experimental verification of deflection of beam using ... deflection distance of a beam under a load...

International Journal of Advance Research in Engineering, Science & Technology(IJAREST),

ISSN(O):2393-9877, ISSN(P): 2394-2444, Volume 2, Issue 3, March- 2015

All Rights Reserved, @IJAREST-2015

1

Experimental verification of deflection of beam using theoretical and numerical

approach

Biltu Mahato1, Anil

2, Harish H.V

3

1, 2 Pre-final Year Undergraduate Students Department of Aeronautical Engineering, NMIT Bangalore, [email protected]

3 Assistant Professor Department of Aeronautical Engineering, NMIT Bangalore, [email protected]

Abstract

This study investigates the maximum deflection of simply supported beam and cantilever beam under point loading. Experiments

on these beams have been carried out and maximum deflection has been noted. The experiment has been carried out for different

loads. The results obtained have been validated through theoretical and numerical approach. Numerical approach includes

mathematical and simulation approach. Euler–Bernoulli beam equation is considered for theoretical, finite element methods

(FEM) for mathematical and ANSYS 14.0 for simulation approach. The results obtained through theoretical, FEM and simulation

is very near to experimental results.

Keywords: Simply supported beam, Cantilever beam, Maximum deflection, FEM, ANSYS

I. INTRODUCTION

A beam is a member subjected to loads applied transverse to

the long dimension, causing the member to bend. A beam

which is fixed at one end and free at other end is known as

cantilever beam. Simply supported beam is a beam

supported or resting freely on the supports at its both ends.

The deflection distance of a beam under a load is directly

related to the slope of the deflected shape of the member

under the load. It can be calculated by integrating the

function that mathematically describes the slope of the

member under that load. Deflection can be calculated by

standard formula calculated using Euler–Bernoulli beam

equation, virtual work, direct integration, Castiglione’s

method and Macaulay's method or the direct stiffness

method.

Gargi Majumder et al [1] have conducted finite element

analysis of the beam considering various types of elements

under different loading conditions in ANSYS 11.0. The

various numerical results were generated at different nodal

points by taking the origin of the Cartesian coordinate

system at the fixed end of the beam. The nodal solutions

were analyzed and compared. On comparing the

computational and analytical solutions it was found that for

stresses the 8 node brick element gives the most consistent

results and the variation with the analytical results is

minimum.

Amer M. Ibrahim et al [2] have described a nonlinear finite

element analyses which have been carried out to investigate

the behavior up to failure of simply supported composite

steel-concrete beams with external pre-stressing, in which a

concrete slab is connected together with steel I-beam by

means of headed stud shear connectors, subjected to

symmetrically static loading. ANSYS computer program

(version 12.0) has been used to analyze the three

dimensional model. They studied load deflection behavior,

strain in concrete, strain in steel beam and failure modes.

The nonlinear material and geometrical analysis based on

Incremental-Iterative load method, is adopted. Three models

have been analyzed to verify its capability and efficiency.

The results obtained by finite element solutions have shown

good agreement with experimental results.

From literature survey, we identified that theoretical and

ANSYS simulations have only been carried out. Further in

this study we have taken FEM as a mathematical approach

to validate the results. In this approach we have considered

global stiffness matrix to find the maximum deflection [3].

Along with this Euler–Bernoulli beam equation is

considered for theoretical validation [4…7] and ANSYS

14.0 workbench for simulation [8, 9].

II. METHODOLOGY

The experiment and validation processes have been carried

out in two steps. For experiment, beam deflection test rig

setup was made as shown in figure 1 and figure 2. Steel

beam with rectangular cross section was used. Maximum

deflection was measured at different locations for different

loading varying from 0.2 Kg to 1.0 Kg. The corresponding

deflection was noted at loading and unloading condition

with the help of digital dial gauge for all setup of beam. All

experiment was repeated for various numbers of times at

different time and conditions. The setup and condition

giving minimum percentage error is considered here for

validation.

III. EXPERIMENTAL SETUP AND

RESULTS

The specifications of beam experimental setup are given in

table 1.

http://en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_equation



http://en.wikipedia.org/wiki/Virtual_work

http://en.wikipedia.org/wiki/Direct_integration_of_a_beam

http://en.wikipedia.org/wiki/Castigliano%27s_method

http://en.wikipedia.org/wiki/Castigliano%27s_method

http://en.wikipedia.org/wiki/Macaulay%27s_method

http://en.wikipedia.org/wiki/Direct_stiffness_method

http://en.wikipedia.org/wiki/Direct_stiffness_method





2

Table 1. Specifications of Experimental Setup

Parameters Dimensions

Material Steel

Young's Modulus 2.1*105 N/mm

2

Thickness 5 mm

Breadth 25 mm

Length 600 mm

Moment of Inertia 260.41667 mm4

Figure 1. Simply supported beam experimental setup

Figure 2. Cantilever beam experimental setup

The experimental results obtained for simply supported and

cantilever beam at different load are presented in the table 2,

3 and 4.

Table 2. Experimental results for simply supported beam

SI No. Load in Kg Maximum deflection in mm

1. 0.2 0.17

2. 0.4 0.33

3. 0.6 0.51

4. 0.8 0.68

5. 1.0 0.86

Table 3. Experimental results for cantilever beam with

center load


1. 0.2 0.77

2. 0.4 1.54

3. 0.6 2.35

4. 0.8 3.36

5. 1.0 4.23

Table 4. Experimental results for cantilever beam with end

load


1. 0.2 2.63

2. 0.4 5.07

3. 0.6 8.12




3

4. 0.8 10.83

5. 1.0 13.44

IV. VALIDATION

Validation of the above experimental results was done by

two approaches. They are:

4.1. Theoretical Approach

First uniform rectangular cross-sectional beams of linear

elastic isotropic homogeneous material have been

considered. The beam is taken mass-less and inextensible

hence have developed no strains. It is subjected to a vertical

point load at the tip of its free end and centre [3…6].

Using the Bernoulli-Euler’s elastic curve equation [3] the

following relationship is obtained:

EI (d2 y/dx

2) =M (1)

Where E is modulus of elasticity which is of constant value,

I is moment of inertia=bh3 /12, b=width of beam, h=height

or thickness of beam, y=deflection due to loading, M=

moment due to applied force.

On solving and applying boundary condition on equation (1)

for simply supported beam as shown in figure 3 we get

y =

(2)

Where W=Force applied on the beam, L=Length of the

beam.

Figure 3. Schematic representation of simply supported

beam

For cantilever beam with centered loading as shown in

figure 4 we get

y =

(3)

Figure 4. Schematic representation of cantilever beam at

center load

For cantilever beam with end loading as shown in figure 5

we get

y =

(4)

Figure 5. Schematic representation of cantilever beam at

end load

4.2. Numerical Approach

4.2.1. Mathematical validation

Mathematical validation means validation using FEM. In this

approach we use global stiffness equation to get the

deflection at the nodes [3].

{f} = [K] {q} (5)

Where {f} is force vector = {F1 M1 F2 M2 ... Fn Mn} T, [K]

is stiffness matrix, {q} is displacement vector = {q1 ɵ1 q2

ɵ2 ... qn ɵn} T, n = number of nodes, F1…Fn is force at node

1…n, M1…Mn is moment due to applied force at node 1…n,

q1… qn is linear deflection at nodes 1…n, ɵ1… ɵn is angular

deflection.

4.2.2. Simulation

Simulation has been performed using ANSYS 14.0

workbench tool [8, 9].

V. RESULTS AND DISSCUSSIONS

The results obtained from validation approach are shown in

table 6, 7 and 8 for simply supported and cantilever beam

for centre and end loading respectively. Figure 6 to figure 21

shows the results obtained using ANSYS simulation.

Similarly all the values obtained from experimental,

theoretical, mathematical and simulation are plotted in load

vs. deflection graph as shown in figure 21, 22 and 23.

Table 5. Validation Results for Simply Supported Beam


Theoretical Mathematical Simulation

1. 0.2 0.1614 0.1614 0.1615

2. 0.4 0.3229 0.3229 0.3231

3. 0.6 0.4843 0.4843 0.4829

4. 0.8 0.6458 0.6458 0.6461




4

5. 1.0 0.8072 0.8072 0.8076

Table 6. Validation Results for Cantilever Beam with

Center Loading



1. 0.2 0.8072 0.8072 0.7996

2. 0.4 1.6144 1.6144 1.5991

3. 0.6 2.4217 2.4217 2.3987

4. 0.8 3.2289 3.2289 3.1983

5. 1.0 4.0361 4.0361 3.9978

Table 7. Validation Results for Cantilever Beam with End

Loading



1. 0.2 2.5831 2.5831 2.6972

2. 0.4 5.1662 5.1662 5.1393

3. 0.6 7.7493 7.7493 7.7031

4. 0.8 10.3325 10.3325 10.2710

5. 1.0 12.9156 12.9156 12.8380

Figure 6. Deflection of simply supported beam at 0.2 Kg

load


load


load




5


load


load

Figure 11. Deflection of cantilever beam with 0.2 Kg load at

center


center




6


center


center


center


end




7


end


end


end


end




8

Figure 21. Load vs. deflection graph for simply supported

beam

Figure 21. Load vs. deflection graph for cantilever beam

with center load

Figure 22. Load vs. deflection graph for cantilever beam

with end load

VI. CONCLUSION

From above validation results, experimental results have

been validated where maximum deflection profiles are

clearly matching. There is a good agreement between the

experimental, theoretical and numerical approach results for

maximum deflection. Although there are some small

discrepancies due to some experimental imperfection,

effects of temperature, creep and shrinkage. The final result

shows an error of around 7% for simply supported beam and

around 5% error for cantilever beam. Though FEM is an

approximation method its results are exactly matching with

the theoretical results whereas structural analysis using

ANSYS 14.0 gives result with an error of less than 1%.

Further from load vs. deflection graph it was clearly

observed that deflection was more in experimental results

when compared to that of the theoretical and numerical

approach results. As the error is within acceptable range we

conclude that FEM and ANSYS simulation tool that can be

used in the future for structural analysis.

ACKNOWLEDGEMENT

The authors want to acknowledge the department of

aeronautical engineering, Nitte Meenakshi Institute of

technology, Bangalore for providing the technical support

regarding the experimental setup and the faculties like Dr.

Vivek Sanghi, Srikant H.V., Mahendra M.A., and Nishant

Deshai for their proper guidance.

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REFERANCES

[1] Gargi Majumder; Kaushik Kumar, “Deflection and Stress

Analysis of a Cantilever and its Validation Using

ANSYS”. International Journal of Mechanical

Engineering and Research, 2013, ISSN 2249-0019,

Volume 3, Number 6, pp. 655-662. [2] Amer M. Ibrahim; Saad k. Mohaisen; Qusay W. Ahmed,

“Finite element modeling of composite steel-concrete

beams with external prestressing”. International Journal of

Civil and Structural Engineering Volume 3, No 1, 2012.

[3] S. B. Halesh, Finite Element Methods, 1st edition, Sapna

Book House, Jan. 2014, pp.311-364.

[4] Dr. R. K. Bansal, A textbook of Strength of Material

(Mechanics of Solids), 5th edition, Laxmi Publication,

2013, pp.515-582.

[5] Timoshenko, S.P. and D.H. Young. Elements of Strength

of Materials, 5th edition. (MKS System).

[6] E.A. Witmer (1991-1992). "Elementary Bernoulli-Euler

Beam Theory". MIT

[7] Unified Engineering Course Notes.Ballarini, Roberto,

"The Da Vinci-Euler-Bernoulli Beam Theory?".

Mechanical Engineering Magazine Online. Retrieved

2006-07-22, April 18, 2003.

[8] ANSYS 14.0 documentation.

[9] Victor Debnath, Bikramjit Debnath, “Deflection And

Stress Analysis Of A Beam On Different Elements Using

ANSYS APDL" International Journal Of Mechanical

Engineering And Technology, 2014, ISSN 0976 – 6359

Volume 5, Issue 6, pp. 70-79.

experimental verification of deflection of beam using ... deflection distance of a beam under a load...

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