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International Journal of Mechanical Civil and Control Engineering
Vol. 1, Issue. 2, April, 2015
ISSN (Online): 2394-8868
Experimental Study on Shallow Funicular Five Layered
GFRP Shells over Square Ground Plan
P. Sachithanantham1, Khalid Bashir2, Rayees Ahmad Bala3, Arif Ahad4, Riya Gungnia5
1
Assistant Professor, Department of Civil Engineering, Bharath University, Chennai, Tamilnadu, India.
2
Student, Department of Civil Engineering, Bharath University, Chennai, Tamilnadu, India. .
3
Student, Department of Civil Engineering, Bharath University, Chennai, Tamilnadu, India.
4
Student, Department of Civil Engineering, Bharath University, Chennai, Tamilnadu, India.
5
Student, Department of Civil Engineering, Bharath University, Chennai, Tamilnadu, India.
Abstract— Shells, stressed skin structures because of their
geometry and small flexural rigidity of the skin, tend to carry loads
primarily by direct stresses acting in their plane. Composite
shallow funicular shells of square ground plan, double curvature
with different rises are loaded to failure with a concentrated
central force. A form consisting of square steel frame and poly
urethane membrane are used for casting the pre moulds. From the
pre mould, moulds are prepared. Using these moulds the
specimens are cast with various rises. Specimens of size 50 cm x 50
cm in plan are prepared with composite materials. The specimens
are prepared having same size with various rises. They are
subjected to ultimate loads and the corresponding deflections are
measured. Failure patterns for shells with different rises are
observed. From the experimental investigations a relation between
span to rise ratio and ultimate load is arrived. It is concluded that
the ultimate loads are function of the rise of the shell.
Keywords—Shells, Composite shell,
Ultimate load, Span to rise ratio.
Funicular
macroscopic scale. A composite is when two or more different
materials are combined together to create a superior and unique
material. For example, concrete is made up of cement, fine
aggregate, coarse aggregate and water. If the composition occurs
on a microscopic scale (molecular level), the new material is
then called an alloy for metals or a polymer for plastics.
Generally, a composite material is composed of reinforcement
(fibres, particles, flakes, and/or fillers) embedded in a matrix
(polymers, metals, or ceramics). The matrix holds the
reinforcement to form the desired shape while the reinforcement
improves the overall mechanical properties of the matrix. When
designed properly, the new combined material exhibits better
strength than would each individual material.
shell,
Composites have been used extensively in industries
such as marine and transportation for more than fifty years. Yet
in some industries composites are just now becoming a primary
material of choice. The use of composites in the building
industry is growing rapidly. Traditional benefits offered by
composites are being recognized and utilized to address design
limitations and can be used to reduce life cycle environmental
and cost impacts. Although the use of structural fibre
composites in critical load-bearing applications is relatively rare
one of its most common uses in the construction industry is
repair of existing structures.
I. INTRODUCTION
Shells belong to the class of stressed skin structures
which, because of their geometry and small flexural rigidity of
the skin, tend to carry loads primarily by direct stresses acting in
their plane. In the design of new forms of composite shell
structures the conventional practice is to select the geometry of
shell first and then making the stress analysis. In this process no
deliberated effort is taken to ensure the desirable state of stress
in the material. Perhaps it is more logical to reverse this process.
The material is also used as a replacement for steel in
reinforced and stressed concrete and in very rare cases to
produce new civil structures almost entirely out of fibre
composites. Some traditional rehabilitation and retrofit methods
use concrete or external steel sheets to re-introduce or improve
structural properties such as strength and ductility. The ability
of concrete to form complex shapes and its suitability to
submerged installation has seen it used for encapsulation of
elements such as bridge piers. Steel can be bonded or bolted to
deteriorated concrete structures to provide strength and stiffness
improvements with relatively little additional weight. In the last
decade the number of instances of fibre composites used as a
surface layer that either protects and improves on the response
of the encapsulated element has been increasing. In these cases
In most of the shell roof is the predominant load is the
dead weight. Hence it is advantageous to select the shape of
shell in such a way that, under this condition of loading, the
shell is subjected to pure compression without bending. This
can be achieved by shaping the shell in the form of a catenary
which the funicular shape is corresponding to the dead weight.
Shell of rectangular and square ground plans are very frequent
occurrence in practice. An attempt is made to study the
influence of rise on the ultimate load of the Shallow Funicular
Glass Fibre Reinforced Polymer (GFRP) Shells of Square
Ground Plan.
A typical composite material is a system of materials
composing of two or more materials (mixed and bonded) on a
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International Journal of Mechanical Civil and Control Engineering
Vol. 1, Issue. 2, April, 2015
ISSN (Online): 2394-8868
the materials are usually bonded externally to the structure in
the form of tows (fibre bundles), fabrics, plates, strips and
jackets. The advantages offered by composites in these forms
include their ability to bond well too many substrate materials
and to follow complex shapes. Composites also offer a potential
benefit over isotropic retrofit materials, such as steel, by
allowing enhancement of strength without increasing stiffness
and vice versa.
John W Weber et al., observed that the mathematical
investigations of shallow funicular shells with large
concentrated loads should be based on large deflection theory
and the deflection characteristics of a shell vary closely with its
rise parameter. Patricia M Belles et al., conclude that the
analysis of the stresses and deformations of concrete shell with
the anti funicular shape found with the homeostatic model
technique (HMT) allows the verification of quasi membrane
behaviour. Vafai and Farshad studied that the experimental
failure loads are found to be directly related to the amount of
reinforcement and the age of concrete shells. Sachithanantham
concluded that the deflection of shallow funicular concrete
shells decrease with increase in rise within elastic range and
also concluded that the ultimate load carrying capacity increases
with increase in rise.
Figure 1 Casting of shallow funicular pre mould
C. Casting of Shallow Funicular Moulds
Concrete funicular moulds of size 50cm x 50cm in plan
are prepared by applying GFRP materials on the pre-mould and
cement concrete with adequate reinforcement. Resin, Gel Coat,
Pigment, Catalyst and Accelerator are mixed thoroughly and the
obtained mixture is placed on the Pre Moulds from the central
axis so that it gets spreaded in the uniform manner. It is then left
for overnight to get a better surface finish. The first layer of
normal Glass Fibre is placed on the Pre Mould and the mixture
of above obtained proportion is smudged on the Glass Fibre
with the help of brushes. Consecutive Five layers of Woven
Glass Fibre are placed and the mixture is laid with the help of
brushes. The layers are uniformly compacted with the help of
rollers.
II. METHODOLOGY
A. Materials
Composite funicular shell specimens of various rises are
prepared with Glass Fibre, Resin, Gel Coat, Pigment, Catalyst
and Accelerator. To investigate the influence of different rises
on the ultimate strength of shallow funicular shells, specimens
are prepared and designated as follows.
A cement concrete with high workability is poured
over the Fibre Layers. Adequate reinforcement is provided for
the moulds. By repeating this process, shell moulds of various
rises are prepared as shown in Fig 2.
i) SFS I – Shallow Funicular Shell with rise (R1) – 3.7 cm
ii) SFS II – Shallow Funicular Shell with rise (R2) – 6.8 cm
iii) SFS III – Shallow Funicular Shell with rise (R3) – 9.8 cm
B. Casting of Shallow Funicular Pre moulds
Concrete funicular pre moulds of size 50cm x 50cm in
ground plan are prepared using cement concrete. A steel frame
along with polyurethane membrane is used for casting the pre
moulds. The edges of poly urethane membrane is stretched
between the boundaries of the steel frame of square ground plan
and clamped. Concrete with very high workability is poured
over the membrane due to which the membrane sags
downwards and forms a funicular shape. The downward sag is
controlled by adjusting the membrane with the clamps provided
at the edges. After the concrete has set the pre moulds are
removed from the frame.
Figure 2 Moulds of Shallow Funicular Shells
D. Casting of Shallow Funicular Shells
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International Journal of Mechanical Civil and Control Engineering
Vol. 1, Issue. 2, April, 2015
ISSN (Online): 2394-8868
Composite funicular shells of size 50cm x 50cm in plan are
prepared by placing five layers of Glass Fibre over the mould.
Resin, Gel Coat, Pigment, Catalyst and Accelerator are
thoroughly mixed and the obtained mixture is poured on the
edges of Mould and spreaded uniformly with the help of
brushes. The first layer of normal Glass Fibre is placed on the
Mould and the mixture of above obtained proportion is smudged
on the Glass Fibre with the help of brushes. Three layers of
Woven Glass Fibre are placed alternatively after the mixture is
being poured on each layer and then laid with the help of
brushes. The layers are uniformly compacted with the help of
roller. Finally one more layer of normal Glass Fibre is placed
over three layers of Woven Glass fibre. Square Shell specimens
are prepared with shell moulds with various rises of 3.7 cm (R1),
6.8 cm (R2) and 9.8 cm (R3) as shown from Fig 3 and Fig 4.
Care is taken to maintain the uniform thickness of funicular
shell as 4 mm with the help of measuring gauge.
Figure 4 Rolling over Glass fibre
Figure 5 Specimen of Rise R2
III. EXPERIMENTAL SETUP AND TESTING
The self-straining load frame and the Hydraulic
loading jack along with Load cell are arranged in such a way to
apply the concentrated force over the centre of the shell
specimen as shown in Fig. 12(a) and 12(b). Care is taken to
avoid eccentricity during loading. Linear Variable Differential
Transformer (LVDT) is mounted where the deflections are
required in the specimen. To facilitate the locations of LVDT
the specimens are specially painted and the surface of the shell
is discretized. Specimens are marked. The rise of the shell
specimens cast is measured using Total Station and it is
observed that the rises are almost equal to the predetermined
values. Shells of SFS I, SFS II and SFS III are placed on loading
frame and subjected to central concentrated force and the
corresponding deflections are measured within the elastic range
using data logger.
Figure 3 Placing of Glass fibre on the mould
Figure 6 (a) Test Programme
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International Journal of Mechanical Civil and Control Engineering
Vol. 1, Issue. 2, April, 2015
ISSN (Online): 2394-8868
Figure 6 (d) Discritized Shell Specimen
After the elastic range, all the specimens are subjected to failure
and hence the ultimate loads are recorded. Visible crakes first
appeared at the centre of the shell‘s outer surface and then
propagated towards the corners along the diagonals. As the load
is increased apparent zones of tension near and approximately
parallel to the supports are also cracked by which the shell
eventually failed. The crack patterns of shell specimens are
shown in figure 7.
Figure 6 (b) Experimental Setup
Figure 6 (c) Experimental Setup
Figure 7 Crack patterns of shell specimens
IV. RESULTS AND DISCUSSIONS
From the experimental investigations of SFS I, SFS II and SFS
III a plot is made between the load and the corresponding
deflection as shown in figure 8, 9, 10 and 11 for rise R1, R2 and
R3 respectively.
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International Journal of Mechanical Civil and Control Engineering
Vol. 1, Issue. 2, April, 2015
ISSN (Online): 2394-8868
Load vs Deflection
DEFLECTION
R1
1
0.5
DEFLECTION
R2
0
DEFLECTION
R3
0
0.2
0.4
0.6
LOAD (kN)
LOAD (kN)
1.5
DEFLECTION (mm)
Figure 8 Load vs Deflection, W1
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
DEFLECTION
R1
DEFLECTION
R2
DEFLECTION
R3
0
2
4
6
DEFLECTION (mm)
Figure 11 Load vs deflection, W
Load vs Deflection
From the figures 12 and 13 it is observed that the
deflection of shallow funicular composite shell decreases with
increase in rise. The ultimate loads for the specimens are
tabulated in table I.
LOAD (kN)
1.5
1
DEFLECTION
R1
0.5
DEFLECTION
R2
0
0
0.2
0.4
0.6
TABLE I Test results of ultimate load (Pu) for
shells
DEFLECTION
R3
Type
Rise (h)
DEFLECTION (mm)
Span/Rise
ratio (λ)
Ultimate
Load,
(Pu)
(kN)
13.51
7.35
5.10
2.92
3.73
3.91
(cm)
Figure 9 Load vs Deflection, W2
SFS I
SFS II
SFS III
A plot is made between ultimate load and the rise of
the shell as shown in Figure 12. It is observed that the ultimate
load increases with increase in rise.
Load vs Deflection
LOAD (kN)
1.5
DEFLECTION
R1
1
0.5
DEFLECTION
R2
0
0
0.2
0.4
0.6
DEFLECTION (mm)
3.7
6.8
9.8
DEFLECTION
R3
Figure 10 Load vs deflection, W3
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International Journal of Mechanical Civil and Control Engineering
Vol. 1, Issue. 2, April, 2015
ISSN (Online): 2394-8868
is observed in SFS III when compared with SFS II and SFS I
respectively.
Ultimate Load vs Rise
Ultimate Load (kN)
5
4
REFERENCES
3
2
[1] IS: 2210 - 1988, ―Criteria for Design of reinforced
Concrete shell Structures and folded plates‖, Bureau of Indian
Standards, 1989.
Rise cm
1
[2] Ramasamy G. S, ―Design and Construction of Concrete
Shell Roofs‖, CBS publishers, 1986.
0
0
5
10
15
[3] John W Weber et al, ―Ultimate Loads for Shallow
Funicular Concrete Shells‖, Northwest, Vol. 58, No. 3, 1984, pp
187 – 194.
Rise (cm)
[4] Patricia M Belles et al, ‗ Virtual Simulation of Shape
Generation of Homeostatic Shell Models‗, Association
Argentina, Mechanica Computational, Vol 25, Nov, 2006, pp
549 – 560
Figure 12 Ultimate load vs Rise
Ultimate load vs Span /rise
ratio
[5] Vafai. A and M. Farshad, (1979) "Theoretical and
Experimental Study of Prefabricated Funicular Shell Units",
Building and Environment, Vol. 14, No. 3, pp. 209-216.
Span/Rise ratio,
(λ)
[6] Karbhari, V. M., Zhao, L., 2000, ―The Use of Composites
for Twenty-First Century Civil Infrastructure‖, Fibre
Composites: Who Needs Them? A Workshop for Civil and
Structural Engineers, University of Southern Queensland,
Australia.
Poly.
(Span/Rise
ratio, (λ) )
[7] Gowripalan N., 1999, ―Fibre reinforced Polymer (FRP)
Applications for Prestressed Concrete Bridges‖, Proceedings of
ACUN 1 Conference Composites: Innovations and Structural
Applications, UNSW, Australia
Ultimate Load (kN)
5
4
3
2
1
0
0
5
10
15
[8] Gowripolan, N., 2000, ―Design Considerations for
Prestressed Concrete Beams with Fibre Reinforced Polymer
(FRP) Tendons‖, Fibre Composites: Who Needs Them?, A
Workshop for Civil and Structural Engineers, University of
Southern Queensland, Australia.
Span/Rise Ratio
Figure 13 Ultimate load vs Span /rise ratio (λ)
[9] Sachithanantham.P, Elavenil.S and Sankaran.S, ( 2011) ―
Study on Shallow Funicular Concrete Shells over Square
Ground Plan Subjected to Ultimate loads‖, International Journal
of Earth
From Figure 13 it is observed that the ultimate load
(Pu) increases with the decrease in span to rise ratio (λ). From
the figure 13 the relationship between Pu and λ can be
approximated by the equation (1) where λ value lies (5 < λ <
15).
Pu = -0.0061 - 0.0038 λ + 4.0885 …….. (1)
[10] IS: 2386 (Part I – IV) - 1963, ―Methods of Test for
Aggregates for Concrete‖, Bureau of Indian Standard, 1963.
[11] IS: 10262-1982, ―Recommended guidelines for concrete
mix design‖, Indian Standards Institution, 1982.
V. CONCLUSIONS
[12] Farshad.M, and G. Ahmadi,( 1979) "Influence of Loading
Behavior on the Stability of Cylindrical Shells", J. of Sound and
Vibration, Vol. 62, No. 4, pp. 533-540.
From the experimental investigations it is concluded
that the deflection of shallow funicular composite shell
decreases with the increase in rise within the elastic range. The
ultimate load carrying capacity increases with the increase in
rise. It is also concluded that an increment of 27.73% of
ultimate load (Pu) is observed in SFS I when compared with
SFS II., an increment of 4.82% and 33.9% of ultimate load (Pu)
[13] Kai-uwe bletzinger, Linhard J, Wüchner R, Bletzinger
KU,( 2009)―Isogeometric shell analysis with Kirchhoff-Love
elements‖. Comp. Meth. Appl. Mech. Engng.198: pp 39023914.
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International Journal of Mechanical Civil and Control Engineering
Vol. 1, Issue. 2, April, 2015
ISSN (Online): 2394-8868
[14] IS: 2210 - 1988, ―Criteria for Design of reinforced
Concrete shell Structures and folded plates‖, Bureau of Indian
Standards, 1989.
[15] IS: 2204 - 1962, ―code of practice for construction of
reinforced concrete shell roof‖, Bureau of Indian Standards,
1962.
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International Journal of Mechanical Civil and Control Engineering
Vol. 1, Issue. 2, April, 2015
ISSN (Online): 2394-8868
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