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1James Bernasconi, consulting engineer, Brisbane, Australia, Email: [email protected]
BUCKLING STUDIES INTO LARGE SCALE HYPERBOLIC PARABOLOID
SHELL AND LATTICE STRUCTURES
James Bernasconi1
ABSTRACT: This paper covers some aspects of the bifurcation and elastic geometric non-linear buckling behaviour
of hyperbolic paraboloid lattice and concrete shells up to 100 metres in plan size. The paper summaries the important
design parameters discovered by earlier research and used variations in the rise to span ratio and edge beam flexural
stiffness to undertake a parametric study. The bifurcation buckling of lattices was found to be similar to that of shells.
The elastic geometric non-linear analysis was always found to be unstable for shells and stable for rectangular lattices.
However, diagonal lattices behaved in a similar fashion to the unstable shells.
KEY WORDS: hyperbolic paraboloids, shells, bifurcation buckling, elastic geometric non-linear buckling
1. INTRODUCTION
This conference paper presents some of the research
results from a recent PhD thesis [3] into the design and
parametric study of large-scale lattice hyperbolic
paraboloids.
WHAT ARE HYPERBOLIC PARABOLOIDS?
A hyperbolic paraboloid ('hypar' for short and used
hereafter) is the name given to a mathematical
description of a well-known physical shape. The name
gives away the principal elements, hyperbolas and
parabolas. Hypars are anticlastic surfaces and unlike
cups cannot hold a fluid no matter how they are
orientated in space. As well as possessing maximum
and minimum curvatures called principal curvatures,
they also have the property of being surfaces of
translation. The most obvious format is a saddle shape
such as in the velodrome for the London Olympic
Games in 2012 shown below in Figure 1.
Figure 1: Velodrome for the London summer Olympics,
2012 [1]
However sometimes architectural demands require that
only a portion of the basic shape be used. In this case,
the centre portion is a segment of a hyperbolic
paraboloid, used on the aquatic centre for the same
Olympic Games, Figure 2.
Figure 2: Aquatic centre for the London summer
Olympics, 2012 [1]
Not so obvious in the aerial view, Figure 2, but during
construction of the centre portion, Figure 3 clearly
shows the hyperbolic paraboloid shape.
Figure 3: Construction of aquatic centre for London
summer Olympics, 2012 [2]
The same general saddle shape is presented again, but
this time with a hole in the middle and the ends
missing, as used in the Sydney Olympic Games in
2000, see Figure 4.
Figure 4: Artist impression of the stadium for the
Sydney Olympics, 2000
It will be obvious from viewing these buildings, that
apart from the pleasing lines created by the hypar, the
shape has an unique ability. The ability to span large
areas with no intermediate columns and this feature
was recognised as early as the 1930s. However, it was
Felix Candela who popularised the shape during the
1960s in the new working material of the time,
reinforced concrete, to create structural shells. The
increasing use of the form, lead to a rise in research
interest and most research was conducted with concrete
in mind and involved the use of small shell models. In
the beginning, the concrete was proportioned by a few
simple formulae based on membrane theories all of
which had some known inconsistencies in their theory.
After some years, the shape fell out of favour along
with the construction of large concrete shells. In recent
times, the shape has been revived but in the form of
lattices in steel or timber. However, there was little
research into large-scale lattice hypars available, only
the earlier, mainly concrete, work.
In usual construction, shells fabricated generally
conform to the geometric requirements for 'shallow
shells' when there rise is small to their span. Vlasov
[7] considered shells to be shallow if the ratio of the
rise to the shorter side is less than or equal to 0.20. The
research project described here has a range of rise to
span ratios up to 0.33. Shells are considered to be 'thin
shells' when they conform to Novozhilov's [4]
requirement for if the ratio of thickness to radius of
curvature is less than 0.05. Shells in civil construction
always meet this requirement. According to
Novozhilov [4] if these requirements are met, an
engineering accuracy giving errors of about 5% is
achieved.
2. GEOMETRY AND RESEARCH
PROPOSAL
HYPAR GEOMETRY
All of the previous examples generate the shape in the
same basic way - by sliding or translating vertical
parabolas over each other. This is shown in Figure 5,
taken from the out of print book by Schueller [5].
.
Figure 5: Hypar with parabolic shape together with
straight line generators, Schueller [5]
However there are other ways to look at things. The
shape allows us to draw grids upon it and create
something different by rotating the grid by 45 degrees.
It is then possible to create a straight-sided rectangular
or square plan shape of translation using either
parabolas or straight lines. The illustration in Figure 6
is again taken from the book by Schueller and shows
the square shape inscribed with parabolic lines,
hyperbolic lines (obtained by cutting the figure in the
horizontal plane) and straight lines.
Figure 6: Relationships between parabolic, straight and
hyperbolic curves, Schueller [5]
The shapes shown in Figure 6 actually were the design
basis for one of the main initial formats of the hypar.
The straight-line mathematical formulation extended
very nicely into concrete construction. Here the curved
shape of the hypar could be formed in concrete by
using straight-line formwork.
RESEARCH PROPOSAL
The previous hypar shell research had been curtailed
some decades ago and had been based on small
laboratory bench models, with even the most ambitious
up to about 3 metres in plan size. There was no point
in conducting more small-scale research. It is also
impossible to research a project specific shape such as
those depicted in Figures 1 to 4. Therefore, a more
generic hypar shape was chosen, such as those shown
in Figure 6. This new research would be based on
investigating square hypar models formed using
straight-line generators in three sizes, based on the
following plan dimensions:
18 metres
50 metres
100 metres
There had been no research undertaken before into
hypar models (shell or lattice) of this size. Of course,
laboratory models of this size could not be physically
built, so the logical modelling choice was using the
computer finite element method. Many finite element
programs now exist that provide convenient and
reliable results in both linear and non-linear work.
This research was undertaken using the program,
Strand7 [6], developed by Strand7 Pty Ltd, Sydney,
Australia. The research emphasis would be a
parametric investigation into hyperbolic paraboloid
shells and lattices to discover what implications
become apparent for real structures of this size.
RESEARCH SET UP AND MODELLING
The large quantity of concrete hypar research available
from the 1960s would provide a platform of useful
parameters that would form the starting point for this
current research project. The main design parameters
that would be worthwhile to investigate in research
program would be:
A comparison between concrete and lattice
models to determine differences and
similarities between the two groups
Size effects that may come into play as the
plan size increased to 100 metres
Effects arising due to changes in the rise to
span ratio, changing the apparent curvature of
the models
Changes in the orientation of lattice bars
Changes in the flexural stiffness of the edge
beam as this had been shown to be significant
Effects arising from asymmetric loading
Effects arising from induced defects in lattice
construction
Variations in the method of support offered to
the corners and edge beams.
Concrete provides a continuous surface, a continuum.
Lattices however can come in a variety of forms and
not just in one layer, sometimes two and three layer
forms are used. For a research thesis, useful
comparisons could be obtained between concrete and
lattice if the configurations were similar. Therefore,
only single plane lattices were considered. Lattice
joints were considered fully rigid. Concrete and lattice
edge beams were proportioned to give similar ratios of
flexural stiffness. Lattice members were based on cold
formed steel circular hollow sections (chs) in actual
sizes available in the Australian market. Overseas steel
markets would allow a greater choice in some cases.
The concrete shell thickness is chosen to be close to a
real life concrete thickness incorporating the usual
allowances for cover and thus durability requirements
are met.
Models were created in two different arrangements:
Corner supported (restrained in translation
only about the 3 axes)
Simply supported along the entire edge beam
Loads were established firstly on the surface of the
concrete models. Here the base load (15 Pa) was
modelled on the projected area of the model. The load
was set as 'global face pressure' on the surface and thus
was independent of the change in curvature. To create
the lattice models, the total equivalent load was placed
at the nodes and the overall reactions checked between
the concrete and lattice models to ensure compatibility.
In all, the research program required the generation of
over 400 computer models to investigate the various
parameters. Figure 7 to Figure 11 show the different
shell and lattice configurations.
Figure 7: A typical hypar shell (using plate elements)
Figure 8: A typical hypar rectangular lattice grid (using
beam elements)
Figure 9: A typical hypar diagonal lattice grid (using
beam elements)
Figure 10: A typical hypar shell with edge beam detail
Figure 11: A typical hypar lattice with edge beam detail
Strand7 offers the full range of analysis types including
the static solvers: linear elastic, elastic geometric non-
linear and bifurcation analysis also known as linear
buckling. In this paper only results from geometric
non-linear and bifurcation, buckling will be considered,
comparing concrete hypar shells with hypar lattices.
This is a valid technique paralleling the technique
called continuum modelling. Continuum modelling is
most often used in preliminary stages as a structural
analysis time saver, being much easier to construct a
shell made of plate elements rather than building a
lattice arrangement possibly containing tens of
thousands of members. The plate model is then used to
examine the macro effects in the structure.
3. SHELL AND LATTICE
COMPARISONS
This paper presents only some of the results obtained
from bifurcation buckling and elastic geometric non-
linear buckling, comparing the response of hypar shells
to hypar rectangular and diagonal lattices.
BIFURCATION BUCKLING
Bifurcation buckling modes were similar between
shells and lattices in both corner supported versions
and simply supported edge beam versions. The
comparison illustrates that some useful macro effects
can be observed in continuum modelling with plate
models.
Corner supported models
Figures 12 and 13 compare the buckling modes of
similar sized shell and lattice models with a standard
rise to span ratio still within the definition of a shallow
shell (<0.20). Figures 14 and 15 compare bifurcation
buckling modes of models with comparable edge beam
flexural ratios.
Figure 12: 50 m shell model, 0.16 rise to span ratio
Figure 13: 50 m lattice model, 0.16 rise to span ratio
Figure 14: 50 m shell model, 50.56 edge beam flexural
stiffness ratio
Figure 15: 50 m lattice model, 51.80 edge beam
flexural stiffness ratio
Simply supported edge beam models
Figures 16 and 17 compare similar sized shell and
lattice models with a standard rise to span ratio within
the definition of a shallow shell. Figures 17 and 18
compare buckling modes with comparable edge beam
flexural ratios.
Figure 16: 50 m shell model, 0.16 rise to span ratio
Figure 17: 50 m lattice model, 0.16 rise to span ratio
Figure 18: 50 m shell model, 50.56 edge beam flexural
stiffness ratio
Figure 19: 50 m lattice model, 51.80 edge beam
flexural stiffness ratio
It is also possible to examine the bifurcation buckling
carrying capacity of shells and lattices as the rise to
span ratios change (space does not allow for edge beam
charts). It is also possible to compare corner supported
models with simply supported edge beams. Figures 20
and 21 show a similar buckling capacity improvement
for both changes in rise ratio with little effect due to
changes in the plan size of the model.
Figure 20: Corner supported shell bifurcation analysis,
rise to span ratio
Figure 21 for rectangular lattices demonstrates some
buckling capacity improvement and more size variation
than similar shells.
Figure 21: Corner supported lattice bifurcation analysis,
rise to span ratio
Figure 22 for a simply supported lattice shows a large
variation in buckling capacity with a variation in lattice
plan size.
Figure 22: Simply supported lattice bifurcation analysis,
rise to span
For the variation in the edge beam flexural stiffness,
concrete shells recorded normalised buckling capacity
improvements of only 1.00 (18 m) to 1.25 (100 m).
However rectangular lattices recorded improvements of
2.00 (100 m) to 25.00 (18 m), a vastly difference
response to the shells.
ELASTIC GEOMETRIC NON-LINEAR
BUCKLING
Shells
The elastic geometric non-linear behaviour is depicted
by use of load vs. deflection curves. Here the
structures were loaded until instability stopped the
solver progressing. All hypar shells exhibited
instability as the load increased. This instability was
independent of the changes in support from corner
supported to edge supported. The instability was also
independent of changes in rise to span ratio (effectively
curvature) and independent of changes in edge beam
0.00
2.00
4.00
6.00
8.00
10.00
12.00
0.05 0.08 0.12 0.16 0.20 0.33
no
rmal
ise
d lo
ad in
ten
sity
rise to span ratio
shell bifurcation analysis - normalised load intensity vs rise to span ratio
18 m shell
50 m shell
100 m shell
0.0
2.0
4.0
6.0
8.0
10.0
12.0
0.05 0.08 0.12 0.16 0.20 0.33 no
rmal
ise
d lo
ad in
ten
sity
rise to span ratio
rectangular lattice bifurcation analysis - normalised load intensity vs rise to span ratio
18 m lattice
50 m lattice
100 m lattice
0.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
0.05 0.08 0.12 0.16 0.20 0.33
no
rmal
ise
d lo
ad in
ten
sity
rise to span ratio
simply supported perimeter rectangular lattice bifurcation analysis - normalised load intensity vs
rise to span ratio
18 m lattice
50 m lattice
100 m lattice
flexural stiffness, see Figures 23 and 25. Figures 24
and 26 show the computer plots of the unstable shell
behaviour.
Figure 23: Corner support shell elastic geometric
analysis with variation in rise ratio
Figure 24: 50 m shell, 0.16 rise ratio, 1st mode
buckling
Figure 25: Corner support shell elastic geometric
analysis with variation in edge beam
Figure 26: 50 m shell, 50.65 edge beam ratio, 1st
mode buckling
In the case of simply supported shells, low rise to span
ratio models (0.05) exhibited stability that was
independent of shell size, see Figure 27. However
once the rise ratio increased (0.16 for example),
buckling again occurred, Figure 27 and the computer
plot, Figure 28. Variations in edge beam flexural
stiffness always allowed instability to develop as
shown in Figures 29 and the buckled shell shown in
Figure 30.
Figure 27: Simply supported shell elastic geometric
analysis with variation in rise ratio
deflection/span (/L) (x104)
loa
d i
nte
ns
ity
(k
N/m
2)
50 m and 100 m corner supported shell geometric non-linear responseload intensity vs deflection/span for various rise to span ratios
(all curves truncated at 20 000 (kN/m2)
-350 -300 -250 -200 -150 -100 -50 0 500
5000
10000
15000
20000
50m rise to span 0.0550m rise to span 0.0850m rise to span 0.1250m rise to span 0.1650m rise to span 0.2050m rise to span 0.33100m rise to span 0.05100m rise to span 0.08100m rise to span 0.12100m rise to span 0.16100m rise to span 0.20100m rise to span 0.33
deflection/span (/L) (x104)
load
in
ten
sit
y (
kN
/m2)
50 m and 100 m corner supported shell geometric non-linear responseload intensity vs deflection/span for various edge beam ratios
-35 -30 -25 -20 -15 -10 -5 0 5 10 15 200
5000
10000
15000
20000
25000
30000
35000
40000
50m eb stiffness 1.050m eb stiffness 10.150m eb stiffness 51.750m eb stiffness 101.7100m eb stiffness 1.0100m eb stiffness 10.1100m eb stiffness 51.7100m eb stiffness 101.7
deflection/span (/L) (x104)
load
in
ten
sit
y (
kN
/m2)
50 m and 100 m simply supported shell geometric non-linear responseload intensity vs deflection/span for various rise to span ratios
-500 -450 -400 -350 -300 -250 -200 -150 -100 -50 0 500
50000
100000
150000
200000
250000
300000
350000
400000
450000
500000
50m rise to span 0.0550m rise to span 0.0850m rise to span 0.1250m rise to span 0.1650m rise to span 0.2050m rise to span 0.33100m rise to span 0.05100m rise to span 0.08100m rise to span 0.12100m rise to span 0.16100m rise to span 0.20100m rise to span 0.33
Figure 28: 50 m simply supported shell, 0.16 rise ratio,
1st mode buckling
Figure 29: Simply supported shell elastic geometric
analysis with variation in edge beam
Figure 30: 100 m shell, 50.65 edge beam ratio, 1st
mode buckling
Rectangular lattices
In contrast to shells, rectangular lattices exhibited
complete stability as the loads increased. The curves
were very predictable. Changes in rise to span ratio
and edge beam flexural stiffness for corner supported
lattices did not induce any instability, see Figures 31
and 32. Figures 33 and 34 show the simply supported
lattice behaving in a stable fashion with the changes in
the rise ratio and the edge beam stiffness as well.
Figures 35 and 36 show typical computer models for
corner supported lattices, but these responses were also
typical for simply supported as well, as the load
increased the lattice remained stable.
Figure 31: Corner support lattice elastic geometric
analysis with variation in rise ratio
Figure 32: Corner support lattice elastic geometric
analysis with variation in edge beam
Figure 33: Simply supported lattice elastic geometric
analysis with variation in rise ratio
deflection/span (/L) (x104)
load
in
ten
sit
y (
kN
/m2)
50 m and 100 m simply supported shell geometric non-linear responseload intensity vs deflection/span for various edge beam ratios
-400 -350 -300 -250 -200 -150 -100 -50 0 500
50000
100000
150000
200000
250000
300000
350000
400000
450000
500000
50m eb stiffness 1.050m eb stiffness 10.150m eb stiffness 51.750m eb stiffness 101.7100m eb stiffness 1.0100m eb stiffness 10.1100m eb stiffness 51.7100m eb stiffness 101.7
deflection/span (/L) (x102)
load
in
ten
sit
y (
kN
/m2)
18, 50, 100 m corner supported lattice geometric non-linear responseload intensity vs deflection/span for various rise to span ratios
-16 -14 -12 -10 -8 -6 -4 -2 00
1
2
3
4
5
6
18m rise to span 0.0518m rise to span 0.0818m rise to span 0.1218m rise to span 0.1618m rise to span 0.2018m rise to span 0.3350m rise to span 0.0550m rise to span 0.0850m rise to span 0.1250m rise to span 0.16
50m rise to span 0.2050m rise to span 0.33100m rise to span 0.05100m rise to span 0.08100m rise to span 0.12100m rise to span 0.16100m rise to span 0.20100 m rise to span 0.33
deflection/span (/L) (x102)
loa
d i
nte
ns
ity
(k
N/m
2)
18, 50, 100 m corner supported lattice geometric non-linear responseload intensity vs deflection/span for various edge beam ratios
-12 -10 -8 -6 -4 -2 00
0.5
1
1.5
2
2.5
3
3.5
18m eb stiffness 1.018m eb stiffness 10.118m eb stiffness 35.718m eb stiffness 101.750m eb stiffness 1.050m eb stiffness 9.950m eb stiffness 51.850m eb stiffness 90.6100m eb stiffness 1.0100m eb stiffness 2.0
100m eb stiffness 3.9100m eb stiffness 6.8
deflection/span (/L) (x102)
load
in
ten
sit
y (
kN
/m2)
18, 50, 100 m simply supported lattice geometric non-linear responseload intensity vs deflection/span for various rise to span ratios
-45 -40 -35 -30 -25 -20 -15 -10 -5 00
2500
5000
7500
10000
12500
15000
17500
18m rise to span 0.0518m rise to span 0.0818m rise to span 0.1218m rise to span 0.1618m rise to span 0.2018m rise to span 0.3350m rise to span 0.0550m rise to span 0.0850m rise to span 0.1250m rise to span 0.16
50m rise to span 0.2050m rise to span 0.33100m rise to span 0.05100m rise to span 0.08100m rise to span 0.12100m rise to span 0.16100m rise to span 0.20100m rise to span 0.33
Figure 34: Simply supported lattice elastic geometric
analysis with variation in edge beam
Figure 35: 50 m lattice typical deflection for variation of
rise ratio and edge beam
Figure 36: 100 m lattice typical deflection for 0.33 rise
ratio
Diagonal lattices
However if the bar arrangement was changed from
rectangular to diagonal, the load deflection behaviour
of the lattices changed. It was found that the corner
supported lattices buckled to an increasing extent as the
rise to span ratio increased, Figure 37. This effect also
occurred in the case of increasing edge beam flexural
stiffness, Figure 39. Figures 38 and 40 show typical
buckling effects in the lattice.
For the cases of simply supported edges, the result was
slightly different. The lower the rise to span ratio
(lower effective curvature), the more stable and
predictable the result. As the rise ratio increased to the
shallow shell limit, 0.20 (and just beyond 0.33),
instability became a factor, see Figure 41. Figure 42
shows the buckled lattice as this very high degree of
curvature. In the models possessing increasing edge
beam flexural stiffness, the extra beam stiffness
reintroduces stability to the lattice as Figure 43
displays. Figure 44 shows the steady deflection in the
centre of the lattice, more load provides more
deflection.
The change in boundary condition support to simply
supported has a beneficial effect in reducing or
eliminating buckling under both increasing rise to span
ratio (within reasonable limits). Increasing the edge
beam flexural stiffness has a beneficial effect in
diagonal lattice arrangements where the bar orientation
parallels the principal axes of the hypar shell.
Figure 37: Corner diagonal lattice elastic geometric
analysis with variation in rise ratio
Figure 38: Corner supported 18 m diagonal lattice, 0.16
rise ratio, 1st buckling mode
deflection/span (/L) (x102)
loa
d i
nte
ns
ity
(k
N/m
2)
18, 50, 100 m simply supported lattice geometric non-linear responseload intensity vs deflection/span for various edge beam ratios
-45 -40 -35 -30 -25 -20 -15 -10 -5 00
2500
5000
7500
10000
12500
15000
17500
18m eb stiffness 1.018m eb stiffness 10.118m eb stiffness 35.718m eb stiffness 101.750m eb stiffness 1.050m eb stiffness 9.950m eb stiffness 51.850m eb stiffness 90.6100m eb stiffness 1.0100m eb stiffness 2.0
100m eb stiffness 3.9100m eb stiffness 6.8
deflection/span (/L) (x104)
load
in
ten
sit
y (
kN
/m2)
18 m corner supported diagonal lattice geometric non-linear responseload intensity vs deflection/span for various rise to span ratios
-350 -300 -250 -200 -150 -100 -50 0 500
2
4
6
8 rise to span 0.05rise to span 0.08rise to span 0.12rise to span 0.16rise to span 0.20rise to span 0.33
Figure 39: Corner diagonal lattice elastic geometric
analysis with variation in edge beam
Figure 40: Corner supported 18 m diagonal lattice,
101.67 edge beam ratio, snap buckling
Figure 41: Simply supported diagonal lattice elastic
geometric analysis with variation in rise ratio
Figure 42: Simply supported 18 m diagonal lattice, 0.33
rise ratio, 1st buckling mode
Figure 43: Simply supported diagonal lattice elastic
geometric analysis with variation in edge beam
Figure 44: Simply supported 18 m diagonal lattice, 50.7
edge beam ratio, steady deflection
4. CONCLUSIONS
The bifurcation buckling modes are similar between
shells and lattices for both corner supported and simply
supported arrangements. The bifurcation buckling
capacity of shells and corner supported rectangular
deflection/span (/L) (x104)
load
in
ten
sit
y (
kN
/m2)
18 m corner supported diagonal lattice geometric non-linear responseload intensity vs deflection/span for various edge beam ratios
-45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 300
2
4
6
8
10
12
14
eb stiffness 1.0eb stiffness 10.1eb stiffness 35.7eb stiffness 101.7
deflection/span (/L) (x104)
load
inte
nsi
ty (
kN/m
2 )
18 m simply supported diagonal lattice geometric non-linear responseload intensity vs deflection/span for various rise to span ratios
-1000 -900 -800 -700 -600 -500 -400 -300 -200 -100 00
50
100
150
200
250
300
350
rise to span 0.05rise to span 0.08rise to span 0.12rise to span 0.16rise to span 0.20rise to span 0.33
deflection/span (/L) (x104)
load
in
ten
sit
y (
kN
/m2)
18 m simply supported diagonal lattice geometric non-linear responseload intensity vs deflection/span for various edge beam ratios
-900 -800 -700 -600 -500 -400 -300 -200 -100 00
50
100
150
200
250
300
350
eb stiffness 1.0eb stiffness 10.1eb stiffness 35.7eb stiffness 101.7
lattices was similar. Simply supported lattices
demonstrated more variation in the bifurcation
buckling capacity result across the model sizes.
Shells buckle under elastic geometric non-linear
analysis independently of shell size, support conditions
and changes in rise ratio and edge beam flexural
analysis. In contrast to hypar shells, lattice models
exhibited stability as the load increased. The stability
of hypar lattices was independent of the parametric
changes in rise to span ratio and edge beam flexural
stiffness.
Corner supported diagonal hypar lattices behaved in a
similar unstable fashion to hypar shells. However, the
introduction of complete edge beam support returned
the lattice to a more stable behaviour.
ACKNOWLEDGEMENTS
This research was conducted in conjunction with the
School of Civil Engineering, University of Queensland,
Brisbane, Australia as a thesis submitted for the degree
of Doctor of Philosophy in 2012. The principal advisor
was Associate Professor Faris Albermani.
REFERENCES
[1] London Olympics photographs,
http://london2012.com
[2] London aquatic centre roof construction
photograph, http://zahahadid.com, photograph by
Helene Binet
[3] Bernasconi, JG 2012, A design and parametric
study in large scale hyperbolic parabolic shell and
lattice structures, PhD thesis, University of
Queensland
[4] Novozhilov,VV 1964, Thin Shell Theory, 2nd
Edition, Groningen, P.Noordhoff.
[5] Schueller, W 1983, Horizontal-Span Building
Structures, Wiley-Interscience Publication
[6] Strand7, 2005, Using Strand7, 2nd Edition,
Strand7 Pty Ltd, Sydney
[7] Vlasov, VZ 1964, General Theory of Shells and its
application in engineering, Vol TTF 99,NASA