International Journal of Civil, Structural,
Environmental and Infrastructure Engineering
Research and Development (IJCSEIERD)
ISSN 2249-6866
Vol. 3, Issue 3, Aug 2013, 113-118
© TJPRC Pvt. Ltd.
ESTIMATION OF ERROR IN DEFLECTION OF A SIMPLY SUPPORTED BEAM
NIKHIL SHARMA, SAKET SAURABH, VISHNU JOSHI & A. S. SANTHI
School of Mechanical and Building Sciences, VIT University, Vellore, Tamil Nadu, India
ABSTRACT
A beam is a key structural member used in most constructions. As such, the analysis of a beam under loading is of
utmost importance. If a beam is supported at two points, and a load is applied anywhere on the beam, the resulting
deflection can be mathematically estimated using the bending equation. Such estimation is based on certain assumptions
which may not always be true. Due to this the actual results may vary from the theoretical results. A little consideration
will show that the error will vary depending on the magnitude of the load as well as the location of the load on the beam.
The objective of this study is to estimate this error as a function of the load applied and the position of the load. The main
focus of this project is on simply supported beams only.
KEYWORDS: Engineering and Architectural Fields, Geometric Relations
INTRODUCTION
General Background
Any element which can withstand loads by resisting bending is regarded as a beam. A beam finds its application
in many engineering and architectural fields as an integral element of the load carrying assembly. If load is applied on the
beam (load may be point load or uniformly distributed load) it gets deflected from its original position .This deflection is
calculated using the fact that:
(M/I) = (E/R)
(1)
Where
M -Bending moment
I – Moment of inertia
E – Young’s modulus of elasticity
R – Radius of curvature
The above formula is obtained based on the following assumptions:

The material of the beam is homogeneous and isotropic

The values of Young’s Modulus of elasticity is the same in tension and compression

The transverse sections, which were plane before bending, remain plain after bending also

The beam is initially straight and all longitudinal filaments bend into a circular arcs with a common centre of
curvature.

The radius of curvature is large compared with the dimensions of the cross section.

Each layer of the beam is free to expand or contract independently of the layer below or above it.
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Nikhil Sharma, Saket Saurabh, Vishnu Joshi & A. S. Santhi
These assumptions need not be always valid. Deviations from these assumptions are observed more often than
not. As such the deflection values obtained on the basis of this theory will contain errors. The objective of this study is to
minimize or remove this error.
Using simple geometric relations it can be determined that M = EI (d2y/d2x) this basic equation is used to
determine deflection of beams. The above equations can be used to calculate the deflection of the beam theoretically. As
mentioned earlier, the aim of this study is to compare the theoretical results with the experimental results and to estimate
the error by formulating an error function. This function will give us the error at different loads and at different locations
on the beam. The key point to understand is that, a part of the error obtained will arise out of human error and (or)
experimental error and the other part will be due to the fact that the assumptions made in the bending theory are not strictly
valid in reality. Thus, the deflection values obtained from the bending theory inherently contain an error arising out of the
invalidity of the assumptions made in bending theory.
The focus of this study will be on simply supported beams. These are quite commonly used in many structures.
Figure 1 shows a simply supported beam under loading. In case of simply supported beams at the supports there are only
vertical reactions as shown in figure 1
Figure 1: Deflection of Simply Supported Beam with Point Load at Centre
EXPERIMENTAL SETUP
The beam specifications are: Length = 980mm; Cross section = 32 x 10mm; Material = Steel (E = 2x105 N/mm2)
In the experiment the load was applied at the centre. Figure 2 shows the setup for the central loading. Four dial
gauges were placed at four different locations. The results are tabulated and given in Table 1.
Table 1: Deflections at Distance X from the LHS Support
Left
Load
Support
(kgs)
(X=0)
1
0
2
0
3
0
4
0
5
0
Deflection in mm
Dial
Gauge 1
(X=190)
0.2
0.41
0.62
0.83
1.04
Load at Centre
Dial
Dial
Gauge 2
Gauge
(X=340)
3(X=490)
0.32
0.37
0.65
0.76
0.99
1.15
1.34
1.56
1.65
1.96
Dial
Guage4
(X=640)
0.32
0.65
0.99
1.34
1.65
Dial
Gauge 5
(X=790)
0.2
0.41
0.62
0.83
1.04
Right
Support
(X=980)
0
0
0
0
0
115
Estimation of Error in Deflection of a Simply Supported Beam
Figure 2: A Photograph of the Experimental Setup
The theoretical results were obtained using Ansys 12.0. Figure 3 shows the results for different load values
obtained using Ansys 12.
A
B
C
D
E
Figure 3: Ansys Results for Deflection of Beam. A – Deflection at 1kg Load ; B – Deflection at 2kg Load;
C – Deflectionat 3kg Load; D – Deflection at 4kg Load; E – Deflection at 5kg Load. (Deflection Values are in mm)
As expected the theoretical and experimental results do not match. The percentage error is obtained according to
the following formula.
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Nikhil Sharma, Saket Saurabh, Vishnu Joshi & A. S. Santhi
Δ = 100 * (experimental deflection value – theoretical deflection value)/experimental deflection value
Table 2: % Error in the Deflection Values at Different Loads
Load
(kgs)
1kg
2kg
3kg
4kg
5kg
X=0
0
0
0
0
0
X=190
0.37
2.80
3.58
3.97
4.20
X=340
1.52
3.03
4.50
5.90
4.48
% Error
X=490 X=640
2.52
1.52
5.09
3.03
5.91
4.50
7.50
5.90
8.01
4.48
X=790
0.37
2.80
3.58
3.97
4.20
X=980
0.00
0.00
0.00
0.00
0.00
RESULTS AND DISCUSSIONS
The variation of the percentage error with distance from the left support has been plotted in Figure 4. From this
figure it can be concluded that the % error is maximum at the centre and decreases as the distance from the centre increases
on either side.
Figure 4: % Error V/s % of Total Length
From Figure 4 it can be concluded that the error is maximum at the point where the deflection is maximum. Now
as the load increases the peak value of the % error also increases. The bottom most curve is for the least load (1kg) and the
successive curves are for the other loads i.e. 2, 3, 4 and 5kg.
From the bending equation we see that the function for the deflection includes the term x3, ‘x’ being the distance
from the support at which the deflection is to be calculated. Thus we have assumed that the error in the deflection also
takes the form of a cubic curve. The equation of this cubic can be obtained by the following procedure:
Figure 5: Maximum Percentage Error V/s Load
Estimation of Error in Deflection of a Simply Supported Beam
117
Figure 5 shows the variation of the maximum value for the % error with the load. The maximum error varies with
the load according to the following equation.
Δmax = 3.4114 *ln(W) + 2.5407;
(2)
From equation 2 it s clear that as the load increases the % error (Δ) will also increase.
Where Δmax  maximum % error, W – load applied (kgs)
Assuming the equation for the cubic error curve as Δ = a*x3 + b*x2 + c*x + d;
The values of the constants a, b, c and d is obtained from the following boundary conditions:
x = 0 and x = L; Δ = 0 at x = L/2, d (Δ)/dx=0 (as maximum load occurs at the centre); Δmax = 3.4114*ln (W) +
2.5407. Using these conditions the values of a, b, c and d are obtained. These values when substituted in the equation for Δ
will give the final equation for Δ.
The final equation for the % error obtained is as follows:
(3)
Here x – distance from the support to the point to calculate the error;
L – the length of the beam under consideration
Consider a load of 100kg, applied at the centre of a 1000mm long beam then;
Δmax = 3.4114*ln(100) + 2.5407 = 18.25 %;
This means that if the theoretical value of the maximum deflection obtained when a load of 100kg( 981N) is
applied at the centre of a 1m long beam is say 1mm, then the actual value will be 1.1825mm. This is quite significant even
though it appears to be a very small difference. The following points are to be noted as far as the error function is
concerned:

The error function gives the value of the % error at any section of the beam when it is centrally loaded.

The error function will vary depending on the material selected and on the point of application of the force.
This procedure may also be followed for eccentric loads i.e. loads not applied at the centre. For this we have to
find the point where the deflection will be maximum and obtain an equation like equation 2. From there on the procedure is
the same as that for the central loading.
CONCLUSIONS

The experimental deflection values are generally greater than the theoretical deflection values.

The error is maximum at the centre i.e. where the load is maximum. The maximum error obtained in case of
central loading = 8.01% (at 5kg)

The error may be small, but the error increases with an increase in the value of the applied load. Thus at higher
values of loads (of the order of kNs), which are observed in real life problems, the error percentage will play a
significant role.
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Nikhil Sharma, Saket Saurabh, Vishnu Joshi & A. S. Santhi

At smaller loads the % errors obtained have lesser values. But an increasing trend for the values of % errors is
observed with increasing load. The % error as a function of load and the distance of the section under
consideration from the support was obtained as Δ = (-4* Δmax*x2)/L2 + (4* Δmax*x)/L which does not take the
material properties into account. Hence for different materials the function will differ. This equation is obtained
for central loading but the methodology may be extended to eccentric loading problems as well. As a further
extension, the procedure may be applied to fixed and cantilever beams. Thus repeating this procedure for as many
materials as possible and with loads applied at different locations a database of such error functions may be
created which will help us predict the values of deflection of a simply supported beam with much better accuracy
as a result of which in many situations it will allow for a relaxation in the Factor of Safety, which will lead to
reduction of cost.
REFERENCES
1.
Dr. R. K. Bansal (2010) - Strength of Materials. India: Laxmi Publication
2.
Mechanics of Materials Laboratory, Beam deflection test, David Clark, Jacob Parton, Zachary Tyler, Andrew
Smith. (www.scribd.com/doc/136561/Mechanics-of-Materials-Beam -Deflection-Test)
3.
Ansys Mechanical APDL (Version 12). [Computer Software] Pennsylvania, Ansys Inc.