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Mechanics of Materials Laboratory
Beam Deflection Test
Date Performed: 2/28/11
Date Due: 3/14/11
Richard Dyar
Group B:
Yazmin Ince
Richard Dyar
Abstract
If a beam is supported at two points, and a load is applied anywhere on the
beam, the resulting deformation can be mathematically estimated. Due to improper
experimental setup, the actual results experienced varied substantially when compared
against the theoretical values. The following procedure explains how the theoretical and
actual values were determined, as well as suggestions for improving upon the
experiment. The percent error remained relatively small, around 11%, for locations close
to supports. Error was experienced when analyzing positions closer to the beam, with
the exception of odd values on gage 4.
1
Background
If a beam is supported at two points, and a load is applied anywhere on the
beam, deformation will occur. When these loads are applied either longitudinally
outside or inside of the supports, this elastic bending can be mathematically predicted
based on material properties and geometry.
Curvature at any point on the beam is calculated from the moment of loading
(M), the stiffness of the material (E), and the first moment of inertia (I.) The following
expression defines the curvature in these parameters as 1/ρ, where ρ is the radius of
curvature.
Equation 1
Equation 1 does not account for shearing stresses.
Curvature can also be found using calculus. Defining y as the deflection and x as
the position along the longitudinal axis, the expression becomes
Equation 2
Central Loading
Central loading on a beam can be thought of as a simple beam with two supports
as shown below.
2
Figure 1
Applying equilibrium to the free body equivalent of Figure 1, several expressions
can be derived to mathematically explain central loading.
Equation 3, 4, and 5
Figure 2 and 3 act as free body diagrams for the section between AB and BC
respectively.
Figure 2
Figure 3
Solving the reactions between AB and BC, equation 1 can be expressed as
3
Equation 6, 7
Integrating twice, Equation 6 becomes
Equation 8, 9
To determine the constants, conditions at certain positions on the beam can be
applied. Knowing the deflection at each of the supports, as well as the slope at the top
of the curve is zero, the constants can be derived to
Equation 10, 11, 12, and 13
Combining Equations 8 and 9 with 10 through 13, the expressions for deflection
can be expressed as
Equation 14, 15
Overhanging Loads
Overhanging loading on a beam is similar to that of central loading. In
overhanging loading, a simple beam is supported with two supports and two loads as
shown below.
4
Figure 4
Using similar methods used previously for central loading, the equation for
determination of deflection as a function of position, load, length, stiffness, and
geometry can be derived as
Equation 16
Procedure
See lab manual section 11
Central Loading
Gage 1 Gage 2 Gage 3 Gage 4
Load
5
Overhanging Loads
Data & Calculations
Central Loading
Table 1 and 2 catalog the dimensions of the beam, as well as the position of the
gages as measured from one of the two fixed supports.
Table 1
Table 2
Gage 1 Gage 2 Gage 3
Load
Load
6
Table 3
Overhanging Loads
Table 4
Table 5
7
Table 6
Results
The theoretical results were not as expected or experienced. There was
significant error between the actual results and theoretical value, especially as the
distance studied approached the midpoint of the beam.
The main source of error within this experiment occurs due to the improper
testing procedure. As seen in Figure 9, the theory used within this exercise is based
upon a beam with one fixed support allowing one degree of freedom, a second support
allowing two degrees of freedom, and a central load.
8
Figure 5
This produces dramatically different results when compared against the actual
setup. When using two knife supports, the setup contains two supports allowing two
degrees of freedom and a central load. This is pictured in Figure 10.
Figure 6
Since both ends are under-constrained, the analysis for the experiment with the above
theory is not accurate.
Another cause of error in the theoretical is the effect of gravity on the beam.
With no applied load, the equations above would return a zero result. This is inaccurate
for beams that are not specifically supported such that gravitational factors are
overcome. Also the measurements and position of gages for overhanging loading may
be incorrect, and the experiment most likely needs to be re-run.
Conclusions
When a load is applied to a beam, either centrally over at another point, the
deflection can be mathematically estimated. Due to the error that occurred in this
exercise, it is clear that margins in safety factors, as well as thorough testing, is needed
when utilizing beam design. It is also important to ensure the scope of the testing
closely models real-world practicality.
9
References
Gilbert, J. A and C. L. Carmen. "Chapter 11 – Beam Deflection Test." MAE/CE 370 –
Mechanics of Materials Laboratory Manual. June 2000.
10