1. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 3, March (2014), pp. 10-19, © IAEME
10
NON LINEAR DYNAMIC AND STABILITY ANALYSIS OF BEAM USING
FINITE ELEMENT IN TIME
Prabhat Kumar Sinha1
, Ishan Om Bhargava2
, Saifuldeen Abed Jebur3
Department of Mechanical Engineering (SSET)
Sam Higginbottom Institute of Agriculture Technology and Sciences, Allahabad
ABSTRACT
In this article the main focus is to predict nonlinear dynamic response of a beam using finite
element in time under given condition. To do the nonlinear dynamic analysis of a beam, a distributed
load is being applied and the beam is experiencing bending. The given beam is homogeneous in
composition and isotropic in nature. Here, considering the stiffness of the beam and its effect on the
deflection, under the distributed load conditions. The use of this method is in determining the
variation of the beam under the given load and the corresponding load conditions. The result of this
entire analysis will be appropriate and this will facilitate the complete knowledge about the nonlinear
dynamic analysis.
INTROUCTION
The need of accurate prediction of nonlinear dynamic response of beam, isotropic in nature
subjected to distributed loads is carried on through graphical analysis of various parameters
applicable to the beam. The load applied is uniformly distributed, has changing point of application.
The case discussed here is about transverse deflection and its effect on the dynamic response. Many
researches have been conducted using different techniques taking into consideration the finite
element method and its application to the solution of the problem acceptable results have been
obtained. This is achieved by using Finite Element in Time, though the importance of the paper is
unanimous and not under any discussion even though we have to consider some studies. Chen [1]
have studied the instability behavior of beams with variable cross section subjected to sub-tangential
non conservative follower forces, and their solution is numerically attained by using a Runge-Kutta
method based on the above work . The dynamic stability of buckled beam considering the snap
through motion under sinusoidal loading has been investigated by Poon et al [2]. Non-linear steady
state response of beams, frames and shallow arches has been analyzed in the frequency domain using
the h-version straight beam finite element by Chen et al [3]. The geometrically non-linear thermo
elastic vibration analysis of straight and curved beams has been carried out using p-version
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ISSN 0976 – 6340 (Print)
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Volume 5, Issue 3, March (2014), pp. 10-19
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2. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 3, March (2014), pp. 10-19, © IAEME
11
hierarchical finite elements by Ribeiro and Manoach [4]. The influence of temperature variation and
curvature on the non-linear dynamics of curved beam has been studied. Periodic as well as a periodic
motions were observed. The vibration of a curved beam in anti-symmetric mode, due to auto-
parametric excitation, has been studied by Lee et al [5]. Chen and Yang [6] have studied, both
theoretically and experimentally, the non- linear vibration of shallow arch under the harmonic
excitation at one end. Based on the above formulation Mata et al.[7] have developed fully geometric
and constitutive non-linear model for the dynamic behavior of beam structures considering an inter-
mediate curved reference configuration. The model has been used to study the transient response of
straight beams, frames and curved beams and [8, 9]. The solution of the problems was achieved
using the analog equation method of Katsikadelis [10] as it was developed for the non-linear
dynamic analysis of beams. It may be noted that the solution of banded system of equations is
computationally more efficient. It can also be noted from the literature that the finite element based
system of equations for beams/plates/shells are solved by the traditional shooting method in which
the second order governing equations of motion are transformed to first order equations. This results
in doubling the number of equations and the banded nature of the system of equations is destroyed.
In this article, this issues has taken care of by directly applying the shooting method to the second
order governing equations [10,11] . In spite of the importance of non-linear dynamic analysis of
curved beams, non-linear forced response studies are relatively few.
Sheinmann [11] has studied the dynamic buckling behavior of shallow and deep circular
arches. The solution methodology used could not capture the converged solutions adequately for the
deep arches.[12,14] .Vast literatures for the dynamics of axially moving continua have been
comprehensively reviewed by Wickert and Mote, Abrate and Chen [15] The system of semi-discrete
dynamic equations of motion is derived from the modified Hamilton principle in which only the
strain variables are interpolated. Such a choice of the interpolated variables is an advantage over
approaches, in which the displacements and rotations are interpolated, since the field consistency
problem and related locking phenomena do not arise. Finite element dynamic analysis of
geometrically exact planar beams has been done by Gams et al.[17] A new finite-element-based
approach along with an iterative incremental method is developed to study the dynamic response of
sandwich beam with hybrid composite face sheets and flexible core by Shariya and others. As a point
of fact the literature suggest that a few response analysis have been carried out on assuming the
solution which is a priori. The finite element in time is applied to carry the distributed load analysis.
It has been observed that properly applied transient application of finite element method has been
carried on partially fixed or supported beams under distributed load conditions. The further
interpretation of various literatures suggest effectiveness of Finite Element Method in analyzing
partially partially fixed or supported beams. Though solving a large number of problems is
cumbersome this issue can be sorted out by using a small bandwidth of equations resulting in smaller
matrix formation in large dynamical systems. In this paper repetition of equations is solution of the
problem discussed Finite Element in Time methodology has been utilized. Its efficacy has been
demonstrated for the non linear dynamic analysis of isotropic and composite curved beams.
POTENTIAL ENERGY APPROACH
Beams with cross sections that are symmetric with respect to plane of loading are considered
here. A general horizontal beam has been shown with the cross section and the bending stress
distribution. For small deflections,

3. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 3, March (2014), pp. 10-19, © IAEME
12
L
P Pm
m Mk
k
x
0
(a)
Y
LOADED BEAM
x
v v'
v'
v
(b)
DEFORMRED NEUTRAL AXIS
,v is the deflection of the centroidal axis at x axis, and I is the moment of inertia of the section about
the neutral axis (z- axis passing through the centroid).

4. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 3, March (2014), pp. 10-19, © IAEME
13
M
Ne utra l a xi s
y
x
V
y
Centroid
z
dA
y
y
SECTIONS OF THE BEAM AND ITS STRESS DISTRIBUTION
FINITE ELEMNT ANALYSIS
The strain energy in an element of length dx is
=
The total strain energy in the beam is given by:
U =

5. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 3, March (2014), pp. 10-19, © IAEME
14
The potential energy of the beam is:
=
Where p is the distributed load per unit length, is the point load at point m, is the
moment of the couple applied at point k, is the deflection at point m, and is the slope at
point k
1 2 3 4 51 2 3 4
Q1 Q3 Q5 Q7 Q9
Q2
Q4
Q6 Q8
Q1 0
Q2 i– 1
Q2 i
DISCRETIZED BEAM ELEMENT
1 2
e
1
v1
v'1
v2
v'2
q1
q2
q3
q4
AN ELEMENT BEING ANALYZED
THE FINITE ELEMENT FORMULATION
The beam is divided into four elements. Each node has two degree of freedom. Typically the
degrees of freedom of node i are . The degree of freedom is the transverse
displacement and is slope or rotation.
Q =

6. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 3, March (2014), pp. 10-19, © IAEME
15
The vector represents the global displacement vector. For a single element, the local degrees
of freedom are represented by
The local and global displacement are in correspondence with each other. The local vector q
is same as . The shape functions for interpolating v on an element are defined in
terms of on -1 to +1. The shape functions for beam elements here are different as we are using a
third degree polynomial. Since nodal values and nodal slopes are involved, Hermite shape functions
have been taken, which satisfy both the nodal value and the slope continuity. Each of the shape
functions is of a cubic order represented by:
Slope = 0
1
= –1 = 0 = +1
Slope = 0
H1
1
–1 0 +1
Slope = 0
H3
1
1
2
Slope = 0
HERMITE SHAPE FUNCTIONS
–1 0 +1
Slope = 0
H2
1
Slope = 1
–1 0 +1
H4
1
Slope = 0 Slope = 1
2
NON LINEAR HERMITE SHAPE FUCTIONS

7. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 3, March (2014), pp. 10-19, © IAEME
16
The conditions given in the following table are satisfied:
0
1
0
0
0
0
0
1
1
0
0
0
0
0
1
0
= -1
= 1
The coefficient and can be easily obtained by imposing those conditions on these
The Hermite shape functions can be used to write v in form
Hence the coordinate transform is given by the relationship
As
The chain rule
Noting that evaluated at nodes 1 and 2 are , respectively, thus

8. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 3, March (2014), pp. 10-19, © IAEME
17
This is denoted as
Where
In the total potential energy of the system, the integrals are considered as summations over
the integrals over the elements. The element strain energy has been given by:
Then, substituting v = Hq following equation is obtained
On substituting we get
Each term in the matrix has been integrated. Knowing that
This result in the form of element strain energy is given by

9. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 3, March (2014), pp. 10-19, © IAEME
18
Where the element stiffness matrix as
From the matrix the following has been observed
Where
which is the set of generalized virtual displacement on the element v = Hq and
RESULT
In this paper finite element in time develop as a analyzing tool for the supported beam
element which has been discretized into finite elements and this element presents a dynamic model
for an isotropic beam considering the bending of beam when the load has been applied .The
formulation of the dynamic model has been carried and analyzed by taking into consideration the
potential energy approach and the relevant energy formulations for the stress analysis of the beam
which has been described and validated through the finite element in time. The formulation has been
done using the Hermite shape functions which result in the most approximate stiffness matrix of the
beam considering the degree of freedom and their orientation with respect to the differential
equations of deflection. These equations have been considered over the boundary conditions and
taken into consideration by the Hermite third degree polynomial. Though the curve so obtained are
nonlinear they result in an augmented stiffness matrix. During the initial modeling and experimental
work, emphasis has been made on dynamic analysis of beam and has been improved by increasing
the nodal points and by analyzing the degree of freedom.
REFRENCE
[1] Chen, Y.Z., “Interaction between compressive force and vibration frequency for a
varyingcross-section cantilever under action on generalized follower force,” Journal of Sound
andVibration, 259, 991-999, (2003).
[2] W.Y. Poon, C.F. Ng, Y.Y. Lee, Dynamic stability of a curved beam under sinusoidal loading,
Proceedings of Institution of Mechanical Engineers, Part G: Journal of Aerospace
Engineering 216 (2002) 209–217.
[3] S.H. Chen, Y.K. Cheung, H.X. Xing, Nonlinear vibration of plane structures by finite
element and incremental harmonic balance method, Nonlinear Dynamics 26 (2001) 87–104.
[4] P. Ribeiro, E. Manoach, The effect of temperature on the large amplitude vibrations of curved
beams, Journal of Sound and Vibration 285 (2005) 1093–1107.
[5] Y.Y. Lee, W.Y. Poon, C.F. Ng, Anti-symmetric mode vibration of a curved beam subject to
auto parametric excitation, Journal of Sound and Vibration 290 (2006) 48–64.

10. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 3, March (2014), pp. 10-19, © IAEME
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[6] J.S. Chen, C.H. Yang, Experiment and theory on the nonlinear vibration of a shallow arch
under harmonic excitation at the end, ASME Journal of Applied Mechanics 74 (2007) 1061–
1070.
[7] P. Mata, S. Oller, A.H. Barbat, Dynamic analysis of beam structures considering geometric
and constitutive nonlinearity, Computer Methods in Applied Mechanics and Engineering 197
(2008) 857–878.
[8] P. Ribeiro, Non-linear forced vibrations of thin/thick beams and plates by the finite element
and shooting methods, Computers and Structures 82 (2004) 1413–1423.
[9] P. Ribeiro, Forced large amplitude periodic vibrations of cylindrical shallow shells, Finite
Elements in Analysis and Design 44 (2008) 657–674 .
[10] J.T. Katsikadelis, The analog equation method. A boundary-only integral equation method
for nonlinear static and dynamic problems in general bodies, Appl. Mech. 27 (2002) 13–38.
[11] I. Sheinman, Dynamic large displacement analysis of curved beams involving shear
deformation, International Journal of Solids and Structures 16 (1980) 1037–1049.
[12] U.Lee, H.Oh, Dynamics of an axially moving viscoelastic beam subject to axial tension,
InternationalJournalofSolidsandStructures42 (2005)2381–2398.
[13] U.Lee, J.Kim, H.Oh, Spectral analysis for the lateral vibration of an axially moving
Timoshenkobeam, JournalofSoundandVibration271 (2004) 685–703.
[14] F.Fung,H.C.Chang, Dynamic and energetic analyses of a string/slidernon- linear coupling
system by variable grid finite difference, Journal of Sound and Vibration 239(3)(2001) 505–
514.
[15] L.Q. Chen, D.Hu, Natural frequencies of nonlinear vibration of axially moving beams,
Nonlinear Dynamics 62(2011)125–134.
[16] Finite element dynamic analysis of geometrically exact planar beams Original Research
Article Computers & Structures, Volume 85, Issues 17–18, September 2007, Pages 1409-
1419M. Gams, M. Saje, S. Srpčič.
[17] Non-linear dynamic analysis of a sandwich beam with pseudoelastic SMA hybrid composite
faces based on higher order finite element theory Original Research Article Composite
Structures, Volume 96, February 2013, Pages 243-255 S.M.R. Khalili, M. Botshekanan
Dehkordi, E. Carrera, M. Shariya.
[18] Prabhat Kumar Sinha, Vijay Kumar, Piyush Pandey and Manas Tiwari, “Static Analysis of
Thin Beams by Interpolation Method Approach to MATLAB”, International Journal of
Mechanical Engineering & Technology (IJMET), Volume 4, Issue 2, 2013, pp. 254 - 271,
ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359.
[19] Jn Mahto, Sc Roy, J Kushwaha and Rs Prasad, “Displacement Analysis of Cantilever Beam
using FEM Package”, International Journal of Mechanical Engineering & Technology
(IJMET), Volume 4, Issue 3, 2013, pp. 75 - 78, ISSN Print: 0976 – 6340, ISSN Online: 0976
– 6359.
[20] Prabhat Kumar Sinha, Chandan Prasad, Mohdkaleem and Raisul Islam, “Analysis and
Simulation of Chip Formation & Thermal Effects on Tool Life using FEM”, International
Journal of Mechanical Engineering & Technology (IJMET), Volume 4, Issue 2, 2013,
pp. 53 - 78, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359.
[21] Prabhat Kumar Sinha and Rohit, “Analysis of Complex Composite Beam by using
Timoshenko Beam Theory & Finite Element Method”, International Journal of Design and
Manufacturing Technology (IJDMT), Volume 4, Issue 1, 2013, pp. 43 - 50, ISSN Print:
0976 – 6995, ISSN Online: 0976 – 7002.