30120140506008

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

70

DEFLECTION AND STRESS ANALYSIS OF A BEAM ON DIFFERENT

ELEMENTS USING ANSYS APDL

Victor Debnath1, Bikramjit Debnath

2

Lovely Professional University1, SRM University

2

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

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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),

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analytically.[2]

where on other hand Sparse solver is used to solve the Finite Element Model through

Ansys 14.0 APDL.[4]

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/dx

2)=M ---- (1)

Using the equation: (M/I)=(E/R)=(σ/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,ѵ =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)=(σ/Y), Stress developed σb 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,ѵ =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)=(σ/Y), Stress developed σb is 75.003N/m2.

Figure 2: Simply Supported Beam with Single Point Load at Centre

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

72

CASE 3: Cantilevered Beam with Uniformly Distributed Load W per unit run over the whole length

Assuming L=100m, b=10m, h=10m, ѵ =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)=(σ/Y), Stress developed σb is 150.0060N/m2.

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, ѵ =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)=(σ/Y), Stress developed σb 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-3

m and Maximum Von-Mises

Stress obtained= 37.4925 N/m2.

B. 189 element- (Fig.7 & Fig.8) Maximum Deflection obtained=0.4e-3

m and Maximum Von-Mises

Stress obtained= 37.5025 N/m2.

C. 185 element- (Fig.9 & Fig.10) Maximum Deflection obtained=0.401e-3

m and Maximum Von-

Mises Stress obtained= 37.0599 N/m2.

D. 285 element- (Fig.11 & Fig.12) Maximum Deflection obtained=0.394e-3

m 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

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

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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-3

m and Maximum Von-

Mises Stress obtained= 74.2500 N/m2.

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

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B. 189 element- (Fig.15 & Fig.16) Maximum Deflection obtained=0.644e-3

m and Maximum Von-

Mises Stress obtained= 75 N/m2.

C. 185 element- (Fig.17 & Fig.18) Maximum Deflection obtained=0.655e-3

m and Maximum Von-

Mises Stress obtained= 111.6060 N/m2.

D. 285 element- (Fig.19 & Fig.20) Maximum Deflection obtained=0.644e-3

m and Maximum Von-

Mises Stress obtained= 74.2500 N/m2.

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

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

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

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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

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

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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.

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

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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

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.

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

Results

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

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

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[3] Timoshenko, S., (1953), History of strength of materials, McGraw-Hill New York.

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