This document analyzes the bending and vibration of isotropic and composite beams using finite element analysis in ANSYS. It summarizes the deflection and natural frequencies of isotropic beams under different loading conditions and boundary constraints. The maximum deflection occurs for a cantilever beam with a concentrated end load. Minimum deflection is found for a fixed-fixed beam under uniform load. Natural frequencies are also reported for beams with hinged-hinged and fixed-fixed boundaries. Deflection results from ANSYS are presented for various composite beam configurations with different loadings.

1. 1

2. 2
ABSTRACT:-
Beams are an integral part of every structural system. The beam is a structural element that is capable of
withstanding load primarily by resisting against bending. They are characterized by their profile (shape of cross-
section), their material and their length. Study of beams is very fundamental to the learning of complicated
structures and their behavior under stresses. This report is mainly based on the bending and free vibration
analysis of beams.
Finite Element Method (FEM) is a numerical technique for finding the approximate solutions to the
boundary value problems for partial differential equations. It is also referred to as Finite Element Analysis. It
divides a larger problem into simpler parts called Finite Elements.FEM then uses variational methods from the
calculus of variations to approximate a solution by minimizing associated error function. Softwares like
ANSYS use FEM to solve the structural problems. This report is mainly based on the analysis using ANSYS.
In this report we have mainly used steel as the material of the isotropic beam. The properties of steel are
listed below:-
 Modulus of elasticity(E)=210GPa=210*109Pa
 Density(ρ)=7800kg/m3
 Poisson’s Ratio(γ)=0.3
The details presented in the report are:-
 Maximum deflection of isotropic beams under different boundary conditions.
 Frequency of vibration of isotropic beams under different boundary conditions.
Also in the section of composite beams, a few results on the deflection are shown. The data used for the
calculation of deflection of beams have been shown. Also deflection based on Finite Element Analysis are
shown.

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CONTENTS:-
 Deflection of isotropic beams…………………………………………………….4
1. Concentrated load at one end of a cantilever beam……………………....5
2. Uniform load over a cantilever beam…………………………………….6
3. Concentrated load at centre of beam hinged at both ends……………......7
4. Uniform load over a beam hinged at both ends…………………………..8
5. Uniform load over a beam fixed at both ends…………………………….9
6. Concentrated load at centre of a beam fixed at both ends……………….10
7. Table showing deflection of beams………………………………………11
 Free vibration of isotropic beams……………………………………………..12-13
1. Vibration of beams hinged at both ends…………………………………13
2. Vibration of beams fixed at both ends…………………………………...13
3. Table showing frequency of vibrations…………………………………..14
 Composite beams…………………………………………………………………15
1. Deflection of a hinged-hinged composite beam………………………16-17
2. Deflection of a fixed-fixed composite beam………………………….18-19
3. Deflection of a fixed-free composite beam…………………………....20-21
 References………………………………………………………………………..22

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DEFLECTION OF ISOTROPIC BEAMS:-
Isotropic materials are the materials which have identical properties in all planes. Thus isotropic beams are the
ones in which the properties doesn’t vary upon varying the plane i.e. E1=E2=E3. In this section we are mainly
interested to find out the deflection in such beams.
The general section of the beam is shown below along with the required data:-
Figure 1: Isometric view of an isotropic beam
 Length of the beam(L)= 1m
 Area of cross section of the beam(A)= (20mm)*(30mm)=6*10-4m2
 Moment of Inertia(I)=
1
12
(20*10-3)(30*10-3)3 =45*10-9m4
Moment-Area Method[1]:-
 Theorem 1:- The angle θ between tangents at any two points A and B on the elastic line is equal to the
total area of the corresponding portion of the bending moment diagram, divided by EI .
 Theorem 2:- The deflection of B away from the tangent at A is equal to the statical moment, with respect
to B, of the bending moment area between A and B, divided by EI.
These theorems have been widely used in the derivation of the deflection of beams under different boundary
conditions.

5. 5
CONCENTRATED LOAD AT ONE END OF A CANTILEVER
BEAM:-
Figure 2: Cantilever beam with bending moment diagram
F=100N is the applied force at the end of the cantilever beam.
So the maximum deflection, δ=
1
2
FL.L×
2L
3
×
1
EI
=
1
3EI
FL3= 3.53mm
Figure 3: Deflection as shown in ANSYS

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UNIFORM LOAD OVER A CANTILEVER BEAM:-
Figure 4: Cantilever beam and bending moment diagram
The uniform load applied over the beam is =w=100N/m
So the maximum deflection, δ=
𝟏
3
×
w
2
L2×L×
3L
4
×
1
EI
=
w
8EI
L4 = 1.32mm
Figure 5:Deflection as shown in ANSYS

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CONCENTRATED LOAD AT CENTRE OF BEAM HINGED
AT BOTH ENDS:-
Figure 6: Hinged-hinged beam and bending moment diagram
F=100N is the force applied at the centre of the beam.
So the maximum deflection, δ=
F
48EI
L3= 0.22mm
Figure 7: Deflection as shown in ANSYS

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UNIFORM LOAD OVER A BEAM HINGED AT BOTH ENDS:-
Figure 8: Hinged-hinged beam and bending moment diagram
w=100N/m is the force applied uniformly all over the beam.
So the maximum deflection, δ=
5w
384EI
L4 = 0.137mm
Figure 9: Deflection as shown in ANSYS

9. 9
UNIFORM LOAD OVER A BEAM FIXED AT BOTH ENDS:-
Figure 10: Fixed-fixedbeam with uniform loading
w=100N/m is the force applied uniformly all over the beam.
So the maximum deflection, δ=
w
384EI
L4 =0.027557mm
Figure 11: Deflection as shown in ANSYS

10. 10
CONCENTRATED LOAD AT CENTRE OF A BEAM FIXED
AT BOTH ENDS:-
Figure 12:Fixed-fixedbeam and bending moment diagram
F=100N is the force applied at the centre of the beam.
So the maximum deflection, δ=
F
192EI
L3 = 0.055114mm
Figure 13: Deflection as shown in ANSYS

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Table 1: Deflections in various beams with boundary conditions as per calculations of ANSYS:-
Loading Condition Beam Type Deflection on the data
Concentratedload Fixed-Free 3.53mm
Concentratedload Fixed-Fixed 0.055114mm
Concentratedload Hinged-Hinged 0.22m
UniformlyDistributedLoad Fixed-Free 1.32mm
Uniformly DistributedLoad Fixed-Fixed 0.027557mm
UniformlyDistributedLoad Hinged-Hinged 0.137mm
In the above table the row‘LoadingConditions’define the type of loadof loadappliedtothe beamandthe nextrow
‘BeamType’define the pointwhere itisapplied.Foreveryconditionthe deflectionamounthasbeengivenasperthe
data obtainedfromANSYS.The dataobtainedfromtheoretical calculationshave alsobeencheckedandshownbefore.
From the data obtained,we cansee thatdeflectionismaximumwhenloadisappliedatone endof a cantileverbeam.
Andthe deflectionisminimumwhenauniformlydistributedloadisappliedtoafixed-fixedbeam.
So these datacan be usedtodevelopastructure where correspondingdeflectionsare required.Forexample ina bridge,
we want minimumdeflectionandsowe use a fixed-fixedbeaminsteadof ahingedone (more orless).
(The data of this table hasbeendevelopedbyanalysisusingFEMin ANSYS.These are checkedwiththe theoretical
values).

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FREE VIBRATION OF ISOTROPIC BEAMS:-
A continuous isotropic beam has uniform distribution of mass and stiffness along its length. Transverse
vibrations result in bending deformation of beams (both linear and angular deformation). The Euler-Bernoulli
beam theory is considered which assumes a plane cross-section remains plane after bending and remains
perpendicular to the neutral axis of the beam before and after bending.
In this section, different boundary conditions have been taken and corresponding vibration analysis has been
performed both theoretically and also using ANSYS.
Basic theory has been explained as below:-
Consider the free body diagram of a section of beam as shown:-
Figure 14: Forces acting in sections of beams while vibrating
Consider the force acting on the beam is f(x,t). Due to vibration the beam is displaced to w(x,t) from the x-axis
which is the datum level.
From the free body diagram of the elemental part, we can say-
Net force on the elemental beam is=+V-(V+dV)+f(x,t)dx=ρAẅ (x,t)
We also know that:- V(x,t)=
dM(x,t)
dt
Thus on simplification, we can write-
For free vibration, we can write f(x,t)=0.
So,

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The free vibration solution can be found using the method of superposition of variables as-
w(x,t)=W(x)T(t)
Solving for the 2 variables, we get-
T(t)= A1cosωt+A2sinωt where A and B are constants that can be found from the initial conditions.
And, W(x)= Asinβx+Bcosβx+Csinhβx+Dcoshβx where A,B,C and D are constants that can be found using the
boundary conditions.
The natural frequencies of beam can be computed as:-
The function w(x) is called the characteristic function of the beam and ω is called the natural frequency of
vibration. For any beam, there will be infinite number of normal modes with one frequency associated with
each normal mode.
Using the boundary conditions, a table has been developed on the various boundary conditions of beams.
VIBRATION OF BEAMS HINGED AT BOTH ENDS:-
Boundary conditions are- at x=0, w=0 and M=0
at x=L, w=0 and M=0
Simplifying the characteristic equation becomes- sinβL=0 => βL=nп
VIBRATION OF BEAMS FIXED AT BOTH ENDS:-
Boundary conditions are- at x=0, w=0 and
𝑑𝑤
𝑑𝑥
=0
at x=L, w=0 and
𝑑𝑤
𝑑𝑥
=0
Simplifying the characteristic equation becomes- cos(βL).cosh(βL) =1
=>βL=4.73, 7.85, 10.99, 14.14

14. 14
Table-2: Vibration of beams under different boundary conditions [2] -
The above table has been prepared for the use of the natural frequency of vibration of beams under different
boundary conditions. These values have been checked using ANSYS software.

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COMPOSITE BEAMS:-
A composite material is defined as a material which is composed of two or more materials at a microscopic
scale and has chemically distinct phases. We mix different materials in order to get a new material with
improved properties like strength, toughness or corrosion.
In a composite, there are typically two types of components- one is called reinforcement and the other is
called the matrix. The matrix forms the bulk of the composite and reinforcements are the fibres. However the
components do not mix chemically in a composite.
Beams made out of such composite materials are called composite beams and have enhanced features.
However, the simple isotropic beams were 2 dimensional (length along x-axis and width along y-axis), but in
case of composite beams, there are many shell formation and hence 3-dimensional.
Figure 15: Composite beam development using ANSYS

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DEFLECTION OF COMPOSITE BEAMS-
Data mainly used here has been listed as below:-
h=0.05m, b=0.025m, a=20×h=1m
F0=2.5×105 N/m, q0=1×107 N/m2
E2=2GPa, E1=50GPa, G12=G13=1GPa, G23=0.4GPa, γ=0.25
Deflection for various boundary conditions are shown below:-
 HINGED-HINGED:-
In this type of beam, both ends are hinged and loads have been applied-one uniformly distributed and
the other a load along the center line of the beam. ANSYS have been used to analyze the deflection and
the results are shown below:-
Here the load has been uniformly distributed:-
Figure 16: Deflection of a hinged-hinged composite beam under a uniformly distributed load over the area.

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Here the load is concentrated along a line:-
Figure 17: Deflection of a hinged-hinged beam under a concentrated load over a line.

18. 18
 FIXED-FIXED:-
In this type of beam, both ends are hinged and loads have been applied-one uniformly distributed and
the other a load along the center line of the beam. ANSYS have been used to analyze the deflection and
the results are shown below:-
Here the load has been uniformly distributed:-
Figure 18: Deflection of a fixed-fixedbeam under a uniformly distributed load over the area.

19. 19
Here the load is concentrated along a line:-
Figure 19: Deflection of a fixed-fixedbeam under a concentrated load along a line.

20. 20
 FIXED-FREE(CANTILEVER BEAM):-
In this type of beam, both ends are hinged and loads have been applied-one uniformly distributed and
the other a load along the center line of the beam. ANSYS have been used to analyze the deflection and
the results are shown below:-
Here the load has been uniformly distributed:-
Figure 20: Deflection of a cantilever beam under a uniformly distributed load over the area.

21. 21
Here the load is concentrated along a line:-
Figure 21: Deflection of a cantilever beam over a concentrated load along a line.

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References:-
1. Elements of Strength of Materials (Fifth Edition) by S.P.Timoshenko and D.H.Young.
2. Mechanical Vibrations (Fifth Edition) by S.S.Rao.
The other things which I have used to produce this report are:-
3. Internet Websites (like Wikipedia, Google).
4. Various books on Mechanics of Materials.