
Generally literature at the level required for students at
this level is uniform with theories and ideas ingrained over many years and
therefore it is quite difficult to find opposing views on any theory
The strain caused can also decrease or increase due to the
radius of the bend with the strain being higher the smaller the bend. When testing
a material many factor must be considered including the thickness, width and
materials properties e.g. young’s modulus and poisons ratio. (2.1 Byars EF,
2.1.1 Initial restrictions
In the experiment a concentrated load is used which is a
load of which has a small area of contact compared to the area of the metal
When experimenting with bending there are many other forms
of failure which have the possibility of occurring if care is not taken
including buckling and twisting which could be a problem when demonstrating to
student as at this time students are unfamiliar with these concepts and the
purpose of the experiment is to demonstrate bending and as such bending should
be the only form of failure observed.
To ensure that these effects are controlled restrictions
must be placed on the beams geometry:
The beam must be straight with a constant square
cross section, made from homogeneous material.
Loads must be applied in the longitudinal plane
Internal reactions in a cross section of any of the beam
could incorporate a resultant shear force or a resultant normal force. To
ensure that bending effects are examined alone restrictions must be placed to
restrict the loading to which the resultant shear and normal forces are zero on
any one section which is perpendicular along the axis that is longitudinal on
The lack of the shear force means that every cross section
of the beam is since the beam is loaded with only pure
couples at the end applied in the same plane of symmetry. This means that the
beam is considered to be in pure bending and the plane of symmetry is also
known as the plane of bending.
This means the beam has the correct
geometry to allow for deformation and reasonable conclusions can be drawn since
there is both no other forms of deformation and there is negligible undesirable
forces on the bar.
In the diagram below U is the distance between the neutral
axis which is shown as EF and the parallel line GH which is the plane of symmetry.
It can be assumed that the difference between these lines is the same Loaded as
when it is loaded.
Therefore the definition for the neutral surface is assumed
The deformation on a fibre of which the original position
was G’H’ then becomes
The definition of axial strain
Since this case considers pure bending
the radius is constant for the length of the entire beam. This means the axial
strain on the longitudinal axis is directly proportional with the distance “u'”
from the neutral surface. The negative symbol shows ta negative strain for a
positive value which is of u. (Benham,
Crawford and Armstrong, 1996) (Gere and Goodno, 2009)
Hookes law is used for stress in this
For elastic strain:
In this case is the normal stress which is due to bending
known as flexure stress.
(Benham, Crawford and
Armstrong, 1996) (Hibbeler, n.d.)
2.1.4 Euler-Bernoulli bending theory
In Euler bending theory the fibres in a bent beam form arcs,
the top fibres are compressed and the bottom fibres are stretched as mentioned
above. The Euler-Bernoulli theory of slender beams the assumption must be made
that Plane sections remain plane.
It also means that there isn’t any deformation due to shear.
This linear distribution only applies if the maximum shear stress which the
material undergoes is less than the yield stress. If stresses exceed the yield
stress it is then known as plastic bending.
The equation for the bending of beaming of beams is:
E I d4w(x) /
dx 4 = q(x)
In the equation; E is youngs modulus, I is the area moment
of inertia, W(x) is also the deflection which the neutral axis of the
beam. When a solution has been found for
the displacement the beam is under has been found, the shear force (Q) and also
the bending moment (M) can be found using the equations below.
M(x) = -EI d2w(x) /
Q(x) = dm / dx
Beam bending generally analysed using the Euler Bernoulli
beam equation. The conditions necessary for using the Euler-Bernoulli simple
bending equations are;
(Cook and Young, 1999)
(Benham, Crawford and Armstrong, 1996)
The beam must be subject to pure bending as to
ensure that the shear force is zero with no torsional and no axial loads would
The material used must be homogeneous and
The materials used must obey Hookes law.
And the bar must be initially straight with a
cross section that is uniform throughout the bar
It must have axis of symmetry in the plane of
The dimensions must be that it wouldn’t fail by
sideways buckling, crushing or wrinkling but by bending.
Bending stress can be determined
with the formula:
= M y / I
In this formula Is the bending stress, y is the perpendicular
distance to the neutral axis M is the moment around the neutral axis, I is the
second moment of area around the neutral axis (x). this equation is considered
a classical equation which gets extended if considering other variations of
(Hibbeler, n.d.) (Boresi
and Schmidt, n.d.)
2.2 Teaching methodology in Laboratories
2.2.1 The goals of Laboratory experiences
Laboratories have a number of goals for students. Most of
these goals are also the goals of science educations in general. (Lunetta, 1998;
Hofstein and Lunetta, 1982). These goals can vary however the core set is
generally consistent. These goals were reviewed then discussed, a list of goals
and desired outcomes for laboratory experiences was compiled (Anderson, 1976;
Hofstein and Lunetta, 1982; Shulman and Tamir, 1973; Lazarowitz and Tamir, 1994).
To enhance the mastery of a subject. – The experiences
may enhance a student’s comprehension and understand of concepts and facts.
To Develop Scientific reasoning – These laboratory
experiences can help develop a student’s ability to understand and identify
questions and concepts.
Developing practical skill – These Laboratories
are used to allow students to learn then use the tools and the conventions
commonly used in science. (Using scientific equipment in the correct manner,
making observations based on findings, carrying out procedures and taking
measurements in the correct manner).
Developing Teamwork abilities – these experiences
can help promote the ability of the student to collaborate effectively with
2.2.2 Recent research in laboratory experiences and its
Generally laboratory experiences have lacked clear learning
goals and in general this approach is still common. Due to this Researchers
tested students in experiments or other scientific activities to determine if
the students underlying understanding of the activity had increased. This was
compared with other methods including videotapes, discussions and lectures. To determine
which were the most effective when compared.
It was found that the best method was to integrate laboratories
in support of other teaching methods was the best methodology. (National
Research Council, 1999), (Glaser, 1994; National Research Council, 1999)
(National Research Council, 2005)
2.2.3 common laboratory experiences
It is commonly claimed that laboratory classes help students
in understanding scientific concepts largely are a resultant of the argument
that these opportunities to observe and manipulate materials can be used to
directly help students to gain a much more substantial grasp of scientific
concepts as a whole. Many researchers and teachers believe that these experience
would help force many students in identifying any misunderstandings they may
have in a concept and therefore change their thoughts towards a more
scientifically accurate way.
However these claims have no direct evidence. These laboratory
experiences that are generally isolated from any flow of instruction and
therefore have no evidence to back up claims that they aid in the learning of
any content of a scientific nature (Hofstein and Lunetta, 1982, 2004;
Lazarowitz and Tamir, 1994).
2.2.4 Development of investigative skills
Research on development of investigative skills has shown the
possibility that students can learn certain aspects of scientific reasoning
using laboratory instruction (Reif and St. John, 1979, cited in Hofstein and
However recent research has shown that during Laboratories
Teaches tend to prioritise procedures in laboratories rather than discussion
for how to plan an investigation or to interpret the results that laboratories
findings have given (Tobin, 1987). Therefore it may be beneficial to
encourage the design and interpretation of the laboratories to encourage the
development of this skills.
This is because Average Laboratories appear to have a low
impact on improving more complex aspects related to scientific reasoning. This can
include the capacity of students to find their own research questions and to
then design experiments and drawing conclusions from the data observed then use
these conclusions to make inferences (Klopfer, 1990, cited in White, 1996).
2.2.5 Developing Teamwork Abilities
Commonly in laboratories teamwork is an integral aspect and
is beginning to be seen as an outcome itself, these laboratory experiences are
now being viewed as an ideal opportunity for developing these skills which are
essential in any scientific pursuit. These team working skills are now seen as
an essential skill for workers in any industry.
2.2.6. Principles for the design of effective experiences in
The research found shows that there are a number of key
factors in the development of good laboratory lessons these are believed to be
Clearly Communicated purposes – Laboratories with clear
goals to guide the experience. These goals should be clearly communicated to
allow students to gain a clear understanding for the reasons behind the
Ongoing Discussion and reflection – Laboratory experiences
shown are generally more effective when focus is placed on the discussion of
activities done during a laboratory and the reflection of the reasons for them rather
than the actual performing of the laboratory activities themselves. The
findings found from research shows that the focus of the laboratories and the
activities surrounding the emphasis should be placed on the development of
explanations to understand the data further rather than confirming the ideas