Deflection of Beams and Cantilevers
1-Introduction
The deflection of beams and cantilevers experiment gives a
visualization and proof of the basic concepts of beam deflection, end
fixing conditions and young modulus. In this experiment, the flexural
stress-strain diagram for an aluminum, brass and steel alloys will be
obtained by loading a beam. With the dimensions of the beam known, the
stress and the deflection as a function of the applied load can be
calculated quite accurately from the Flexure formula.
2-Theory
The deformation of a beam is usually expressed in terms of its
deflection from its original unloaded position. The deflection is measured
from the original neutral surface of the beam to the neutral surface of the
deformed beam. The configuration assumed by the deformed neutral
surface is known as the elastic curve of the beam.
1
Numerous methods are available for the determination of beam
deflections. These methods include:
1. Double-integration method
2. Area-moment method
3. Strain-energy method (Castigliano's Theorem)
4. Conjugate-beam method
5. Method of superposition
Of these methods, the first two are the ones that are commonly used.
2-1 double integration method
The double integration method is a powerful tool in solving deflection
and slope of a beam at any point because we will be able to get the
equation of the elastic curve. In calculus, the radius of curvature of a
curve y = f(x) is given by
In the derivation of flexure formula, the radius of curvature of a beam is
given as:
Deflection of beams is so small, such that the slope of the elastic curve
dy/dx is very small, and squaring this expression the value becomes
practically negligible, hence:
2
Thus, EI / M = 1 / y''
If EI is constant, the equation may be written as:
where x and y are the coordinates shown in the figure of the elastic curve
of the beam under load, y is the deflection of the beam at any distance x.
E is the modulus of elasticity of the beam, I represent the moment of
inertia about the neutral axis, and M represents the bending moment at a
distance x from the end of the beam. The product EI is called the flexural
rigidity of the beam.
The first integration y' yields the slope of the elastic curve and the
second integration y gives the deflection of the beam at any distance x.
The resulting solution must contain two constants of integration since EI
y" = M is of second order. These two constants must be evaluated from
known conditions concerning the slope deflection at certain points of the
beam. For instance, in the case of a simply supported beam with rigid
supports, at x = 0 and x = L, the deflection y = 0, and in locating the point
of maximum deflection, we simply set the slope of the elastic curve y' to
zero.
3
2-2 Beam Deflections (Area-Moment Method)
Another method of determining the slopes and deflections in beams is
the area-moment method, which involves the area of the moment
diagram.
Theorem I
The change in slope between the tangents drawn to the elastic curve at
any two points A and B is equal to the product of 1/EI multiplied by the
area of the moment diagram between these two points.
Theorem II
The deviation of any point B relative to the tangent drawn to the elastic
curve at any other point A, in a direction perpendicular to the original
position of the beam, is equal to the product of 1/EI multiplied by the
4
moment of an area about B of that part of the moment diagram between
points A and B.
and
Rules of Sign
The deviation at any point is positive if the point lies above the tangent,
negative if the point is below the tangent. Measured from left tangent, if θ
is counterclockwise, the change of slope is positive, negative if θ is
clockwise.
2-2-1 Deflection of Cantilever Beams | Area-Moment Method
Generally, the tangential deviation t is not equal to the beam deflection.
In cantilever beams, however, the tangent drawn to the elastic curve at the
wall is horizontal and coincidence therefore with the neutral axis of the
beam. The tangential deviation in this case is equal to the deflection of
the beam as shown below.
5
From the figure above, the deflection at B denoted as δB is equal to the
deviation of B from a tangent line through A denoted as tB/A. This is
because the tangent line through A lies with the neutral axis of the beam.
2-2-2 Deflections in Simply Supported Beams | Area-Moment Method
The deflection δ at some point B of a simply supported beam can be
obtained by the following steps:
1. Compute
2. Compute
3. Solve δ by ratio and proportion (see figure above).
3-Description of the apparatus
The apparatus consists of a backboard with a digital dial gage which
have a sliding movement along the test beam. Two rigid clamps mount on
the backboard and can hold the beam in any position. Two knife- edge
6
supports also fasten anywhere along the beam. scales printed on the
backboard allow quick and accurate positioning of the digital dial test
indicator, knife-edges and loads.
Fig.1: A schematic for the apparatus
The masses supplied with the equipment give maximum flexibility and
ease of use (Fig.2).
7
Fig.2: Hanger loaded with masses
Figure (3) shows the deflection of beams and cantilevers experiment
assembled in the structures test frame.
Fig.3: The assembled beam tested apparatus.
8
4-Experimental Procedures
1. place the assembled test frame on the workbench.
2. there are two securing nuts in each of the side members of the
frame (on the inner track). Slide them to roughly the positions
indicated by the thumbscrews shown in figure3.
3-lift the backboard into position and have an assistant secure the back
board by threading the thumbscrews into the securing nuts .if necessary,
level the backboard by loosening the thumbscrews on one side ,
repositioning the backboard, and tightening the thumbscrews.
In the first part of this experiment, the deflection of a simply supported
beam subjected to an increasing point load must be measured to establish
again the load deflection relationship.
9
Mass
Actual deflection
Theoretical deflection
(g)
(mm)
(mm)
0
0
0
100
0.40
0.43
200
0.78
0.85
300
1.18
1.28
400
1.60
1.70
500
2.00
2.13
Typical results for experiment (fixed beam length variable load)
10
In the second part of the experiment, the beam length must be varied to
establish the relationship between the deflection and the cube of the
beams length.
length
𝑙𝑒𝑛𝑔𝑡ℎ3
deflection
(m)
(𝑚)3
(mm)
200
0.008000
0.27
260
0.017576
0.55
320
0.032768
1
380
0.054872
1.7
440
0.085184
2.6
500
0.125000
3.87
560
0.175616
5.5
Typical results for experiment (fixed beam load variable length)
11
5-Discussion
1- Discuss the proportional relationship between the applied load and
the beam deflection.
2- Examine the fundamental relationship between beam deflection
and the cube of the beam length.
3- Why is stress not directly measurable?
4- Is there any deviation between experimental and theoretical
results? Discuss that.
12