“TENSEGRITY BASED TOWER STRUCTURES”

Project Report on
“TENSEGRITY BASED TOWER
STRUCTURES” Submitted by:
APURVA PRAKASH
2003CE10243
Under the Guidance of:
PROF. ASHOK GUPTA & DR. SURESH BHALLA
In partial fulfillment of the requirements of the degree of
Bachelor of Technology
to the
Department Of Civil Engineering
Indian Institute Of Technology Delhi
April, 2007
CERTIFICATE
“I do certify that this report explains the work carried out by me in the Course
CED411 Project-Part 1 & CED412 Project-Part 2 under the overall supervision of
Prof. Ashok Gupta and Dr. Suresh Bhalla. The contents of the report including text,
figures, tables, computer programs, etc. have not been reproduced from other sources
such as books, journals, reports, manuals, websites, etc. Wherever limited
reproduction from another source had been made the source had been duly
acknowledged at that point and also listed in the References.”
Himanshu Pande
2003CE10251
i
CERTIFICATE
“This is to certify that the report submitted by Himanshu Pande describes the work
carried out by him in the course Courses CED411 Project-Part 1 & CED412 Project
Part 2, under my overall supervision.”
Dr. Suresh Bhalla
Assistant Professor
Prof. Ashok Gupta
Professor
ii
ACKNOWLEDGEMENT
I would like to express my gratitude to the Department of Civil Engineering, IIT
Delhi for providing me with the opportunity to work on this challenging project as my
B.TECH Project. I would especially like to thank my supervisors, Dr. Ashok Gupta
and Dr. Suresh Bhalla, Department of Civil Engineering, IIT, Delhi, for their
guidance. Their constant support and encouragement and my frequent discussions
with them have helped me to gain a lot of insight and knowledge which in turn has
helped me greatly in my work for this project. Working under them has been a
rewarding experience and without their enthusiasm and help, this work would not
have been possible.
I would also like to thank my fellow batch mate, Apurva Prakash along with structural
lab and workshop staff for their help in various parts of this project.
A special thanks to M.tech Student Mr.Vijaykumar for his constant support in
carrying out the project.
Himanshu Pande
2003CE10251
iii
ABSTRACT
The qualities of the tensegrity structures, which make the technology attractive for
human use, are their resilience and ability to use material in a very economical way.
Thus the construction of the structures using tensegrity principle will make them
highly resilient and reasonably economical at the same time. Though a lot of research
has focused on the theoretical aspect such as form finding, only a few practical works
have been done on how to use these structures. Hence this project aimed at
fabrication, experimentation and analysis of tensegrity grid structures using the
tensegrity prototype half-cuboctahedron. The purpose of this project was to study the
behavior of grid structure based on tensegrity by comparing the theoretical results
with experimental results using the technique of model updating. Focus of the project
was on new fabrication techniques and overcoming the previous flaws.
-
New kind of joints are designed
ƒ
-
More material efficient
ƒ
Require less labour
ƒ
Lighter in weight
ƒ
Improve Connectivity of cable and struts
Layered Fabrication
ƒ
Structures have been constructed in layers
ƒ
Fabrication made very easier
ƒ
Higher Precision
Once the fabrication was completed, non-destructive static and dynamic testing of
the structure was done in the lab and the appropriate model updating was done.
iv
TABLE OF CONTENTS
CERTIFICATE.....................................................................................................
i
CERTIFICATE.....................................................................................................
ii
ACKNOWLEDGEMENT....................................................................................
iii
ABSTRACT...........................................................................................................
iv
TABLE OF CONTENTS......................................................................................
v
LIST OF FIGURES..............................................................................................
vii
List of Tables.........................................................................................................
x
CHAPTER ONE: LITERATURE REVIEW.....................................................
1
Principle of Tensegrity................................................................................
1
The Benefits of Tensegrity..........................................................................
3
Some Disadvantages...................................................................................
7
CHAPTER TWO: PROTOTYPE AND MODEL DETAILS..........................
8
2.1 Description of the model prototype............................................................
8
2.2 Description of the model.............................................................................
8
CHAPTER THREE: ANSYS MODELING.......................................................
10
3.1 Introduction.................................................................................................
10
3.2 Assumptions taken while modeling the structure.......................................
10
3.3 ANSYS tensegrity grid model....................................................................
11
3.4 Material Properties and Characteristics......................................................
12
3.5 Analysis of the grid structure using ANSYS..............................................
12
CHAPTER FOUR: FABRICATION OF THE STRUCTURE........................
19
4.1 Methodology...............................................................................................
19
4.2 Joints Description........................................................................................
20
v
4.3 Design and detailing of joints......................................................................
24
4.4 Fabrication of joints....................................................................................
26
4.5 Fabrication and erection of structure...........................................................
29
CHAPTER FIVE: TESTING OF THE STRUCTURE.....................................
31
CHAPTER SIX: MODEL UPDATING & COMPARATIVE ANALYSIS.....
33
CONCLUSIONS....................................................................................................
40
RECOMMENDATIONS......................................................................................
41
REFERENCES......................................................................................................
42
APPENDIX A Newton Raphson Method..............................................................
43
APPENDIX B Comparison of Various Joints........................................................
45
APPENDIX C Dynamic Testing............................................................................
47
vi
LIST OF FIGURES
Figure 1.1 Tensegrity — a balance of continuous pull and discontinuous push....
3
Figure 1.2 Complexity in construction for large tensegrity structures...................
7
Figure 2.1 Half-cuboctahedron...............................................................................
8
Figure 2.2 Top View of the Grid Structure to be fabricated..................................
9
Figure 3.1 Top View in Ansys...............................................................................
11
Figure 3.2 Perspective view in Ansys....................................................................
12
Figure 3.3 Convergence graph...............................................................................
13
Figure 3.4 Numbering of Nodes.............................................................................
14
Figure 3.5 Vertical displacement variations for nodes 12, 13 & 14.......................
14
Figure 3.6 General Load vs. Vertical displacement Graph....................................
15
Figure 3.7 Deformed shape of the Structure..........................................................
15
Figure 3.8 Member Forces in the Grid Structure at failure load............................
17
Figure 3.9 Variation of reaction force for node 2 and node 3................................
18
Figure 4.1 Eye bolt Joint........................................................................................
20
Figure 4.2 Box-type Joint.......................................................................................
20
Figure 4.3 Circular plates Joint..............................................................................
20
Figure 4.4 Cable hydraulic pressed joint................................................................
20
Figure 4.5 Cable bolt joints....................................................................................
21
Figure 4.6 Proposed cable joints............................................................................
21
Figure 4.7 Proposed Main Joint.............................................................................
22
Figure 4.8 Proposed Strut Connections with a Groove..........................................
22
Figure 4.9 Adopted side joint (a) Top View (b) Front view..................................
23
Figure 4.10 Adopted mid-joint (a) Top View (b) Front View...............................
23
vii
Figure 4.11 Side Joint Details................................................................................
24
Figure 4.12 Middle Joint Details............................................................................
24
Figure 4.13 Cutting of angles into pieces (a) cutter one (b) cutter two..................
26
Figure 4.14 Edge Smoothening..............................................................................
27
Figure 4.15 Marking of holes.................................................................................
27
Figure 4.16 Drilling of holes..................................................................................
27
Figure 4.17 Grooving of Struts..............................................................................
28
Figure 4.18 A Cable Joint......................................................................................
28
Figure 4.19 A Turn Buckle...................................................................................
29
Figure 4.20 Turn Buckle attached to top rope........................................................
29
Figure 4.21 A single prototype...............................................................................
29
Figure 4.22 Top view of tensegrity grid structure fabricated in lab.......................
30
Figure 4.23 Front view of the same grid structure.................................................
30
Figure 5.1 Strain Gauges applied to ropes.............................................................
31
Figure 5.2 Strain Gauges applied to struts.............................................................
31
Figure 5.3 Static testing of structure......................................................................
32
Figure 5.4 Monitoring System...............................................................................
32
Figure 6.1 Post- testing Ansys Model (updated)....................................................
33
Figure 6.2 A loaded ANSYS model with DOF constraints...................................
34
Figure 6.3 Side cable and strut for which analysis was done.................................
35
Figure 6.4 Top cable and bottom cable for which analysis was done....................
35
Figure 6.5 Theoretical and Experimental comparison for a strut...........................
36
Figure 6.6 Theoretical and Experimental comparison for a top cable...................
37
Figure 6.7 Theoretical and Experimental comparison for a side cable..................
38
Figure 6.8 Theoretical and Experimental comparison for a bottom cab................
39
viii
Figure C.1 Accelerometer......................................................................................
47
Figure C.2 Digital multimeter................................................................................
47
Figure C.3 Hammering the structure......................................................................
47
Figure C.4 Dynamic response of the structure.......................................................
48
ix
LIST OF TABLES
Table 3.1 Material Properties and Characteristics............................................
12
Table B.1 Disadvantages of previously used nodal joints................................
45
Table B.2 Comparison of different nodal joints...............................................
45
Table B.3 Disadvantages of previously used cable joints................................
46
Table B.4 Comparison of different cable joints...............................................
46
x
________________________________________
CHAPTER ONE
LITERATURE REVIEW
Tensegrity is a portmanteau of tensional integrity. It refers to the integrity of
structures as being based in a synergy between balanced tension and compression
components. Tensegrity structures are built of struts and cables. The struts can resist
compressive force and the cables cannot. Most cable–strut configurations which one
might conceive are not in equilibrium, and if actually constructed will collapse to a
different shape. Only cable–strut configurations in a stable equilibrium will be called
tensegrity structures. If well designed, the application of forces to a tensegrity
structure will deform it into a slightly different shape in a way that supports the
applied forces. Tensegrity structures are very special cases of trusses, where members
are assigned special functions. Some members are always in tension and others are
always in compression. A tensegrity structure’s struts cannot be attached to each other
through joints that impart torques. The end of a strut can be attached to cables or ball
jointed to other struts.
1.1. Principle of Tensegrity
‘Tensegrity’ is a pattern that results when ‘push’ and ‘pull’ have a win-win
relationship with each other. Pull is continuous where as push is discontinuous. The
continuous pull is balanced by the discontinuous push, producing the integrity of
tension and compression. These fundamental phenomena do not oppose, but rather
complement each other. Tensegrity is the name for a synergy between a co-existing
pairs of fundamental physical laws of push and pull, or compression and tension, or
repulsion and attraction [7].
1
If one pushes a ping-pong ball on a smooth table with the point of a sharp pencil, the
ball would always roll away from the direction of the push, first rolling one way then
the other. Push is divergent. On the other hand, if a string be attached to the ping pong
ball with tape, then pulling it leads to convergence. So Pull is convergent [2].
Another example from common experience occurs when pulling a trailer with a car.
When driving uphill, one is pulling against gravity, and a trailer will converge toward
the same course behind the car. If the trailer begins to sway, increasing pull by
increasing acceleration can dampen the swaying motion. Driving downhill, however,
the trailer may begin to push, and the trailer will begin to sway from side to side [5].
Two tensegrity are easily recognizable in the systems of the human body. The
muscular-skeletal system is a tensegrity of muscles and bones, the muscles provide
continuous pull, the bones discontinuous push. This forms the basis for all human
physical mobility. The central nervous system can also be seen as using the analogy
of tensegrity where motor neurons and sensor neurons, complement the other in a
balance [5].
A more common example of a tensegrity is in a child's balloon. When examined as a
system, the rubber skin of the balloon can be seen as continuously pulling (against the
air inside) while the individual molecules of air are discontinuously pushing against
the inside of the balloon keeping it inflated. All external forces striking the external
surface are immediately and continuously distributed over the entire system, hence the
balloon is quite strong despite its thin material [2].
2
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Molecules of air discontinuously
pushing against the continuously
pulling rubber skin of the balloon.
Figure 1.1 Tensegrity — a balance of continuous pull and discontinuous push
The automobile tire works the same way. It is the tensional integrity in the tire that
yields a low failure rate despite the wear of high speeds and long miles [7]. Thus a
tensegrity is any balanced system composed of two elements, a continuous pull
balanced by discontinuous push. When these two forces are in balance, it results in a
stabilized system that is maximally strong [1].
1.2. The Benefits of Tensegrity
A large amount of literature on the geometry, art form, and architectural appeal of
tensegrity structures exists, but there is little on the dynamics and mechanics of these
structures. Form finding results for simple symmetric structures appear and show an
array of stable tensegrity units is connected to yield a large stable system, which can
be deployable. Several reasons are given below why tensegrity structures should
receive new attention from mathematicians and engineers.
(a) Tension stabilizes
A compressive member loses stiffness as it is loaded, whereas tensile member gains
stiffness as it is loaded. Stiffness is lost in two ways in a compressive member. In the
absence of any bending moments in the axially loaded members, the forces act exactly
through the mass center, the material spreads, increasing the diameter of the center
cross section; whereas the tensile member reduces its cross-section under load. In the
presence of bending moments due to offsets in the line of force application and the
center of mass, the bar becomes softer due to the bending motion. For most materials,
the tensile strength of a longitudinal member is larger than its buckling strength.
Hence, a large stiffness-to-mass ratio can be achieved by increasing the use of tensile
members [2].
(b) Tensegrity Structures are efficient
The geometry of material layout is critical to strength at all scales, from nano-scale
biological systems to mega-scale civil structures. Traditionally, humans have
conceived and built structures in rectilinear fashion. Civil structures tend to be made
with orthogonal beams, plates, and columns. Orthogonal members are also used in
aircraft wings with longerons and spars. However, evidence suggests that this
“orthogonal” architecture does not usually yield the minimal mass design for a given
set of stiffness properties. Bendsoe and Kikuchi, Jarre, and others have shown that the
optimal distribution of mass for specific stiffness objectives tends to be neither a solid
mass of material with a fixed external geometry, nor material laid out in orthogonal
components. Material is needed only in the essential load paths, not the orthogonal
paths of traditional manmade structures. Tensegrity structures use longitudinal
members arranged in very unusual (and non orthogonal) patterns to achieve strength
with small mass. Another way in which tensegrity systems become mass efficient is
with self-similar constructions replacing one tensegrity member by yet another
tensegrity structure [4].
4
(c) Tensegrity Structures are deployable
Materials of high strength tend to have a very limited displacement capability. Such
piezoelectric materials are capable of only a small displacement and “smart”
structures using sensors and actuators have only a small displacement capability.
Because the compressive members of tensegrity structures are either disjoint or
connected with ball joints, large displacement, deployability, and stowage in a
compact volume will be immediate virtues of tensegrity structures. This feature offers
operational and portability advantages. A portable bridge or a power transmission
tower made as a tensegrity structure could be manufactured in the factory, stowed on
a truck or helicopter in a small volume, transported to the construction site, and
deployed using only winches for erection through cable tension. Erectable temporary
shelters could be manufactured, transported, and deployed in a similar manner [5].
(d) Tensegrity Structures are easily tunable
The same deployment technique can also make small adjustments for fine tuning of
the loaded structures, or adjustment of a damaged structure. Structures that are
designed to allow tuning will be an important feature of next generation mechanical
structures, including civil engineering structures [3].
(e) Tensegrity Structures can be more reliably modeled
All members of a tensegrity structure are axially loaded. Perhaps the most promising
scientific feature of tensegrity structures is that while the global structure bends with
external static loads, none of the individual members of the tensegrity structure
experience bending moments. Generally, members that experience deformation in
two or three dimensions are much harder to model than members that experience
deformation in only one dimension. The Euler buckling load of a compressive
5
member is from a bending instability calculation, and it is known in practice to be
very unreliable. That is, the actual buckling load measured from the test data has a
larger variation and is not as predictable as the tensile strength. Hence, increased use
of tensile members is expected to yield more robust models and more efficient
structures. More reliable models can be expected for axially loaded members
compared to models for members in bending [4].
(f) Tensegrity Structures Facilitate High Precision Control
Structures that can be more precisely modeled can be more precisely controlled.
Hence, tensegrity structures might open the door to quantum leaps in the precision of
controlled structures. The architecture (geometry) dictates the mathematical properties
and, hence, these mathematical results easily scale from the nano-scale to the megascale, from applications in microsurgery to antennas, to aircraft wings, and to robotic
manipulators [8].
(g) A Paradigm that Promotes the Integration of Structure and Control
Disciplines
A given tensile or compressive member of a tensegrity structure can serve multiple
functions. It can simultaneously be a load-carrying member of the structure, a sensor
(measuring tension or length), an actuator (such as nickel-titanium wire), a thermal
insulator, or an electrical conductor. In other words, by proper choice of materials and
geometry, a grand challenge awaits the tensegrity designer: How to control the
electrical, thermal, and mechanical energy in a material or structure? For example,
smart tensegrity wings could use shape control to maneuver the aircraft or to optimize
the air foil as a function of flight condition, without the use of hinged surfaces.
Tensegrity structures provide a promising paradigm for integrating structure and
control design [7].
6
1.3. Some Disadvantages
• Tensegrity arrangements suffer the problem of bar congestion. As some designs
become larger (thus, the arc length of a strut decreases), the struts start running into
each other [3].See figure 1.2
• The same author stated, after experimental research, “relatively high deflections and
low material efficiency, as compared with conventional, geometrically rigid
structures” [3].
• The fabrication complexity is also a barrier for developing the floating compression
structures. Spherical and domical structures are complex, which can lead to
problems in production. [1]
• In order to support critical loads, the pre-stress forces should be high enough, which
could be difficult in larger-size constructions [6].
Figure 1.2 Complexity in construction for large tensegrity structures
7
CHAPTER TWO
PROTOYPE AND MODEL DETAILS
________________________________________
2.1 Description of the model prototype
The tensegrity prototype to be used in this project for fabrication of the grid structure
was half cuboctahedron. It comprises of 12 cables (4 top cables, 4 bottom cables and
4 side cables) and 4 struts in all. The diagrammatic representation of the prototype is
as shown in figure 2.1 with perspective view and top view.
4
7
7(0.5,1,0.5)
6(0,1,0)
8(1,1,0)
5
2
5(1,0.5,0.5)
4(0,0.5,0.5)
6
8
1
3(1,0,0)
1(0,0,0)
2(0.5,0,
3
Figure 2.1 Half-cuboctahedron
Figure 2.1 also gives the co-ordinate system chosen to calculate the desired length of
all cables and struts. The length of the bottom cable is 1 m and that of the top and side
cables are 0.707 meter. The struts are1.224m in length and the height of the prototype
is 0.5 m. All the lengths are measured from centre to centre of joints.
2.2 Description of the model
A tensegrity grid structure is defined as a structure obtained by combining two or
more modules. This project aimed at fabrication, modeling and experimental analysis
of tensegrity grid structure made by combination of four modules of halfcuboctahedron (details of which are given above). Figure 2.2 diagrammatically
represents the top view of the grid structure.
8
2m
2m
Figure 2.2 Top View of the Grid Structure to be fabricated
In all there will be 16 struts and 40 cables in the grid structure with same dimensions
as the prototype half-cuboctahedron.
This grid structure is symmetrical in shape and is expected to show a symmetrical
behavior during analysis. The main purpose behind the fabrication and analysis of the
grid structure is to check whether the structure is a suitable choice for roofing
mechanism. For this purpose the testing of the grid structure will be done under
vertical loads and vertical displacement of the structure will be plotted against the
loads to study the response.
9
____________________________________
CHAPTER THREE
ANSYS MODELLING
3.1 Introduction
A characteristic feature of the tensegrity structures is the presence of geometric
nonlinearities due to the changing geometry as they deflect under loads. That is the
stiffness matrix [k] is a function of the displacement (u). There are four types of
geometric non-linearity - large strains, large rotations, stress stiffening and spin
softening (Cook et al 2003).
In tensegrity structures, stress stiffening is more prominent. In this type of non
linearity both strains and rotations are small. Stress stiffening effect normally needs to
be considered for thin structures, such as cables, thin beams and shells that have very
small bending stiffness as compared to the axial stiffness. In such structures, the in
plane and the transverse displacements are coupled. This effect also augments the
regular non-linear stiffness matrix produced by large strain or large deflection effects.
Generating and then using additional stiffness matrix called as stress stiffness matrix
accounts for the effect of stress stiffening. It may be used for static and transient
analysis.
So while modeling the tensegrity grid in ANSYS 9, geometrical non linearity and
large deformation effects are taken into account.
3.2 Assumptions taken while modeling the structure
(1) All elements are truss elements i.e. there are no bending moments developed. To
take this assumption into account, the elements are modeled as 3-D spar elements
10
which are uniaxial tension-compression elements with three degrees of freedom at
each node i.e. translations in x, y and z directions.
(2) Each element is defined by two nodes, the cross sectional area, an initial strain and
material properties. The initial strain readings are taken from previous
experimental data and can be updated after the actual experiment.
(3) The materials are assumed to be isotropic and linear in nature
(4) At the bottom nodes degree of freedom in Z-direction is locked except for the
central node for which all the degree of freedom are locked and at the top nodes
all the degrees of freedom are released.
3.3 ANSYS tensegrity grid model
Taking the above assumptions into account the modeling for grid structure was done
in Ansys 9. Figure 3.1 and 3.2 shows the top view and perspective view of the model.
Top Cables
Struts
Bottom cables
Figure 3.1 Top View in Ansys
11
Top Cables
Side Cables
Struts
Bottom cables
Figure 3.2 Perspective View in Ansys
3.4 Material Properties and Characteristics
The material which will be used for fabrication of the grid structure is steel. Table 3.1
gives the properties which have been taken for the initial analysis and can be changed
while model updating.
Table 3.1 Material Properties and Characteristics*
S.No
1.
2.
3.
4.
Properties
Area
Young’s Modulus
Poisson Ration
Initial Strain
Steel Struts
Steel Cables
2
160.284 mm
205000 N/mm2
0.25
0.64823E-04
6.53 mm2
95200 N/mm2
0.25
0.17427E-02
*All these values have been taken from previous similar experiments
3.5 Analysis of the grid structure using ANSYS
Now since tensegrity structures are very flexible and undergo large deformation due
to which they experience geometric non linearity so generally serviceability failure
criterion is used for their analysis along with a check on member forces to see
whether there is any material failure or not.
12
As per codes, permissible vertical deformation for the tensegrity structure is L/100
where L is the length of the base which is 2 meters in this case. So permissible
displacement in vertical direction for this grid structure is 2000/100=20mm. So at a
vertical displacement of 20mm the grid structures is considered to be failed as per
serviceability criterion.
Using the technique of trial and error, various vertical loads are applied and maximum
vertical deformation is obtained. All the loads are applied on top nodes such that load
at the middle node is twice the load applied on the node at the periphery. It is found
that when a load of 2500N is applied on each node at the periphery and 5000N on
each middle node than the structure undergoes a maximum vertical displacement of
19 mm. So it is concluded that the loading capacity of the structure will be 16*
2500N= 40000N i.e. 4000 kg. Newton Raphson method is used for the analysis of the
structure (see appendix). Figure 3.3 gives the convergence graph of obtained during
the analysis.
Figure 3.3 Convergence graph
Figure 3.4 shows node numbering. The variation of vertical displacement with time,
where every unit increase in time corresponds to a load increase of 4000N, is shown
in figure 3.5.
Figure 3.4 Numbering of Nodes
Figure 3.5 Vertical displacement variations for nodes 12, 13 & 14
It is found that following node undergo same vertical displacement:
(a) nodes 6, 12, 18, 20
(b) nodes 7, 13, 16, 21
(c) nodes 5, 8, 11, 17
It is observed that the variation of vertical displacement is linear for all the nodes. The
possible reason for this can be that the failure load as per the serviceability failure
criterion falls in the linear region of the actual load vs. vertical displacement graph as
depicted by figure 3.6.
Failure Load
Linear Region
Load
Displacement =L/100
Vertical Displacement
Figure 3.6 General Load vs. Vertical displacement Graph
The deformed shape of the structure is as shown in figure 3.7. As can be seen from
the figure 3.7 there are displacements in X and Y directions also.
Figure 3.7 Deformed shape of the Structure
After this a check for member forces is carried out to see whether any material failure
takes place.
(a) Allowable strength for struts
Assuming hinged-hinged conditions the effective length for strut is equal to the actual
length. So,
Effective length= l =1.224
Radius of gyration r =
(I/A), where, I is the moment of inertia =7109.627x10-12 m4
and A is the area = 160.284x10-6m2. So we get
r= 6.66x10 –3 m,
l= 1.224 m,
Slenderness Ratio=183.78
From table 5.1 of IS: 800-1984, for fy= 240 N/mm2 permissible stress of the strut
comes out to be 31.866 N/mm2. So allowable force in struts = 31.866x160.284 =
5107.61 N
Hence allowable load in the strut is 5107.61
(b) Allowable strength for cables
From previously conducted experiments it was found that the proof stress in the cable
is 1119.575N/mm2
Maximum load the cable can carry is equal to 6.53x1119.575=7310.82N
Now the member forces which are obtained from the analysis are diagrammatically
represented in figure 3.8
16
3.8 Member Forces in the Grid Structure at failure load
As can be seen from figure 3.8 the maximum compression developed is 2356 N which
is much less than the allowable strength of struts. Similarly maximum tension
developed is 2198 which is also much less than the allowable strength of the cables.
So there is no material failure for the load capacity of 4000 kg. It is also clearly
visible from the above figure that all struts are in compression and all cables are in
tension so no cable is acting as a redundant part.
The reaction forces were also calculated and obtained during the analysis. The
reaction forces obtained are 6135.74 N and 3864.26 N. Due to the symmetry of the
structure:
(a) Nodes 2, 4, 9 and 15 have same reaction forces equal to 6135.74 N
(b) Nodes 3, 10, 14 and 19 have same reaction force equal to 3864.26 N
The variation of the reaction forces with time is as shown in the figure 3.9 for node 2
and node 3.
Figure 3.9 Variation of reaction force for node 2 and node 3
As can be seen from the figure 3.9 the variation of the of reaction forces with time or
applied load is also linear. All the reaction forces obtained are in positive Z-direction.
And as a check for equilibrium the summation of these reaction forces on all nodes
mentioned above is equal to total applied load of 40000N in the final step.
So as seen from the theoretical analysis, all the variations obtained are linear in nature
hence from pre-experiment theoretical work it can be concluded that structure is
expected to show a linear response.
CHAPTER FOUR
FABRICATION OF THE STRUCTURE
______________________________________
4.1 Methodology
From the previous experimentation and experiences it was observed that the main
problem which is encountered during the fabrication of these structures is the
fabrication of the appropriate joints. The joints of these structures are subjected to
following constraints:
-
must be as much flexible as possible
-
light in weight so that light weight advantage of tensegrity over other
conventional structures is maintained
-
Easy to fabricate and require less labour
-
Strong enough so that structure should not fail due to failure of these joints
So before the fabrication of the actual structure, fabrication of joints was done. Joints
were basically divided into three kinds of categories:
-
Main joints
-
Cable connections
-
Strut connections
Main joints were fabricated using a T-section in civil engineering workshop along
with cable connections whereas strut connections were fabricated in the central
workshop of IIT Delhi. Once the fabrication of joints was completed, then fabrication
of structure was done. Layered fabrication approach was adopted i.e. the whole
structure was fabricated in two layers: bottom layer and top layer and after that its
erection was done in civil engineering workshop. In all four turn buckles were used in
the top layer to facilitate the deployment of the structure.
19
4.2 Joints Description
The earlier joints which were fabricated in the lab were less flexible, heavier, and
difficult to fabricate as they required technical labour and used to fail before the
failure of the structure. Apart for this, from economic point of view also those joints
were not desirable as they required hired technical labour and involved more
dependence on machines which were not available in IIT labs. Figure 4.1 to 4.5
depicts joints used in previous projects and experimentation.
Figure 4.1 Eye bolt Joint
Figure 4.2 Box-type Joint
Figure 4.3 Circular plates Joint
Figure 4.4 Cable hydraulic pressed joint
Figure 4.5 Cable bolt joints
To overcome the weaknesses of previously mentioned joints, new kind of joints were
fabricated in the lab, which were based on the design proposed by a fellow B.tech
Student.
(a) Proposed Cable Joint: A loop was formed at the end of the cable and joint was
made by winding a ‘winding wire (1.5mm dia)’ around it with 30 turns. It was
then soaked in Epoxy solution (Fevitite) and left to dry for a day.
21
Figure 4.6 Proposed cable joints
(b) Proposed Main Joint: A circular Plate (of diameter 10 cm. and thickness 5
mm.) was taken and a semicircular plate (of same dimensions) was welded at 90
degrees on its diameter. A hole (of 8mm) was made at 1.4 cm. from the top in the
semicircular plate.
Figure 4.7 Proposed Main Joint
(c) Proposed Strut Connection: A GI pipe was taken and a groove was made along
the length so that the pipe could be fitted into the plate. A bore of 8mm was also
drilled so that a bolt could be fastened through it.
Figure 4.8 Proposed Strut Connections with a Groove
22
The proposed main joint was flexible but it was not easy to fabricate as well noneconomical. Hence slight modifications were done to make fabrication easier and
more economical. A T-Joint was adopted which had a similar structure as the
proposed main joint and was much economical and easy to fabricate. So in all twenty
one joints were fabricated and they were of two types:
(a) Side Joints (unidirectional)
(b) Mid Joints (multi-directional)
Figure 4.9 and 4.10 pictorial represents the joints which were used in the fabrication
of the structure in this project. These joints were found to be very flexible and
lightweight without any compromise in strength.
(a)
(b)
Figure 4.9 Adopted side joint (a) Top View (b) Front view
(a)
(b)
23
Figure 4.10 Adopted mid-joint (a) Top View (b) Front View
On the other hand the cable and strut connections, as proposed, were adopted for the
real design. The cable connections were fabricated manually in the lab whereas the
strut connections were fabricated in IDDC lab of IIT Delhi. For a comparative study
of different joints one can refer to the appendix.
4.3 Design and detailing of joints
Design of joints and drilling of holes was done according to IS code
recommendations. As recommended by IS code an edge distance of 19 mm was given
for rough edges whereas an edge distance of 17 mm was given for rolled edges when
the hole drilling was done. Details for a side joint are shown in figure 4.11 and for a
middle joint in figure 4.12.
1.7 cm
1.9 cm
Rolled Edge
3 mm
Rough Edge
5 cm
Rough Edge
3mm
1.7 cm
Rolled Edge
Figure 4.11 Side Joint Details
Rough Edge
Rolled Edge
1.7 cm
1.9 cm
3 mm
1.9 cm
24
1.9 cm
3mm
1.7 cm
Rough Edge
7.62 cm
Figure 4.12 Middle Joint Details
These minimum edge distances as recommended by IS codes provides a safe guard
against bursting and shearing. Bursting takes place when bolts are placed too near the
edge of the plate, the plate may burst out due to the pressure of the bolt or may shear
out. Now due to strut connections the force experienced by bolts and joints can be
calculated using the theory of steel structures.
(a) Bearing of plate or bolt
Resistance in bearing = Fbr * d * t
where,
Fbr = safe stress in bearing
d = diameter of the hole
t = thickness of plate in contact
so, resistance in bearing = 220 * 3 * 2 = 1320 N
(b) Shearing
Now the bolt connection will be in double shear so,
Resistance in double shear = (2 * Fs * 3.14 * d2)/4
where, Fs = Safe stress in shearing and d = diameter of the hole
hence, resistance in shearing = 2 * 80 * 7.07 = 1131.2 N
25
Now as seen from above calculations, the resistance in shearing is slightly less than
the resistance in bearing so shear strength is the governing factor for design
considerations. Since the shear strength of the bolt was coming out to be too low so
minimum edge distances were adopted from IS code recommendations to prevent the
failure due to shearing and bursting.
4.4 Fabrication of joints
(a) Main Joints
Fabrication of main joints was done in civil engineering workshop of IIT Delhi. It
involved the following steps and in all twenty one joints were fabricated out of which
twelve were side joints and nine were middle joints:
-Cutting of T-beam
The T-beam was cut into pieces of two lengths:
(1) Piece of length 5 cm for side joints
(2) Piece of length 7.62 cm for middle joints
Two kinds of cutters were used as shown in figure 4.13 to speed up the fabrication.
(a)
(b)
Figure 4.13 Cutting of angles into pieces (a) cutter one (b) cutter two
-Welding
26
After the cutting of angles, required welding was done for the fabrication of middle
joints. Rectangular Strips of length 3.81 cm were cut and welded to T-sections of
length 7.62 cm. Welding done had throat thickness of 3 mm
- Edge smoothening
After the fabrication, smoothening of edges was done in order to improve the
appearance of joints and to make them harmless. The process of smoothening is
shown in figure 4.14.
Figure 4.14 Edge Smoothening
- Drilling of Holes
Once the smoothening of the edges was completed, holes were drilled using drill bits
of diameter 3 mm. Location of drilling depended upon the type of joint. First the
marking of holes was done as shown in figure 4.15 then the drilling was done as
shown in figure 4.16
27
Figure 4.15 Marking of holes
Figure 4.16 Drilling of holes
- Painting of joints
In the end, joints were painted with red oxide paint to prevent rusting and improve
their appearance.
(b) Strut Connections
For connections of struts a groove of length 3 cm was made on each end of the strut
and then holes of 3 mm diameter were drilled. All these connections were made in
central workshop of IIT Delhi as shown in figure 4.17
Figure 4.17 Grooving of Struts
(c) Cable connections
Cable connections were made manually and simultaneously during the fabrication of
the structure. A loop was formed at the end of each cable and joint was made by
winding a wire of diameter 1.5 mm around it with 30 turns. And then the joint was
soaked in Epoxy
solution
(Fevitite) and left
to dry for a day
as shown in figure
4.18.
28
Figure 4.18 A Cable Joint
4.5 Fabrication and erection of structure
The fabrication of structure was done in civil engineering workshop of IIT Delhi. The
structure was fabricated in two layers: top layer and bottom layer to ease out the
process of fabrication. Once these layers ware completed, the erection of structure
was done and the ropes were tightened using turn buckles which were attached to top
ropes as shown in figure 4.19 and 4.20.
Figure 4.19 A Turn Buckle
Figure 4.20 Turn Buckle attached to top rope
In all four turn buckles were used. Erection of structure was done in steps i.e. first a
single prototype was erected (as shown in figure 4.21) and then the adjacent ones.
29
Figure 4.21 A single prototype
The completely fabricated structure is as shown in figure 4.22 and 4.23
30
Figure 4.22 Top view of tensegrity grid structure fabricated in lab
Figure 4.23 Front view of the same grid structure
CHAPTER FIVE
TESTING OF THE STRUCTURE
________________________________________
After completion of fabrication, the static and dynamic testing of the structure was
conducted in the lab. Though the aim of the project was to fabricate the structure and
to conduct static testing but due to availability of time dynamic testing of the structure
was also done and response was captured using an accelerometer. For further details
on dynamic testing refer to appendix C.
The static testing of the structure was done in the civil engineering workshop of IIT
Delhi. To get the member forces, it was necessary to obtain strain in member elements
due to applied loads and hence the strain of the structural elements was recorded using
electrical strain gauges as shown in figure 5.1 and 5.2.
Figure 5.1
Strain
Gauges
applied
to ropes
Figure
5.2 Strain
Gauges
applied to
struts
Electrical strain gauges are based on the principle that under a mechanical stress, the
electrical resistance of a conductor varies in proportion to the load induced strain. Due
to the symmetry of the structure, in all nine strain gauges were applied, four 4mm
strain gauges were bonded along the circumference of the pipe in order to avoid the
bending effects, two 2mm strain gauges were bounded to the top and bottom cable
each and one 2mm strain gauge on side cable.
Non-destructive static testing of the structure was done i.e. loads were not applied till
failure because the purpose of the testing was not to fail the structure but to record the
structural response so as to compare it with the theoretical model. The structure was
loaded by using wooden planks as shown in figure 5.3 and then the loads was applied
at equal intervals of time.
32
Figure 5.3 Static testing of structure
All the strain gauges were connected to the data logger which in turn was connected
to the computer having the software strain smart version3.1 installed in it to monitor
the strains at different loading as shown in figure 5.4
Figure 5.4 Monitoring System
A total load of 105 kg i.e. 1050 N was applied on the structure including the weight of
the planks and the results obtained have been discussed in next section
__________________________________________
CHAPTER SIX
MODEL UPDATING AND COMPARITIVE ANALYSIS
As discussed in previous section, static and dynamic testing was done on the structure
and the results obtained have been compared with the theoretical results obtained by
analyzing the structure on ANSYS.
The ANSYS model which was used for Pre-testing analyses was updated
appropriately by making the following changes and is as shown in figure 6.1:
33
(a) The material properties in simulated model were updated according to the
properties of the materials actually used.
(b) The slight change in the configuration of the model was done so as to match the
actual configuration.
(c) The degree of freedom constraints were updated appropriately as per the actual
experimental constraints.
Figure 6.1 Post- testing Ansys Model (updated)
Once the theoretical model was updated so as to resemble the actual model, then the
simulated analysis was carried out. Loads were applied in stages as was done in actual
experimentation and for each load application member forces were obtained and
tabulated. A loaded ANSYS model with degree of freedom constraints is as shown in
figure 6.2.
Figure 6.2 A loaded ANSYS model with DOF constraints
As can been seen from the figure above, it was assumed that the load distribution on
the mid joints will be double the load on the corner joints. Once the member force
variation was obtained for the theoretical model, then the analysis of experimental
results was done.
Since strain gauges were used to capture the response of the structure so the results
obtained after experimentation were strains in micro units for members on which
these strain gauges were applied. But for the analysis of the experimental results
member forces were required, so strain readings obtained were used for calculating
the same. Once the member force data from the experimental results was tabulated
then a comparison of theoretical results with experimental results was done in
graphical format. The analysis was done for four different structural members.
(a) Strut
(b) top cable
(c) bottom cable
(d) side cable
The elements for which theoretical and experimental analysis was done are shown in
figur
e 6.3
and
6.4
Side Cable (node 1 and 6)
Figure 6.3 Side cable and strut for which analysis was done
Top Cable (node 6 and 10)
Bottom Cable (node 1 and 9)
Figure 6.4 Top cable and bottom cable for which analysis was done
- Strut
The strut member corresponding to node 9 and 7 was the one used for the
comparative analysis. The graphical variation for the force in this strut for both
theoretical and experimental results is as shown in figure 6.5. Negative values indicate
that the member is in compression
Memeber Force Vs. Load Diagram for Strut
Load Applied
0
0
200
400
600
800
1000
1200
Member Force
-200
-400
Theoretical Results
Experimental Results
-600
-800
-1000
-1200
Figure 6.5 Theoretical and Experimental comparison for a strut
As can be seen from the above graph the experimental results are slightly higher than
the theoretical results but the trend for both the results is same i.e. the force in strut
varies linearly with the loads applied under given range. This similarity is there
maybe because of the use of greater number of strain gauges (four strain gauges were
applied along the circumference of the strut to nullify the effect of bending and to get
more accurate results).
37
- Top Cable
The top cable corresponding to node 6 and 10 was used for the comparative analysis.
The graphical variation of the force in this cable for both theoretical and experimental
results is as shown in figure 6.6
Memeber Force Vs. Load for Top Cable
1400
1200
Member Force
1000
800
600
Theoretical Results
Experimental Results
400
200
0
0
200
400
600
800
1000
1200
Load Applied
Figure 6.6 Theoretical and Experimental comparison for a top cable
As can be seen from the above graph the experimental results are slightly higher than
the theoretical results and the variation trend in quite similar. The theoretical variation
is completely linear whereas the variation obtained from experimental results has
slight non-linearity and steeper slope. These inaccuracies are there maybe because of
the use of lesser number of strain gauges as compared to strut and other experimental
errors. Pre-tensioning effects also accounts for inaccurate results because for the
perfect erection of tensegrity structures the pre-tension in the cables must neither be
too high or too low from theoretically calculated pre-tensions.
38
-Side Cable
The side cable corresponding to node 1 and 6 was used for the comparative analysis.
The graphical variation of the force in this cable for both theoretical and experimental
results is as shown in figure 6.7
Memeber Force Vs. Load for Side Cable
1400
1200
Memeber Force
1000
800
600
Theoretical Results
400
Experimental Results
200
0
0
200
400
600
800
1000
1200
Load Applied
Figure 6.7 Theoretical and Experimental comparison for a side cable
As can be seen from the above graph the experimental results are quite high than the
theoretical results in the beginning, but with the application of load, this difference
decreases. The theoretical variation is completely linear whereas the variation
obtained from experimental results has slight non-linearity and steeper slope. It can
also be observed that the member force variation for this side cable is quite similar to
the force variation for the top cable which was discussed earlier, which practically
should be there because both of these cables have a shared node. And the possible
reasons for inaccuracies in experimental results are same as in the case of top cable.
39
- Bottom Cable
The side cable corresponding to node 1 and 6 was used for the comparative analysis.
The graphical variation of the force in this cable for both theoretical and experimental
results is as shown in figure 6.8
Memeber Force Vs. Load for Bottom Cable
Member Force
1300
Theoretical Results
Experimental Results
1200
1100
1000
0
200
400
600
800
1000
1200
Load Applied
Figure 6.8 Theoretical and Experimental comparison for a bottom cable
As can be seen from the above graph the experimental results are quite different from
the theoretical results. Theoretically, the pre-tension in bottom cables is so high that
on application of load the variation in the member force is negligible, as matter of fact
under given load range there is no variation in the cable force. The force variation in
bottom cables takes place at higher loadings as seen from pre-test Ansys simulation.
But experimentally the pre-tension in this bottom cable was not high enough due to
fabrication errors, as result of which the member force variation obtained from
experimental data diverts a lot from the theoretical one.
40
CONCLUSIONS
(1) The new joints developed are more flexible, easy to fabricate, economic, and
lighter than the joints used previously.
(2) The new nodal joints developed using T-section, are more economic, easy to
fabricate lighter but less flexible than the nodal joints, with the same conceptual
design, developed using circular plates.
(3) Due to the short length of nodal joints, their stability was affected i.e. they were
prone to rotation which made the deployment difficult.
(4) New joints fabricated are also material efficient as compared to previous joints as
very small amount of material is required for their fabrication.
(5) On having a comparative analysis between the experimental and theoretical
results, it was found that the trend of variation is almost similar but there is
difference in the theoretical and the experimental values.
(6) The possible reasons for observed disparities in the theoretical and experimental
results are:
-
inaccuracies of the experimental equipment
-
joints are not completely flexible
-
pre-tension effect in cables
-
fabrication errors
41
RECOMMENDATIONS
(1) Since most of the project work was dedicated to the conceptual design of the grid
structure so it is recommended that more thorough analysis should be done in future
projects.
(2) Due to malfunctioning of the accelerometer used for dynamic testing, an improper
response was recorded as result of which inaccurate results were obtained. Hence it is
recommended to perform again the dynamic testing and analysis of the structure as a
part of future projects (refer to appendix C for more details).
(3) After the fabrication of the structure it was found that the length of nodal joints
was slightly smaller because of which they used to get rotate when the loading was
applied. Thought this feature increased the flexibility factor but yet it also introduced
inaccuracies in the fabrication. So it is recommended to work on the length of these
joints.
(4) Non-destructive static testing of the structure was done and under the given load
range linear response was obtained. Hence it is recommended to perform a destructive
testing also, so that the loading capacity of the structure maybe obtained along with
the mode of failure.
(5) For fine tuning of the structure after the erection has been done, it is recommended
to use turn-buckle on each kind of cable rather than using all four of them on top
cables so that adjustments can be distributed over whole body of the structure and are
not just confined to upper layer.
42
REFERENCES
1. Chu, R., Tensegrity, Journal of Synergetics, 2(l), 1988.7.
2. Fuller, B., Tensile-integrity structures, US Patent, 3, 063, 521, 1962.
3. Hanaor, A., Double-layer tensegrity grids as deployable structures, International
Journal of Space Structures, 8, 1992.
4. Motro, R., et al., Form finding numerical methods for tensegrity systems, IASSASCE International Symposium, Atlanta, GA, April 24–28, 706–713, 1994.
5. Pugh, A., An Introduction to Tensegrity, University of California Press, Berkeley,
1976.
6. Pellegrino, S., Analysis of prestressed mechanisms, International Journal of Solids
and Structures, 26(12), 1329–1350, 1989.
7. Snelson, K., Continuous tension, discontinuous compression structures, US Patent
3, 169, 611, 1965.
8. Williamson, A., and Skelton, R.E., A general class of tensegrity systems:
Equilibrium analysis, Engineering Mechanics for the 21st Century, ASCE
Conference, La Jolla, 1998.
43
APPENDIX -A
Newton Raphson Method
The finite element discretization process yields a set of simultaneous equations:
[K] {u} = {Fa}
(1)
where:
[K] = coefficient matrix
{u} = vector of unknown DOF (degree of freedom) values
{Fa} = vector of applied loads
If the coefficient matrix [K] is itself a function of the unknown DOF values (or their
derivatives) then Equation 1 is a nonlinear equation. The Newton-Raphson method is
an iterative process of solving the nonlinear equations and can be written as
{Kit}{Δui} = {Fa}-{Finr}
(2)
{ui+1}= {ui} + {Δui}
(3)
where:
{Kit}= Jacobian matrix (tangent matrix)
i = subscript representing the current equilibrium iteration
{Finr}= vector of restoring loads corresponding to the element internal loads
Both {Kit} and {Finr} are evaluated based on the values given by {ui}. The right-hand
side of Equation 2 is the residual or out-of-balance load vector; i.e., the amount the
system is out of equilibrium. In a structural analysis, {Kit} is the tangent stiffness
matrix, {ui} is the displacement vector and {Finr} is the restoring force vector
calculated from the element stresses. In a thermal analysis, {Kit} is the conductivity
matrix, {ui} is the temperature vector and {Finr} is the resisting load vector calculated
from the element heat flows. In an electromagnetic analysis, {Kit} is the Dirichlet
44
matrix, {ui} is the magnetic potential vector, and {Finr} is the resisting load vector
calculated from element magnetic fluxes. In a transient analysis, {Kit} is the effective
coefficient matrix and {Finr} is the effective applied load vector which includes the
inertia and damping effects.
More than one Newton-Raphson iteration is needed to obtain a converged solution.
The general algorithm proceeds as follows:
1.Assume {u0}. {u0} is usually the converged solution from the previous time step.
On the first time step, {u0} = {0}.
2.Compute the updated tangent matrix {Kit} and the restoring load {Finr} from
configuration {ui}.
3.Calculate {Δui} from Equation 2
4.Add {Δui} to {ui} in order to obtain the next approximation {ui + 1} (Equation 3).
5.Repeat steps 2 to 4 until convergence is obtained.
The solution obtained at the end of the iteration process would correspond to load
level {Fa}. The final converged solution would be in equilibrium, such that the
restoring load vector {Finr} (computed from the current stress state, heat flows, etc.)
would equal the applied load vector {Fa} (or at least to within some tolerance). None
of the intermediate solutions would be in equilibrium
45
APPENDIX –B
Comparison of Various Joints
A comparative study of various kind of joints used for fabrication of tensegrity
structures till date, has been presented in this appendix. Nodal joints which were
earlier used for the fabrication of these structures had many disadvantages as shown
in table B.1.
Table B.1 Disadvantages of previously used nodal joints
Nodal Joints
Disadvantages
Eye- Bolt Type
Box-Type
Circular Plate Type
1.Rigid
1.Very rigid
1.Breaks Easily
2.Bulky
2.Bulky
So the previously used nodal joints either had very bulky and rigid design or were
prone to structural failure. A comparison of the new joints fabricated and the old ones
is presented in the table B.2.
Table B.2 Comparison of different nodal joints
Similar was the case of previously used cable joints as shown in table B.3. Either they
were very expensive to make or were prone to failure.
Table B.3 Disadvantages of previously used cable joints
Cable Joints
Hydraulic Joint
Disadvantages 1.Slips easily
2.Made in market
Bolt-Type
1.Made in market
2.Very Expensive
3.Expensive
A comparative study of previously used cable joints and newly developed joints has
been shown in the table B.4.
Table B.4 Comparison of different cable joints
Properties
Bolt type
Hydraulic
Pressed
Average load
(kg.)
Manufactured
Cost
70
993.33
Winding wire
with epoxy
(fabricated)
820
in market
Expensive
In market
Very expensive
In the lab
Very cheap
So as inferred from the above table, the newly fabricated cable joints are cheaper and
can easily be fabricated in lab without any compromise in strength as compared to
previously used joints.
APPENDIX –C
Dynamic Testing
For the dynamic loading case it is essential to obtain the resonance modes and modes
corresponding frequencies, for this purpose an accelerometer was used as shown in
figure C.1. The accelerometer was connected to digital multimeter (shown in figure
C.2) to collect the time-domain analysis data for the specified time interval at
excitation given by hammering the structure at a joint as shown in figure C.3.
Figure C.1 Accelerometer
Figure C.2 Digital multimeter
Figure C.3 Hammering the structure
The time domain analysis data obtained was tabulated and its fast Fourier
transformation was done on MATLAB to obtain the frequency distribution. This
frequency distribution is used for obtaining the modes of vibration of the structure but
in this case due to the malfunctioning of the accelerometer used, the response was not
48
captured properly and as a result incorrect frequency distribution was obtained as
shown in figure C.4
Frequency Vs. FFT
Fast Fourier Transform
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
20
40
60
80
100
120
Frequency
Figure C.4 Dynamic Response of the Structure
The above graph is for hundred frequency data obtained by FFT of thousand
frequencies. As can be seen from the graph the response is not clear as no mode shape
is visible from the distribution. Whereas when the theoretical analysis of the structure
was done then it was found that the fundamental frequency of the structure was
37.108 whereas the first overtone occurred at 61.889 and the second overtone at
71.803. But due to the inaccurate experimental results a comparative analysis of the
dynamic data was not possible. Hence it is recommended to perform again the
dynamic testing and analysis of the structure as a part of future projects.
49