Indian Journal of Science and Technology, Vol 9(18), DOI: 10.17485/ijst/2016/v9i18/88147, May 2016
ISSN (Print) : 0974-6846
ISSN (Online) : 0974-5645
Comparative Study of Linear and Geometric
Nonlinear Load-Deflection Behavior of
Flexural Steel Members
K. K. Riyas Moideen and U. K. Dewangan
Department of Civil Engineering, National Institute of Technology Raipur, Opp. Ayurvedic College, GE Road,
Raipur - 492010, Chattisgarh, India; [email protected], [email protected]
Abstract
Background/Objectives: The proper analysis of beam is important for understanding the actual behavior and economical
use of sections. The prime objective of this research paper is the comparative study of linear and geometric nonlinear
load-deflection behavior of beams under vertical load. Methods/Statistical Analysis: In geometrically linear analysis, the
equations of equilibrium are formulated before the deformation state and are not updated with the deformation, while in
geometrical nonlinear analysis updated stiffness matrix is used at each load increment.In this study a three noded steel
beam is formulated for linear and nonlinear analysis. The incremental central point load is applied to the beam. Linear and
geometrically nonlinear deflection is computed for the beams. Three beams of the same length were taken for analysis,
but having different thickness & support conditions. The linear and geometrical nonlinear load-deflection behavior
is studied using STAAD PRO. The linear deflection is also computed by developing a finite element based code using
MATLAB. Findings:The results obtained after the analysis of beams deformations are quite interesting. The percentage
variation of linear and geometrical nonlinear deflection is very high for beam with lesser thickness. It was found that
the support conditions also affect variation of deflection for the linear and nonlinear cases. The variation between linear
and geometrical nonlinear deflection of beam is negligible when the ends are fixed. But for the same beam with simply
supported end conditions the defections were found having variation up to 37 percentages. Geometrical nonlinearity is
more when the load is very high and section is thin. At the initial stages of loading behavior of beam is linear only and it
behaves nonlinear when we go for higher loads. Application/Improvements: Linear analysis is only an approximation, so
for understanding the actual behavior of the structure and for the economical/optimized usage of sections it is suggested
to go for nonlinear analysis.
Keywords: Finite Element Method, Geometrical Nonlinear Analysis, Linear Analysis, Load-Deflection Behavior, STAAD
PRO, MATLAB
1. Introduction
Beam is a structural member that is capable of taking loads
by resisting against bending. It is characterized by their
profile (shape of cross-section), their length, and their
material. Beams are traditionally descriptions of building
or civil engineering structural elements, but smaller
structures such as truck or automobile frames, machine
frames, and other mechanical or structural systems
contain beam structures that are designed and analyzed
* Author for correspondence
in a similar fashion. It is important to study the real
behavior of the beam against the vertical loads. Generally,
for simplicity the linear nature of both material and
geometry of structures are considered. This assumption
is not valid for heavy loads and for thinner sections. So
the nonlinear nature of structures for material as well
as geometry should consider. In this paper the effect of
geometric nonlinearity on load-deflection behavior of
simply supported and fixed beams with different thickness
are considered.
Comparative Study of Linear and Geometric Nonlinear Load-Deflection Behavior of Flexural Steel Members
2. Literature Review
3. Analysis of Flexural Members
Performed geometrically nonlinear analysis of elastic
in plane oriented bodies like beams and frames. The
displacements and rotations are unrestricted in magnitude
and nonlinear equilibrium equations are solved using
the Newton-Raphson method. An incremental total
Lagrangian formulation is presented by2 which allows
the calculation of arbitrarily large displacements and
rotations. 3Discussed and surveyed the principal methods
for numerical solution of the non−linear equations.
Special emphasis is placed upon the description of an
automatic load incrimination procedure with equilibrium
iterations. 3Suggested a simple scalar quantity denoted the
current stiffness parameter, which is used to characterize
the overall behavior of non−linear problems. 4Carried
out an exhaustic study on updated Lagrangian and a
total Lagrangian formulation of a three-dimensional
beam element for large displacement and large rotation
analysis. 4Found that two formulations yield identical
element stiffness matrices and nodal point force vectors,
but updated Lagragian formulation is computationally
more effective than total lagrangian formulation. The
analysis of the geometrically nonlinear behavior of space
structures, using the modified arc length method is
explained5. New stiffness matrix for the analysis of thin
walled beams is derived6.
7
Presented a geometric and material non-linear
analysis procedure for framed structures using a solution
algorithm of minimizing the residual displacements.
7
Introduced the concept of the effective tangent stiffness
matrix and it is found to be efficient, simple and
logical in handling the non-linear analysis of frames.
8
Addressed to the review of advances, techniques and
theoretical background of the non-linear analysis of
steel beam. 9Investigated the geometric and material
nonlinear analysis of three-dimensional steel frames. A
matrix-replacement method was used which takes into
consideration the effects of axial forces on the stiffness
of the member by using the stability functions and the
effects of plastic hinges by systematically changing the
stiffness matrix in each occurrence of the plastic hinges.
10
Studied the behavior of deep beam of various span to
depth ratio by ANSYS 13.0 under two-point loading of
50 KN and also investigated the stress distribution of
deep beam. 11Performed nonlinear analysis of torsion in
reinforced concrete members after developing the initial
crack.
3.1 Linear Analysis
1
2
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Linear analysis in which structure which returns into
original form after the removal of loads and there will
be small changes in shape stiffness and no change in
loading direction or magnitude. A linear FEA analysis is
undertaken when a structure is expected to behave linearly,
i.e. obeys Hook’s Law. In linear elastic analysis, the material
is assumed to be unyielding and its properties invariable
and the equations of equilibrium are formulated on the
geometry of the unloaded structure. In this approach the
primary unknowns are the joint displacements, which
are determined first by solving the structure equation of
equilibrium. Then the unknown forces can be obtained
through compatibility consideration. In geometrically
linear analysis, the equations of equilibrium are
formulated in the un-deformed state and are not updated
with the deformation. This is valid in case of small
deformation only. Three node beam element is selected as
the finite element model in this study, having six degrees
of freedom; one lateral and one rotational at each node.
Two nodes at each supports and one node at the center of
the beam as shown in Figure 1.
Figure 1. Degree of Freedom of the Beam Elements.
The nodal variable vector is
{d} = [d1 d2 d3 d4 d5 d6] T
where d1, d3, d5 are the lateral displacement and
d2, d4, d6 are rotations at nodes one, two and three
respectively. In this work our objective to find out the
deflection at center of the beam (d4). After selecting
elements and nodal unknowns next step in finite element
analysis is to assemble element properties for each
element. These element properties are used to assemble
global properties/structure properties to get system of
equations.
[K] {d} = {F}
F = Load vector
K = Global stiffness matrix
Then the boundary conditions are imposed. The
solution of these simultaneous equations give the nodal
Indian Journal of Science and Technology
K. K. Riyas Moideen and U. K. Dewangan
unknowns. In the linear analysis of beams the value
of stiffness matrix is constant throughout the analysis.
Deflection of beam is calculated from the above
expression.
3.2 Non-Linear Analysis
In order to approach the real behavior of the steel beams,
rather than the approximate solutions with linear analysis, nonlinear analysis is preferred. In nonlinear analysis the structure will not regain its original shape after
the removal of load. Its geometry will changes resulting
in stiffness change. If a structure experiences large deformations, its changing geometric configuration can
cause the structure to respond nonlinearly. Geometric
nonlinearity is characterized by large displacements or
rotations. It arises due to the lateral loading also and
this stretching leads to a nonlinear relationship between the strain and the displacement.
types of beams are analysed using software’s. Linear
behavior is carried out using STAAD Pro and by a finite
element based MATLAB code and nonlinear behavior
by using STAAD Pro. For checking the effect of beam
thickness and supporting condition on geometrical
nonlinear problems following three beams are considered:• Simply supported steel beam with thickness 30 mm,
length 1000mm and width 50 mm shown in Figure 2.
• Simply supported steel beam with thickness 40 mm,
length 1000mm and width 50 mm.
• Both end fixed steel beam with thickness 30 mm,
length 1000mm and width 50 mm.
Figure 2. Simply Supported Beam with Central Point
Load.
[K(d)] {d} = {F}
Nonlinearity is achieved by updating element
stiffness matrices with respect to nodal displacements.
The element stiffness matrix is the function of
displacement. So at each load increment the value of
stiffness matrix will change. Newton Raphson Method
is used for nonlinear analysis. The force is applied in
several increments and these increments are equally
divided. At each increment the iterations are performed
until the convergence is achieved. If the iterations
cannot converge, the applied load is divided into two.
The reasons for non-convergence are either the element
fails or the iterations are not enough. Decreasing load
increments solves both of these problems. By this
approach the failure load is determined. When the
applied load is solved with initial tangent stiffness, the
applied force and the internal force are not equal to
each other. The difference is called unbalanced load and
at each iteration the unbalanced load decreases. When
the unbalanced loads are smaller than a tolerance, the
solution is converged.
4. Methodology
To check the validity of the formulation of linear and
geometrical nonlinear load-deflection behavior, three
Vol 9 (18) | May 2016 | www.indjst.org
Figure 3. Load vs. Deflection Curve for Simply Supported
Steel Beam with Thickness 30 mm.
5. Results and Discussions
Linear Analysis is carried out using STAAD Pro and a finite
element based MATLAB code while nonlinear by STAAD
Pro. In the analysis part we considered only the point load
Indian Journal of Science and Technology
3
Comparative Study of Linear and Geometric Nonlinear Load-Deflection Behavior of Flexural Steel Members
at the center of the beam (30KN) and we neglected the
self-weight of the beam for all cases. Load deflection
behavior of simply supported steel beam with thickness
30 mm, simply supported steel beam with thickness 40
mm and both end fixed steel beam with thickness 30 mm,
are shown in Figure 3, Figure 4 and Figure 5 respectively.
level of thirty percentage of total load (9 kN) while the
beam with 40 mm thickness shows at seventy percentage
of total load (21 kN). The thickness of beams plays a
significant role in geometrical nonlinearity of structures.
Beam with both end fixed shows almost linear behavior
for both cases. The STAAD value and finite element based
MATLAB code value almost coincide for linear case for
three beams. Linear and nonlinear deflection for three
beams under the point load is shown in Table 1, Table 2
and Table 3.
Table 1. Deflection of 30 mm Thickned Simply
Supported Steel Beam
Load
Percen
tage
Figure 4. Load vs. Deflection Curve for Simply Supported
Steel Beam with Thickness 40 mm.
10
20
30
40
50
60
70
80
90
100
Load
in KN
3
6
9
12
15
18
21
24
27
30
Deflection Under the Load Difference
(mm)
Between
Nonlinear Linear Linear Linear &
analysis analysis analysis Nonlinear
using
using
staad
matlab
2.71
2.72
2.71
0.01
5.28
5.44
5.42
0.16
7.51
8.15
8.13
0.64
9.42
10.87
10.84
1.45
11.08
13.59
13.55
2.51
12.53
16.31
16.26
3.78
13.82
19.02
18.97
5.20
14.99
21.74
21.68
6.75
16.06
24.46
24.39
8.40
17.04
27.18
27.10
10.14
Table 2. Deflection of 40 mm Thickned Simply
Supported Steel Beam
Load
Percen
tage
Figure 5. Load vs. Deflection Curve for Fixed Steel Beam
with Thickness 40 mm.
Load-deflection curve for simply support beam with
thickness 30 mm shows its nonlinear behavior at a load
4
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10
20
30
40
50
60
70
80
90
100
Load
in KN
3
6
9
12
15
18
21
24
27
30
Deflection Under the Load Difference
(mm)
Between
Linear &
Nonlinear Linear Linear
analysis analysis analysis Nonlinear
using
using
staad
matlab
1.15
1.15
1.14
0.00
2.31
2.30
2.29
0.01
3.49
3.45
3.43
0.04
4.63
4.60
4.57
0.04
5.74
5.74
5.72
0.01
6.79
6.89
6.86
0.10
7.80
8.04
8.00
0.24
8.76
9.19
9.15
0.43
9.68
10.34
10.29
0.67
10.54
11.49
11.43
0.95
Indian Journal of Science and Technology
K. K. Riyas Moideen and U. K. Dewangan
The difference between linear and nonlinear
deflection is more in case of simply supported beam
with thickness 30 mm. When the thickness of beam
increasing the effect of nonlinearity is decreasing.
The support condition has also significant effect on
the nonlinear behavior of beams. In case of both end
fixed with thickness 30 mm the effect of geometric
nonlinearity is almost negligible Percentage variation
of linear and nonlinear deflection for beam with different
thickness is shown in Table 4. The percentage variation of
linear and nonlinear deflection for different end conditions
with same thickness of beam is shown in Table 5. In full load
condition the percentage variation of linear and nonlinear
deflection is 37.3 for thickness 30 mm while it is only 8.23
for beam with thickness 40 mm. Hence the size of beam is
affecting the behavior of beams significantly.
Table 3. Deflection of 30 mm Thickness Steel Beam with Fixed Support at Both Ends
Load
Percentage
Load in KN
Deflection Under the Load (mm)
Nonlinear
analysis
Linear analysis
using staad
Linear analysis
using matlab
Difference Between Linear &
Nonlinear
10
3
0.69
0.69
0.68
0.00
20
6
1.37
1.37
1.36
0.00
30
9
2.07
2.06
2.03
0.01
40
12
2.78
2.74
2.71
0.04
50
15
3.49
3.43
3.39
0.07
60
18
4.20
4.11
4.07
0.09
70
21
4.91
4.80
4.74
0.11
80
24
5.60
5.48
5.42
0.12
90
27
6.29
6.17
6.10
0.13
100
30
6.97
6.85
6.78
0.12
Table 4. Percentage Variation of Linear and Nonlinear Deflection of Beams with Different Thickness
Load
Load
Percentage in KN
10
20
30
40
50
60
70
80
90
100
3
6
9
12
15
18
21
24
27
30
Vol 9 (18) | May 2016 | www.indjst.org
Percentage Variation of Linear
and Nonlinear (H=30mm)
Percentage Variation of Linear and Nonlinear
(H=40mm)
0.40
2.85
7.85
13.30
18.47
23.15
27.33
31.05
34.35
37.31
0.00
0.61
1.13
0.78
0.12
1.44
2.98
4.67
6.43
8.23
Indian Journal of Science and Technology
5
Comparative Study of Linear and Geometric Nonlinear Load-Deflection Behavior of Flexural Steel Members
Table 5. Percentage Variation of Linear and
Nonlinear Deflection of Beam with Different End
Conditions
Load
Load
Percentage in KN
10
20
30
40
50
60
70
80
90
100
3
6
9
12
15
18
21
24
27
30
Percentage
Percentage
Variation of Variation of Linear
Linear and
and Nonlinear
Nonlinear SSB
Fixed Beam
0.40
0.00
2.85
0.15
7.85
0.68
13.30
1.31
18.47
1.90
23.15
2.21
27.33
2.31
31.05
2.24
34.35
2.04
37.31
1.74
6. Conclusions
In this work the linear and geometrical nonlinear behavior
of flexural members subjected to central point load was
carried out. The studies on software and theoretical results
associated with them lead to the following conclusions:
• Thickness of beams significantly affect the behavior
of beams. The percentage variation of linear and
nonlinear deflection is very high for beam with
lesserthickness.
• Support conditions affect variation of deflection
between linear and nonolinear considerably. For the
fixed ends beam the variation between linear and
geometrical nonlinear deflection is neglegible, but
same beam with the simply supported coditions was
giving variation in deflection upto 37 percentages.
• Linear deflection of beams that obtained from
STAAD Pro and finite element based MATLAB code
are almost same.
• At the initial stages of loading the deflection behavior
of beam is linear. While it behaves nonlinear with the
increasing the load values. Geometrical nonlinearity
6
Vol 9 (18) | May 2016 | www.indjst.org
is more when the load is very high and section is thin.
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Indian Journal of Science and Technology