On the Collapse of Ankara Sport Center in 1958
†1Nahit KUMBASAR1
ABSTRACT
In 1958, Ankara Sports Center, the first long span cylindrical shell structure in Turkey, collapsed two years after its scaffoldings were removed. The underlying reasons for this collapse, in spite of the juridical experts’ report and designer ITU professor’s statement of defense, could not be elucidated. In this study, this collapse is investigated by making use of Bayülke’s paper published in TMH, Turkish Engineering News [1]. Additional concrete load to level out the manufacturing faults, imperfections due to these faults, the reported presence of strong winds during the collapse, unforeseen buckling due to creep, which were stated as probable causes by experts and the designer, are considered in this analysis. An equivalent bar system for the shell, proposed by Hrennikoff [5] is used to overcome difficulties of shell buckling and nonlinearity of the problem.
Keywords: Buckling of shells, creep, wind loads, nonlinear analysis.
1. INTRODUCTION
The first long span shell building constructed in Turkey collapsed on June 2nd of 1958, graduation year of this author. It was extremely fortunate for our country and the connected responsible parties that the incidence occurred without loss of life since the hall was closed during the collapse. No satisfactory technical explanation was provided then and later, and the report of experts and the plea of Ord. Prof. İhsan İnan who carried out the design calculations, remained inconclusive. No scientific or technical investigation was conducted in later years, since the computational design of reinforced concrete shell was not available.
The paper published by Bayülke [1] in Turkish Engineering News (Türkiye Mühendislik Haberleri) is the only publication containing any information on this subject. Bayülke investigated various sources and accessed facts unknown until today. While Bayülke’s paper is available online, parts used in this paper will be presented below.
This paper which aims to investigate to what extent the factors cited as the cause of collapse ie creep, buckling etc, are realistic, is based on a paper by Bayülke on a publication of the architect of the building, Prof. Mesut Evren, and his personal communications with one of the court experts, Dr. Ing. Ali Terzibaşoğlu who prepared the aforementioned report.
This information is sufficient for an approximate evaluation, while not as clear as information obtainable from a reinforced concrete design.
1 Istanbul Technical University, Istanbul, Turkey - [email protected]
† Published in Teknik Dergi Vol. 26, No. 2 April 2015, pp: 7099-7114
On the Collapse of Ankara Sport Center in 1958
2. PRELIMINARY INFORMATION
The evaluations reached in expert’s report are retrieved below as they are given in Bayülke [1].
There is no movement in foundations.
The concrete is poured with low viscosity and with no vibration.
Pores and segregation are observed in concrete.
Concrete and steel are in the strength required in the design.
Reinforcements are not in the exact required positions.
The building is constructed according to the project plans.
There is no excessive load, except the ventilation motors, which were included in the structure afterwards, that were not considered in design.
Project details are not given at the level of detail required by the importance of the building.
Installation of reinforcement in one layer is found dubious.
It is surmised that some of assumptions made during calculations would eliminate each others’ shortcomings and due to the high redundancy, a mistake arising from a calculation error or some deficient assumption would not cause a collapse.
It is stated in the report that reinforcement is installed in one layer and close to bottom face instead of midsection. The cause of collapse is stated as ‘insufficiency of stability safety.’
Shell load on the instant of collapse is found to be 330 kg/m2 whereas the load assumed for design is 275 kg/m2,while the collapse load for the modulus of elasticity used in design, is claimed to be 1000 kg/m2. Nevertheless, it is stated that, experiments made in European countries yield 1/3 reduction in the modulus of elasticity, due to rheological properties of concrete.
The same reference indicates the shell thickness as 10cm in general, 15cm in supports neighborhood and reinforced concrete materials as B160 St-I. The 1/3 reduction in the modulus of elasticity, mentioned in the report, is related to increase of deformation due to creep of concrete and a similar proposition is valid also in TS500, the Turkish Standard
“Requirements for Design and Construction of Reinforced Concrete Structures”.
The scheme of structural system given in Bayülke [1] based on a paper of Prof. Mesut Evren differs from the classical barrel shell roof supported by spandrel walls. In this building cylindrical shell is supported not only by spandrel walls but also by a quite rigid beam sitting on columns connected to the spectators seating system, positioned on linear sides. Therefore, while the classical barrel behaves approximately as a beam with a section of circular segment, here one expects rather a vault behavior. However, static solutions show that, although vertical displacements in this area were not significant, due to lateral displacements near these supports, comportment of the shell diverted from beam behavior and considerable tensile and compressive membrane forces emerged.
3. ON THE EXPERTS’ REPORT
It is certain that the best possible experts committee of the time was selected. Their observations and evaluations are scientific and engineer-worthy. However, disparate evaluations may arise in some points. The committee finding the installation of reinforcement in one layer as dubious, must have overlooked that the installation of reinforcement for shells was generally in this manner at the time. The reasoning for this application is based on the argument that bending moments efficient near supports may be neglected in other parts and membrane forces only may be considered. An example for this reasoning, which is abandoned today, is shown in Fig 1, a shell section reproduced from a reinforced concrete textbook translated from Russian [2]. The same committee stated in the report that reinforcement was installed in one layer and close to bottom face instead of midsection but not specified this being due to construction error as encountered even today due to stamping and changing the position of reinforcements. The creep effect was mentioned obscurely as rheological properties of concrete.
Fig. 1 Reinforcement layout applied formerly
It is known that creep is the increase of concrete strains with no change of stresses. It varies depending on humidity of environment, the age of concrete at the instant of loading, its composition and dimensions. Therefore, here creep will be considered using one third of modulus of elasticity as it was in the experts report, since the state of these factors, especially the age of concrete on the instant of collapse are not known.
On the Colla
4. THE BU The bucklin introduced b numerous pu investigated The case of and boundar obtained wi included thi Figure 90 o cylinder sim approximati However, fr neighborhoo buckling of full spheres.
Using the m used in that long, 20m ra that obtained applications On the othe (also loaded finite eleme method, ove
apse of Ankar
CKLING MA ng matter emp
by the design ublications on d with regards f buckling of s ry conditions.
ith some app s type of appr on page 573 o mply supporte
ion to simula requently, one od of boundar cylindrical or .
mentioned diag era, a critical adius and 10c d a critical loa s in those year er hand, defin d in plane) ha ent method, ertook the wor
Fig.2. Mod
ra Sport Cente
ATTER phasized in co
engineer and n the buckling buckling and shell structure
. However, so roximations.
roximations. F of Pflüger [8]
ed on circular ate a cylindric e of the buckli
ry conditions r spherical she
grams and ado l external pres cm thickness;
ad as 1000 kg/
rs.
nition of an eq as been known
developed in rk of Hrenniko
del of cylindric
er in 1958
ommittee repo implicitly acc g due to impe
imperfection s is not a trivi ome results th
Relationships For example, ] are construc r edges. It ma cal roof from ing modes yie of the analyz ells were base
opting concret ssure of 109M
probably the /m2. There wa quivalent bar
n since 1941 n later years
off.
cal shell and t
ort and additio cepted by the erfection. The
.
ial problem ex hat may elucid s given in sh
Figure 15 on cted for findin
ay appear at f m an unrelated elding very clo zed system. T ed on the buck
te modulus of Mpa is obtaine experts also m as no better ap
system for a from the wo and admitted
the half of buc
onal 5cm con committee, e erefore, the su xcept for simp date daily pra hell literature
page 433 of F ng buckling lo first glance a d system, a f ose eigenvalue Therefore, at t
kling of full c
f elasticity as 2 ed for full cyli made the sam pproximation
continuum su rk of Hrennik d as continua
ckled shape.
ncrete weight evokes recent ubject will be ple geometry ctice may be at that time Flügge [7] or oad of a full a very coarse full cylinder.
es falls in the that time, the cylinders and
21000Mpa as inder of 60m me calculation available for
uch as plates koff [5]. The ation of this
This author has presented a paper on the buckling of profile sections, applying this method to buckling problems in a Mustafa İnan seminar, at ITU. In the computer program used in this application, kumbas26, since the continuum is converted to a three dimensional frame, buckling may be investigated as the stability of a bar system, with sufficient approximation.
The load is applied in increasing small steps and deflections are added to geometry of the system to consider p-delta effect. Since different modulus of elasticity or moment of inertia may be used at each step, the material nonlinearity may also be considered. The buckling of bar system can also be investigated by the frequently used program SAP2000.
To check the level of approximation, the above mentioned method will be applied to a problem with a known solution, before its application to the case of the sport center. When the cylinder mentioned above is reconsidered for E=24000 MPa with charts in [8] for l/a=60/20=3, with k=(t/a)2/12=2.08x10-6, q1=1 x10-4 and p=q1D/a, the critical load
2 4
2 7
/ 5 . 12 10 0 1 . 20 ) 2 . 0 1 (
1 . 0 10 4 .
2 kN m
p
is found. Using the symmetry conditions, the cylinder is modelled in SAP2000 with equivalent beams of (2.0×0.1and 2.09×0.1)m sections, as a mesh 30 parts in semi circle and 30 parts in l=60m. Loads in the direction of cylinder axis are applied to longitudinal beams and it is observed that the section is warped for 14.9kN/m, and the buckling load is reached (Fig 2). The buckling mode in the chart of Pflüger appears to include 5~6 waves.
Similar buckling mode is found in SAP2000.
When the same model is used with abovementioned kumbas26 program, applying 1kN to each node to search the buckling load, adding the load at each step, the buckling is reached at 64th step. Since each node comprises 2.00×2.09m area, the buckling corresponds to
/ 2
3 . 09 15 . 2 00 . 2
1
64 kN m
p
which matches the SAP solution and is close to the Pflüger solution. The difference from the Pflüger solution may be because the boundary conditions used in the computer program do not exactly correspond to the theoretical simple support.
After it was shown that the buckling load of a cylindrical shell could be determined with sufficient approximation, the half of Ankara Sport Center was modeled as a bar system, still using the symmetry conditions. For the tempane wall and auxiliary structural elements, estimated section thickness is used in this modeling; for the shell 0.10m in general and 0.15m thickness close to support, for the beam supporting the shell 0.60/1.10m sections are used. Since the time gap between concrete pouring and scaffoldage removal is not known the modulus of elasticity E is used as its one third value, E=8000Mpa, considering rather unfavorable conditions for creep.
On the Colla
Fig. 3
The bar sy program usi with shell e 2.5kN applie In shell mod In bar mode membrane f compatible.
For shell mo For bar mo acceptable d The structur this analysi program. Th computed as
2 0 . 2
992 . p 0
Due to the simulation p=1000/3=3
apse of Ankar
3 SAP200 mod
stem modelin ing the same elements, and
ed to each nod deling: circula eling: circular forces were fo
At midpoint i odeling: wmax=
odeling: wmax
due to compati re was at first s by applyin he program yi
s
93 . 09 5 . 2
50 .
2 k
conditions o
333kg/m2 (3.3
ra Sport Cente
del prepared f
ng mentioned sections for c analyzed to de, at midpoin ar direction N1
direction N1= found at midp in vertical dire
= 0.0058m
x= 0.0128m ibility of inter
analyzed with ng 25kN at e
ields a coeffic
/ m2
kN
f that time, t
33kN/m2)
er in 1958
for Ankara Sp
d above is al columns and b
check for app nt of the shell:
1=-12.13 kN/m
=-11.98 kN/m point of the sh
ection
were attaine rnal forces.
hout imperfec each node (on cient of 0.992.
the buckling
port Center co
lso repeated beams, this ti proximation. U
m longitudinal longitudinal hell, whereas
ed. Thus, the
ction. The buc ne half for e The uniform
load, that wa
onsidering sym
with SAP200 ime modeling Under the act
l direction N2= direction N2= displacemen
e approximat
ckling load wa edge points) mly distributed
as found mak mmetry
00 computer the cylinder tion of loads
=-33 kN/m
=-36 kN/m ts are not as
ion may be
as pursued in in SAP2000
load may be
king a rough
is quite different from 5.93kN/m2 a value which is more realistic, and the structure appears 1.78 times secure against buckling under the action of existing load.
Since a strong wind was blowing during the collapse, a wind load may be added to 330kg/m2 the existing load at the time of incident, as computed by the committee. However, using one third of modulus of elasticity in this analysis will be excessively on the safe side, because the wind load is not a long term effect. Nonetheless, to provide simplicity, a buckling control is performed adding the wind load to self weight. The distribution of wind load acting on vault structures has been investigated by Blackmore and Tsokri [9]. The wind pressure and the rate of sucking for the height/span ratio of this case, given by Blackmore and Tsokri are close to the values given in EN1991-1-4 Euronorm. With additional loads of 0.64kN/m2 and - 0.16kN/m2, computed at one half of cylindrical roof with a coefficient 0.8 and at other half -0.2, buckling analysis is performed using SAP2000.
At each node 3.33kN for self-load and 0.64kN or -0.16kN for wind load, according to the region, are considered. The program yields a buckling coefficient of 0.698. The uniformly distributed load corresponding to this value is
/ 2
56 . 09 5 . 2 0 . 2
33 . 3 698 .
0 kN m
p
which indicates a minor effect of wind load on buckling of this building. These results are obtained for linear behavior and uncracked sections.
5. NONLINEAR ANALYSIS
The committee report remarks that installation of reinforcement is in one layer and portions of reinforcement are not exactly in required positions. This situation shows that in pure bending and bending with tensile load, bending rigidity of sections fall below a value quite lower than the rigidity considered in design. As is known, bending rigidity of a section depends on factors such as amount of reinforcement, neutral axis position as well as the elastic modulus of concrete. Since the kumbas26 program approximates the solution step by step, it is possible to use different section rigidity at each step. The SAP2000 which was also used here, can also perform nonlinear analysis; but since nonlinear behavior is previously defined, the effect of axial load, which differs for developing loadings in different parts of the system, can not be sufficiently taken into account. With this in mind, determination of the behavior of the sections forming the shell structure of Ankara Sport Center was undertaken. The rotation of a section may be defined in terms of steel and concrete strains (Fig. 4) as
EI M
d
s
c
(1)
One concludes that, if the strains are known for the acting bending moment, the bending rigidity of a section may be defined. This relation is applicable in nonlinear cases also, so far as Bernoulli-Navier hypothesis is valid.
On the Collapse of Ankara Sport Center in 1958
EI=
s c
Md
(2)
Fig.4 Rotation and strains in a reinforced concrete section
On the other hand, stress-strain diagram for concrete, accepted for ultimate stress section design is shown in Fig.5. The initial slope of the parabola for small values of concrete strains may be found as
Ec= fc fc
001 2000 . 0
2 (3)
This relation is correct with admissible approximation for the subject concrete. An appropriate curve may be used for the higher strength concrete. The relation composed by a parabola and a straight line shown in Fig.5 for concrete and a bilinear relation suitable for the yield strain of an elastoplastic material such as steel, are used in the computer programs performing section design.
This means that in order to acquire the stress-strain diagrams of sections whose elastoplastic behavior needs to be established, it suffices to perform section design in adequate quality and quantity so as to reflect various situations and obtain
cand
s. Based on the information given in [1], this process is performed for three types of sections of the cylindrical shell under consideration. The first type, which is specified as used near supports, is 0.15m in thickness and has Φ12/150mm reinforcement at both sides. The second type, which is used generally in other regions, is 0.10m in thickness and has Φ12/150mm reinforcement at midsection. The third type represents a section identical to type 2 except that it has a 20 mm dislocated reinforcement due to manufacturing error. This section must be used in negative bending moment area. The consideration of this fault is abandoned since the location of dislocated reinforcement in the structure is not known and to use this section in every negative moment area seems to be a guided evaluation. Under the action of various axial loads, increasing bending moments are applied to these three sections, the bending moment corresponding to yield of section and strains
c,
s areεc
h
εs
determined, and bending rigidity corresponding to the ultimate case is calculated according to relation (2).
Fig.5 Stress strain relations for concrete and steel
Considering the linear behavior of the sections in the case of small values of M and N, the neutral axis distance is computed for a linear stress distribution and the moment of inertia is attained depending on this value. The ratio of modulus of elasticity of steel and concrete are taken as 7.65. The neutral axis distance and the moment of inertia for linear behavior of combined bending are given in [10]. The third degree equation, which yields the neutral axis distance in terms of reinforcement percentage and eccentricity, is solved for increasing eccentricity values and the corresponding moment of inertia I is determined.
The moment of inertia of a section exposed to any combined bending N, M is computed by interpolation in terms of moment of inertia for linear case and for ultimate stress case, using
Concrete
0 0,2 0,4 0,6 0,8 1 1,2
0 0,5 1 1,5 2 2,5 3 3,5
strain %o
stress/fc
Steel
0 0,2 0,4 0,6 0,8 1 1,2
0 2 4 6 8 10 12
strain %o
stress/fy
On the Collapse of Ankara Sport Center in 1958
the ratio of acting moment to the moment corresponding to the ultimate case for present axial force
EI=EIlin- (EIlin- EItg) Mtg
M (4)
The ultimate moment for increasing values of axial force N is expressed for both sections, by relations similar to those of Çakıroğlu-Özer [11]. Similarly, the coefficient for moment of inertia i expressed depending on exponential functions. A ¼ of moment of inertia corresponding to the ultimate state is used for moment values larger than the ultimate one.
The necessity to make section design for each case, requiring large computing time, is thus eliminated. Interpolation formulae, used for the first type section with 0.15m thickness, are given below.
For linear starting rigidity, the coefficient i which decreases the moment of inertia, with en=M/(Nd)
i=0.256+0.251e-en+0.450×10-en (5)
The ultimate state relation and the coefficient i while the axial force is tensile (-)
0.0465N+0.732M-15.3=0 (6)
i=0.015+0.020e-en+0.006×10-en (7)
The ultimate state relation and the coefficient i while the axial force is compressive (+)
0.000037N2-0.0633N+0.841M-15.3=0 (8)
i=0.015+0.010e-en+0.090×10-en (9)
These relations are also obtained for the second type section. A subprogram attached to kumbas26 determines the section rigidity at each step of loading making use of these relations. The moment of inertia of uncracked section is used if the maximum tensile stress computed for homogenous section is less than tensile strength for tensile axial force or if en eccentricity ratio is less than 0.20 in the case of compressive axial force. For elements with no information, estimated section values and uncracked section rigidity is used for all type of loading.
The variation of i values, which decrease the moment of inertia, is presented in Fig. 6 for the second type section of 0.10m thickness and with its reinforcement at center, with various N values and increasing bending moments. The curve for N=400kN is at the top and the curve N=-100kN is at the bottom of this diagram. It is seen that the moment of inertia decreases with increasing moment values for all N values while the axial force influences the outcome as an important factor. The considered tensile force is at the same magnitude
as the tensil must be con In order to d values for ax presented in real momen curve is at t From this fa increasing th case of lack section that
Fig 6. Mo
The cylindr membrane f entirely. Ye close to this moment of i A small fra geometry is axial force, above and th w to x, y, z c new section matrix for b
le force emerg nsidered in com
determine non xial loads vary n Fig. 7 a-b. Th nt-curvature di
the top. The c act, one may o
he neutral axi k of a signific
has dislocated
Moment of inert thickness
rical shell o forces to a g et, the beam s s beam bear te inertia and bea action of the
defined as da the sections heir rigidity is coordinate of n rigidities are buckling in ter
ging in the sh mputation of e nlinear behavi ying between hese figures a iagrams. In th urve for N=-1 observe that a
s distance. Th cant axial pre d reinforceme
tia coefficient- containing on
f Ankara Sp great extent an supporting th ensile forces.
aring capacity load carried ata in kumbas2
are checked s specified. Th each node, an e calculated us rms of trigono
hell. This situa element rigidi ior reflected b -100~+400 k aim to reflect t hese diagrams 100kN in Type
xial pressure he same diagra
essure load. T nt.
-bending mom ne layer reinfo
ports Center nd circumfere he shell in lon The sharp de y is shown in F
by the system 26 program. U
at each step he next step is nd the interna sing the interp ometric functi
ation reveals t ty.
by the relation kN, for increas the character o
the curves ar e 2 is quite cl load increases ams reflect the This situation
ment relation f forcement at m
is subjected ential compre ngitudinal dir ecreasing effe Fig 6.
m (about 1/50 Under the effe
of computati s prepared by a
l forces to pre polation form ions is used a
that the axial ns (4, 7, 9), th
sing bending m of the relation re arrayed so lose to the hor s the moment e sharp decrea is more app
for the section mid-section
d to axial co essional mem
ection and sh ct of tensile a
0~1/100), and ct of bending on with the m adding displac evious interna mulae. The kno
and the determ
force values e sum of M/i moments, are ns and are not as N=400kN rizontal axis.
of inertia by ase in i in the arent for the
n of 0.10m
ompressional mbrane forces
hell elements axial force in
d the system moment and method cited cements u, v, al forces. The own stiffness minant of this
On the Colla
matrix is c determinant
Fig 7a. Defo
Fig 7b. Defo
In this analy using (4)~(9 somewhat la
apse of Ankar
computed at value to nega
formation-mom
formation-mom thickn
ysis for finding 9) relations t arger than 1.0
ra Sport Cente
each step. T ative zone.
ment relation f thickness and
ment relation f ness and one la
g buckling loa that evaluate 0 due to rein
er in 1958
The buckling
for various ax d two layers r
for various ax ayer reinforce
ad of the struc the sections nforcement. In
g appears wi
xial load value reinforcement
xial load value ement at mids
ctural system, s. The dimini n subsequent
ith the transi
es for the sect
es for the sect ection.
the rigidities ishing factor, steps, first th
ition of this
tion of 0.15m
tion of 0.10m
are found by i is 1.0 or he rigidity of
cracked sect applied at ea
. 2 0 . 2
2 . 0 p 63
An imperfec boundaries concrete, ar the referenc occurred in
. 2 0 . 2
. 0 p 57
is calculated buckling occ
. 2 0 . 2
2 . 0 p 45
tion and later ach node, buck
77 . 09 3
25 kN
Fig 8. T
ction form sh of imperfecti e applied to t es. Contrary t 57th step and
41 . 09 3 .
25 kN
d. If the wind curs at 45th st
69 . 09 2
25 kN
rigidity of the kling arises at
/ m2
N
The imperfecti
hown in Fig 8 ion area, and the system to to expectation for this situat / m2
N
d load is take tep, which cor
/ m2
e section near t 63th step, ie
ion form repre
8., 0.10m at m an additiona search for th ns, an importa
tion a load
en into accou rresponds to a
r plasticizing i for a load
esented in plan
midpoint, linea al load corresp he effect of im ant variation d
unt together w a buckling load
is used. For a
ne mesh.
arly decreasin ponding to th mperfection em did not arise. T
with this impe d
a load 0.23kN
ng toward the he additional mphasized in The buckling
erfection, the
On the Collapse of Ankara Sport Center in 1958
which is less than the load 3.30 kN/m2 specified by the committee as present during the collapse. In other words, the structure may buckle for a modulus of elasticity diminished due to creep E/2.45. The 670 of 892 bars used for modeling the structure are used to form the shell. In the 240 bars of these 670, the coefficient that decreases the moment of inertia, is less than 0.5 (i<0.5) just before buckling. The 132 of these 240 elements are in longitudinal direction. The number of elements for which the coefficient of moment of inertia i is less than 0.10, is 97 and 93 of these elements are in longitudinal direction. The elements with an i value of less than 0.01 are 17 in number and all of them are longitudinal and are exposed to tensile axial force.
This result indicates that in collapse of Ankara Sports Center, the imperfection due to manufacturing fault, the strong wind and the decrease in bending rigidities for the section with center reinforcement, especially when under the effect of tensile force, are the main factors.
6. CONCLUSIONS
A nonlinear buckling analysis is performed for Ankara Sports Center, based on the information given in previous reports, where the causes of the collapse were investigated [1]. In this approximate analysis, the shell which is a continuum, was converted to a three dimensional bar system using the method given in [4], facilitating the use of the capabilities of the bar system for calculations of nonlinear behavior and buckling. The axial membrane forces will affect strength and rigidity of this shell naturally. This effect is considered using relations of interpolation formulae at each loading step. The results reveal that the excessive decrease of rigidity in a small number of elements subjected to tension resulted in the loss of stability of the shell, and it buckled.
The diagrams obtained for Ankara Sports Center shown in Fig 6~7 reveal that in problems for which element rigidities significantly influence the results, the consideration of axial force for definition of rigidities is absolutely necessary.
Symbols
d : The depth of concrete section E : Modulus of elasticity
en : M/(Nd) eccentricity ratio fc : Strength of concrete I : Moment of inertia
Ibrüt : Moment of inertia for full section
i : I/Ibrüt diminishing factor for moment of inertia Ilin :Moment of inertia for linear behavior
Itg : Moment of inertia in ultimate case
k : a parameter defined for buckling of cylindrical shell l : The length of cylinder
p : Uniformly distributed load wmax : The maximum vertical deflection
c : strain of concrete : strain in steel s
References
[1] Bayülke, N. Ankara Spor Sarayı Neden Çöktü, Türkiye Mühendislik Haberleri, 59, 1, 47-57, 2009.
[2] Murashev, V.I.,Sigalov, E.,E., Baikov, V.,N., Design of Reinforced Concrete Structures. Moscow, Mir, 1968.
[3] Kao, R., Nonlinear Creep Buckling Analysis of Initially Imperfect Shallow Spherical Shells, Computures & Structures. 14. I-2. II I-122. 1981
[4] Hamed, E, Bradford, M.A., Gilbert, R.,I. Creep Buckling of Imperfect Thin-walled Shallow Concrete Domes,Journal of Mechanics of Materials and Structures, 5, 1, 107-128, 2010
[4] Hrennikoff, A., Solution of Problems of Elasticity by the Framework Method, J.
Appl. Mech. 12. 169-1175. 1941
[6] Schneider ve diğ. Stability Analysis of Perfect and Imperfect Cylinders Using MSC/Nastran Linear and Nonlinear Buckling, web.mscsoftware.com.
[7] Flügge, W., Stresses in Shells, Berlin, Springer, 1962.
[8] Pflüger, A.,Elastostatiğin Stabilite Problemleri, (Çeviri: Tameroğlu, S., Cinemre, V., Özbek, T) İTÜ, 1970.
[9] Blackmore, P.,A., Tsokri, E., Wind Loads on Curved Roofs, Journal of Wind Engineering and Industrial Aerodynamics 94, 833–844, 2006
[10] Löser, B., Bemessungsverfahren Wilhelm Ernst & Sohn, Berlin, 1951.
[11] Çakıroğlu, A., E.Özer. Eğik Eğilme ve Eksenel Kuvvet Etkisindeki Dikdörtgen Betonarme Kesitlerde Taşıma Gücü Formülleri. Yesa Yayınları, İstanbul 1983. [12] CSI, Three Dimensional Static and Dynamic Finite Element Analysis and Design of
Structures. Berkeley, Ca. USA, 1993
On the Collapse of Ankara Sport Center in 1958
