buckling studies into large scale hyperbolic … 093 buckling studi… · buckling studies into...

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


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


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


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


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


Figure 32: Corner support lattice elastic geometric


Figure 33: Simply supported lattice elastic geometric



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


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


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


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


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


Figure 38: Corner supported 18 m diagonal lattice, 0.16

rise ratio, 1st buckling mode


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


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


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


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


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


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

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