This paper studies the maximum deflection and Von-Misses stress analysis of:- a) Simply Supported Beam and b) Cantilever Beam under two different types of loading. The theoretical calculations are done based on the general Euler-Bernoulli’s Beam Equation. The Computational Analysis is done on ANSYS 14.0 software. Comparing the Numerical Results with that of the ANSYS 14.0, excellent accuracy of the present method has been extracted and demonstrated. In ANSYS 14.0 accuracies of different elements are measured and it has been visualized and concluded that Beam 189 element is most suitable element for Beam Analysis as compared to the Beam 188 element and other Solid elements

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IJMET
© I A E M E
DEFLECTION AND STRESS ANALYSIS OF A BEAM ON DIFFERENT
ELEMENTS USING ANSYS APDL
Victor Debnath1, Bikramjit Debnath2
Lovely Professional University1, SRM University2
ABSTRACT
This paper studies the maximum deflection and Von-Misses stress analysis of:- a) Simply
Supported Beam and b) Cantilever Beam under two different types of loading. The theoretical
calculations are done based on the general Euler-Bernoulli’s Beam Equation. The Computational
Analysis is done on ANSYS 14.0 software. Comparing the Numerical Results with that of the
ANSYS 14.0, excellent accuracy of the present method has been extracted and demonstrated. In
ANSYS 14.0 accuracies of different elements are measured and it has been visualized and concluded
that Beam 189 element is most suitable element for Beam Analysis as compared to the Beam 188
element and other Solid elements.
Keywords: ANSYS, Beam, Beam Analysis, Euler-Bernoulli’s Beam Equation, 188 Element,
189 Element, Solid Elements.
I. INTRODUCTION
Beams belong to the basic structural members used in the modeling abstraction of mechanical
systems [1]. In this paper behavior of beam and solid elements are discussed on the basis of Von-
Misses stress and Deflection occurred on beam due to various types of load i.e point load and
uniformly distributed load applied on rectangular section beam. A member subjected to bending
moment and shear force undergoes certain deformations. The material of the member will offer
resistance or stresses against these deformations.[2] It is possible to estimate these stresses with
certain assumptions. The beam cannot have any translational displacements at its support points, but
no restriction is placed on rotations at the supports. The deflected distance of a member under a load
is directly related to the slope of the deflected shape of the member under that load. While the beam
gets deflected under the loads, bending moment occurs in the same plane due to which stresses are
developed. Here the deflection of the beam element is calculated by using the Euler-Bernoulli’s
beam equation [3] and the bending stresses using the general standard bending equation

2. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 6, June (2014), pp. 70-79 © IAEME
analytically.[2] where on other hand Sparse solver is used to solve the Finite Element Model through
Ansys 14.0 APDL.[4]
71
The effect of elements structure on Maximum Von-Misses stress and Deflection are analyzed
in this paper. Those elements are Beam 188, Beam 189, Solid 185 and solid 285. And it has been
noticed that the most accurate result was measured by Beam 189 followed by Beam 188.
II. THEORETICAL CALCULATIONS
The calculations are done considering a uniform rectangular cross-sectional beam of linear
elastic isotropic homogeneous materials. The beam is assumed to be massless, inextensible having
developed no strains [5].
Using the bending moment curvature relationship the following equation is obtained:
EI (d2y/dx2)=M ---- (1)
Using the equation: (M/I)=(E/R)=(/Y) ---- (2)
Stress is calculated.
Where E is modulus of elasticity, which is constant. I is moment of inertia=bh3/12, b=width of beam,
h=height of beam, M=moment developed.
Four cases are considered in this paper a) Simply Supported Beam with Uniformly Loading
b) Simply Supported Beam with Single Point Load at centre c) Cantilevered Beam with Uniformly
Loading d) Cantilevered Beam with Single Point Load at the end.
CASE 1.Simply Supported Beam with Uniformly Distributed Load W per unit length
Assuming L=100m, b=10m, h=10m, =0.3, E=2×107N/m2, F=500N. The maximum deflection of
beam at a distance x=L/2 from one of the fixed end is 5WL4/384EI and it is calculated as
0.0003906m. With the required boundary conditions the maximum bending moment is obtained as
WL2/8. Using the equation: (M/I)=(E/R)=(/Y), Stress developed b is 37.502N/m2.
Figure 1: Simply Supported Beam with Uniformly Distributed Load
CASE 2: Simply Supported Beam with Single Point Load at Centre
Assuming L=100m,b=10m,h=10m, =0.3,E=2×107N/m2,F=500N. The maximum deflection of
beam at a distance x=L/2 from one of the fixed end is WL3/48EI and it is calculated as 0.00063m.
With the required boundary conditions the maximum bending moment is obtained as WL/4. Using
the equation: (M/I)=(E/R)=(/Y), Stress developed b is 75.003N/m2.
Figure 2: Simply Supported Beam with Single Point Load at Centre

3. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 6, June (2014), pp. 70-79 © IAEME
CASE 3: Cantilevered Beam with Uniformly Distributed Load W per unit run over the whole length
Assuming L=100m, b=10m, h=10m, =0.3, E=2×107N/m2, F=500N. The maximum deflection of
beam at a distance x=L from the fixed end is WL4/8EI and it is calculated as 0.00375m. With the
required boundary conditions the maximum bending moment is obtained as WL2/2. Using the
equation: (M/I)=(E/R)=(/Y), Stress developed b is 150.0060N/m2.
72
Figure 3: Cantilevered Beam with Uniformly Distributed Load W per unit run over the whole
length
CASE 4: Cantilevered Beam with Single Point Load at the end
Assuming L=100m, b=10m, h=10m, =0.3, E=2×107N/m2, F=500N. The maximum deflection of
beam at a distance x=L from the fixed end is WL3/3EI and it is calculated as 0.01m. With the
required boundary conditions the maximum bending moment is obtained as WL. Using the equation:
(M/I)=(E/R)=(/Y), Stress developed b is 300.0120N/m2.
Figure 4: Cantilevered Beam with Single Point Load at the end
III. COMPUTATIONAL RESULT
CASE 1: Simply Supported Beam with Uniformly Distributed Load W per unit length
A. 188 element- (Fig.5 Fig.6) Maximum Deflection obtained=0.4e-3m and Maximum Von-Mises
Stress obtained= 37.4925 N/m2.
B. 189 element- (Fig.7 Fig.8) Maximum Deflection obtained=0.4e-3m and Maximum Von-Mises
Stress obtained= 37.5025 N/m2.
C. 185 element- (Fig.9 Fig.10) Maximum Deflection obtained=0.401e-3m and Maximum Von-
Mises Stress obtained= 37.0599 N/m2.
D. 285 element- (Fig.11 Fig.12) Maximum Deflection obtained=0.394e-3m and Maximum Von-
Mises Stress obtained= 37.6991 N/m2.
Figure 5: Displacement At Different Figure 6: Stress Distribution At Different
Nodes In 188 Element Nodes In 188 Element

4. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 6, June (2014), pp. 70-79 © IAEME
73
Figure 7: Displacement At Different Figure 8: Stress Distribution At Different
Nodes In 189 Element Nodes In 189 Element
Figure 9: Displacement At Different Figure 10: Stress Distribution At Different
Nodes In 185 Element Nodes In 185 Element
Figure 11: Displacement At Different Figure 12: Stress Distribution At Different
Nodes In 285 Element Nodes In 285 Element
CASE 2: Simply Supported Beam with Single Point Load at Centre
A. 188 element- (Fig.13 Fig.14) Maximum Deflection obtained=0.644e-3m and Maximum Von-
Mises Stress obtained= 74.2500 N/m2.

5. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 6, June (2014), pp. 70-79 © IAEME
B. 189 element- (Fig.15 Fig.16) Maximum Deflection obtained=0.644e-3m and Maximum Von-
Mises Stress obtained= 75 N/m2.
C. 185 element- (Fig.17 Fig.18) Maximum Deflection obtained=0.655e-3m and Maximum Von-
Mises Stress obtained= 111.6060 N/m2.
D. 285 element- (Fig.19 Fig.20) Maximum Deflection obtained=0.644e-3m and Maximum Von-
Mises Stress obtained= 74.2500 N/m2.
74
Figure 13: Displacement At Different Figure 14: Stress Distribution At Different
Nodes In188 Element Nodes In 188 Element
Figure 15: Displacement At Different Figure 16: Stress Distribution At Different
Nodes In 189 Ellement Nodes In 189 Element
Figure 17: Displacement At Different Figure 18: Stress Distribution At Different
Nodes In 185 Element Nodes In 185 Element

6. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 6, June (2014), pp. 70-79 © IAEME
75
Figure 19: Displacement At Different Figure 20: Stress Distribution At Different
Nodes In 285 Element Nodes In 285 Element
CASE 3: Cantilevered Beam with Uniformly Distributed Load W per unit run over the whole length
A. 188 element- (Fig.21 Fig.22) Maximum Deflection obtained=0.3797e-2 m and Maximum Von-
Mises Stress obtained= 148.5070 N/m2.
B. 189 element- (Fig.23 Fig.24) Maximum Deflection obtained=0.3797e-2 m and Maximum Von-
Mises Stress obtained= 148.9980 N/m2.
C. 185 element- (Fig.25 Fig.26) Maximum Deflection obtained=0.3761e-2 m and Maximum Von-
Mises Stress obtained= 161.8340 N/m2.
D. 285 element- (Fig.27 Fig.28) Maximum Deflection obtained=0.3703e-2 m and Maximum Von-
Mises Stress obtained 151.3950 N/m2.
Figure 21: Displacement At Different Figure 22: Stress Distribution At Different
Nodes In 188 Element Nodes In 188 Element
Figure 23: Displacement At Different Figure 24: Stress Distribution At Different
Nodes In 189 Element Nodes In 189 Element

7. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 6, June (2014), pp. 70-79 © IAEME
76
Figure 25: Displacement At Different Figure 26: Stress Distribution At Different
Nodes In 185 Element Nodes In 185 Element
Figure 27: Displacement At Different Figure 28: Stress Distribution At Different
Nodes In 285 Element Nodes In 285 Element
CASE:4 Cantilevered Beam with Single Point Load at the end
A. 188 element- (Fig.29 Fig.30) Maximum Deflection obtained=0.010105 m and Maximum Von-
Mises Stress obtained= 298.5000 N/m2.
B. 189 element- (Fig.31 Fig.32) Maximum Deflection obtained=0.010105 m and Maximum Von-
Mises Stress obtained= 300.0000 N/m2.
C. 188 element- (Fig.33 Fig.34) Maximum Deflection obtained=0.010043 m and Maximum Von-
Mises Stress obtained= 321.0030 N/m2.
D. 188 element- (Fig.29 Fig.30) Maximum Deflection obtained=0.010105 m and Maximum Von-
Mises Stress obtained= 298.5000 N/m2.
Figure 29: Displacement At Different Figure 30: Stress Distribution At Different
Nodes In 188 Element Nodes In 188 Element

8. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 6, June (2014), pp. 70-79 © IAEME
77
Figure 31: Displacement At Different Figure 32: Stress Distribution At Different
Nodes In 189 Element Nodes In 189 Element
Figure 33: Displacement At Different Figure 34: Stress Distribution At Different
Nodes In 185 Element Nodes In 185 Element
Figure 35: Displacement At Different Figure 36: Stress Distribution At Different
Nodes In 285 Element Nodes In 285 Element
IV. COMPARISON OF RESULTS
In table.1 and table.3 the Analytical Results of Maximum Von-Mises stress and Maximum
Deflection on different elements after considering the loading conditions as mentioned above are
compared with the Computational Results. And table.2 and table.4 demonstrates the percentage of
error between the Analytical Results and Computational Results.

9. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 6, June (2014), pp. 70-79 © IAEME
DEFLECTION ERROR PERCENTAGE
188ele 189ele 185ele 285ele
CASE 1 2.30 2.30 2.56 0.77
CASE 2 2.20 2.20 3.97 3.17
CASE 3 1.25 1.25 0.29 1.25
CASE 4 1.01 1.01 0.43 1.14
VON MISES STRESS(N/m)
188ele 189ele 185ele 285ele Analytical
CASE 1 37.4925 37.5025 37.0599 37.6991 37.5020
CASE 2 74.2500 75.0000 111.6060 73.4405 75.0030
CASE 3 148.5070 148.9980 161.8340 151.3950 150.0060
CASE 4 298.5000 300.0000 321.0030 303.4560 300.0120
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Table 1: DEFLECTION
DEFLECTION(m)
188ele 189ele 185ele 285ele Analytical
Results
CASE 1 0.400e-3 0.400 e-3 0.401 e-3 0.394 e-3 0.391 e-3
CASE 2 0.644 e-3 0.644 e-3 0.655 e-3 0.610 e-3 0.630 e-3
CASE 3 0.3797 e-2 0.3797 e-2 0.3761 e-2 0.3703 e-2 0.3750 e-2
CASE 4 0.010105 0.010105 0.010043 0.009886 0.01000
Table 2: DEFLECTION ERROR PERCENTAGE
Table 3: VON-MISES STRESS
Results
Table 4: VON-MISES STRESS ERROR PERCENTAGE
VON MISES STRESS ERROR PERCENTAGE
188ele 189ele 185ele 285ele
CASE 1 0.025 0.001 1.179 0.526
CASE 2 1.000 0.004 48.802 2.080
CASE 3 0.999 0.672 7.885 0.926
CASE 4 0.504 0.004 6.997 1.148
V. CONCLUSION
After going through all the tables’ data, it can be concluded that the ELEMENT 189 is the
best element to do BEAM ANALYSIS rather than 188 element and other SOLID ELEMENTS.
Rather, if we define the priority of the elements for beam analysis than it would be as follows 189
element, 188 element, Solid 285 element, solid 185 element. Hence, it is very well justified to
mention that to solve beam type of problem we always need to rely on beam 189 element rather than
any other elements.

10. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 6, June (2014), pp. 70-79 © IAEME
79
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[3] Timoshenko, S., (1953), History of strength of materials, McGraw-Hill New York.
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