Su depolarında rijitlik frekans ilişkisi
The connection between stiffness and frequency at water tank
Yazar: MEHMET ERTUGRUL
Danışman: PROF. DR. ABDUL HAYIR
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / İnşaat Mühendisliği Ana Bilim Dalı / Yapı Mühendisliği Bilim Dalı
Derece: Yüksek Lisans, 2016
Konu: İnşaat Mühendisliği = Civil Engineering

Özet:
Yapılan bu tez çalışmamızda bir su deposunun rijitlik, frekans ilişkisini incelemek
amacıyla üç farklı grupta modellemeler yapılmış ve sonuçları gözlemlenmiştir.
Rijitlik, bir yapının yükler karşısında stabil kalabilme yeteneğine verilen isimdir
yani, elastik cisimlerin ve yapıların ötelenme ve yer değiştirme tesirlerine karşı
koyma derecesini ifade eder. Frekans veya titreşim sayısı bir olayın birim zaman
(genel olarak 1 saniye) içinde hangi sıklıkla, kaç defa tekrarlandığının ölçümüdür,
matematiksel ifadeyle periyodun çarpmaya göre tersidir birimi s-1 veya Hertzdir. Bir
olayın frekansının ölçmek için o olayın belirli bir zaman aralığında kaç kere tekrar
ettiği sayılır, sonra bu sayı zaman aralığına bölünerek frekans elde edilir.
Yapılan bütün çalışmalar, (modelleme, analiz, sonuç tabloları), SAP2000 isimli
bilgisayar programında sonlu elemanlar yöntemi kullanılarak yapıldı. İlk model
gurubunda yapının aks bilgileri değiştirilmeden içerisinde bulunan su miktarı
değiştirilmiş ve analiz yapılmıştır. İkinci gurupta ise yapı içerisinde su bulunmayacak
şekilde değiştirilmiş ve yapının yüksekliği kademeli olarak artırılmıştır. Üçüncü ve
son gurupta ise benzer şekilde yapının içerisinde su bulunmadan, perde kalınlıkları
değiştirilerek analizler yapılmıştır. Yapılan bütün modellemelerin analiz sonuçları
karşılaştırıldığında yapının rijitliğinde görülen değişmenin frekans ve periyot
değerlerine etkisi görülmüştür.
İlk model gurubunu incelediğimizde üstü tamamen açık olan su depomuzun (3 metre
yüksekliğinde 5 metre genişliğinde ve 10 metre uzunluğunda) içerisinde bulunan su
yüksekliği 0, 1,5 ve 3 metre olacak şekilde üç farklı model tasarlanmıştır. Yapılan bu
modellemelerin analiz sonuçları karşılaştırıldığında frekans değerlerinde herhangi bir
değişiklik gözlemlenmemiştir.
Aynı modellemeler bu defa üstü kapalı olacak şekilde tasarlanıp analiz sonuçları
gözlemlenmiştir. Bu modellemelerde de değişen su miktarı frekansa etki etmemiştir.
Üstü kaplı model açık modele göre daha rijit olduğundan dolayı frekans değerleri
daha düşük çıkmıştır.
İkinci model gurubumuzda yapı içerisinde su bulunmamaktadır, yapının yüksekliği
2,4,6,8,10 ve 12 metre olarak değişitirlmiş olup altı farklı model tasarlanmış ve
analiz sonuçları kayıt altına alınmıştır. Sonuçlar göstermiştir ki yapının yüksekliği
arttıkça, yapının kütlesi artıp rijitliği azalacağından dolayı, beklenen şekilde frekans
değerlerinde düşüş gözlemlenmiştir.
Aynı gurubunda ikinci kısmında aynı modeller yapının üstü kapalı olacak şekilde
tekrar tasarlanıp analizler yapılmıştır. Bu analiz sonuçlarına baktığımızda, üstü açık
benzer şekilde frekans değerlerinde, yükseklik arttıkça, düşüş gözlenmiştir. Ayrıca
bu modellemeler, üstü açık modellere göre, kütlesi daha fazla olduğundan dolayı
frekans değerleri daha küçük gözlemlenmiştir.
Üçüncü ve son model gurubunda ise yapının içerisinde su bulunmamaktadır yapının
perde duvarları 5,10,15 ve 20 cm olacak şekilde dört farklı model tasarlanmış ve
analiz sonuçları incelenmiştir. Yapıda bulunan taşıyıcı perde duvar kalınlığı artması
dolayısıyla yapının rijitliği artmış ve beklendiği üzere frekans değerleri artmıştır.
Gurubun ikinci kısmında aynı yapılar üzeri tamamen kapalı olacak şekilde tekrar
tasarlanıp analiz sonuçları kayıt altına alınmıştır. Sonuçlar da benzer şekilde olup,
perde kalınlığı arttıkça frekansta artış gözlenmiştir. İki kısım modelleri
karşılaştırdığımızda, beklenen şekilde üstü kapalı gurubun frekans değerleri daha
yüksek gözlenmiştir.
Sonuç olarak yapılan bu çalışma göstermiştir ki rijitliği artan yapının frekansı her
şart altında artmamaktadır. Bazı durumlarda kütledeki artış daha baskın
olabilmektedir.

Abstract:
The aim of this study is to find a connection between stiffness and frequency at a water
tank.
Stiffness is the rigidity of an object the extent to which it resists deformation in response
to an applied force. The complementary concept is flexibility or pliability; the more
flexible an object is the less stiff it is. The stiffness of a structure is very important in
many engineering applications, so the modulus of elasticity is usually one of the
primary properties when selecting a material. For example, less modulus of elasticity is
needed when flexibility is required.
The stiffness, k, of a body measure of the resistance offered by an elastic body to
deformation. For an elastic body with a single degree of freedom (DOF) is proportion of
force applied on the body and displacement produced by the force along the same
degree of freedom.
Period is the time needed for one complete cycle of vibration to pass a known point.
Period and frequenc are in an inverse proportion. For example, while the period
increasing frequency will decrease. Period eqals to total time divided by the number of
cycles.
Frequency is the number of occurrences of a repeating event per unit time. It is also
referred to as temporal frequency, which emphasizes the contrast to spatial frequency
and angular frequency. Calculating the frequency of a repeating event is accomplished
by counting the number of times that event occurs within a specific time period, then
dividing the count by the length of the time period. The period is the duration of time of
one cycle in a repeating event, so the period is the reciprocal of the frequency.
Frequency is an important parameter used in science and engineering to specify the rate
of vibration phenomena, such as earthquakes.
The unit of frequency is the Hertz (Hz) and also s-1.
Mass is a speciality of a physical body. It is a measure of a object's resistance to
accelerationwhen a force is applied. Mass is not the same as weight, even though we
often calculate an object's mass by measuring its weight with a spring scale, rather than
comparing it directly with known masses. An object on the moon would weight less
than on the earth. But mass never changes.
A water tank is a container for storing liquid. The need for a water tank is as old as
civilization, to provide storage of water for use in many applications, drinking water,
irrigation agriculture, food preparation, etc. during history wood, ceramic and stone
have been used as water tanks. These containers were all naturally occuring and some
man made and a few of these tanks are still in use. Various materials are used for
making a water tank; plastics, fiberglass, concrete, stone, steel. Water tanks are an
efficient way to help developing countries to store clean water. By design a water tank
must not harm to the water. Water is susceptible to a number of amount negative
influences, viruses, changes in pH and accumulated gas.
In this study we analyzed a concrete water tank in three groups.
At the first group of the structure has same measurements, (3 meters height, 5 meters
width and 10 meters length), but the water level changes from 0 to 3 meters. This group
has two sections. Firstly, the top of structure totally open and secondly, the top of
structure totally closed.
The second group of the structure has no water and the height is being changed from 2
to 12 meters. This group has two sections. Firstly, the top of structure totally open and
secondly, the top of structure totally closed.
The third group of the structure has no water and the thickness of the concrete wall is
changed from 5 to 20 centimeters. This group has two sections. Firstly, the top of
structure totally open and secondly, the top of structure totally closed.
All models analyzed with SAP2000 with finite element method.
At the first group there are two sections and six different models. The top surface of
first section is totally open, there is a completely contact between water and air. The
first model of this group has no water. The second model of this group has 1,5 meters
depth of water. The third and last model this group is totally full of water (3 meters of
water depth). The top surface of second section is totally closed. The fourth model of
this group has no water. The fifth model of this group has 1,5 meters
depth of water. The sixth and last model this group is totally full of water (3 meters of
water depth) All of this models are analyzed and observed theirs frequency.
It is seen obviously the change of water depth has no effect to structures frequency. If
we compare two section of this group, with same water level, the closed section is much
more rigid than open section and the frequency of closed sections are smaller. For
example, the first model is totally open and there is no water inside it and the frequency
is 0,061217. The second model is also totally open and there is 1.5 meters water level
and its frequency is 0,061217. The fourth model is totally closed and there is water level
and its frequency is 0,049517. The fifth model is also totally closed and there is 1.5
meters water level and its frequency is 0,049517.
At the second group there are two sections also but twelve models. Both sections have
no water. The top surface of first section is totally open. The first model's height is 2
meters. The second model's height is 4 meters. The third model's height is 6 meters.
The fourth model's height is 8 meters. The fifth model's height is 10 meters. The sixth
and the last models height is 12 meters. The top surface of second section is totally
closed. The seventh model's height is 2 meters. The eighth model's height is 4 meters.
The ninth model's height is 6 meters. The tenth model's height is 8 meters. The
eleventh model's height is 10 meters. The twelfth and the last models height is 12
meters. All of these models are analyzed and observed their frequency.
The result of analyses are as expected, while the height of structure is increased the
stiffness of the model is decreased. Therefore, the frequency is increasing. If we
compare two section of this group generally the closed section is much more rigid than
open section and the frequency of closed sections are bigger. But the first model of two
sections which have 2 meters height, the change of the frequency different. The first
model of this groups frequency is, which has 2 meters height and totally open,
0,031833. The first model of the second group's frequency is, which has 2 meters height
and totally closed, 0,047303. It means the increasing of mass has much more effect on
the frequency than the increasing of stiffness at these two models.
At the third and the last group there are two sections and eight models. Both sections
have no water. The top surface of first section is totally open. The thickness of concrete
wall of structure is 5 cm at the first model. The thickness of concrete wall of structure is
10 cm at the second model. The thickness of concrete wall of structure is 15 cm at the
third model. The thickness of concrete wall of structure is 20 cm at the fourth model.
The top surface of second section is totally closed has same thickness. The thickness of
concrete wall of structure is 5 cm at the fifth model. The thickness of concrete wall of
structure is 10 cm at the sixth model. The thickness of concrete wall of structure is 15 
cm at the seventh model. The thickness of concrete wall of structure is 20 cm at the
eighth model.
The result of analyses are as expected, while the thickness of concrete wall of structure
is increased the stiffness of the model is increased. Therefore, the frequency is
increasing. If we compare two section of this group the closed section is much more
rigid than open section and the frequency of closed sections are bigger. For example, the
first model of this group, the thickness of concrete wall is 5 cm and totally open, its
frequency is 4.0878. The second model of this group, the thickness of concrete wall is
10 cm and also totally open, its frequency is 8.1739. The fifth model of this group, the
thickness of concrete wall is 5 cm and totally closed, its frequency is 12,247.
After all the results above increasing of the stiffness (decreasing of structure height,
increasing the thickness of concrete wall, changing top surface condition open or closed,
etc.) does not increase the frequency allways. If we look the second group of study we
can realize that at some conditions increasing of mass has much more effect on the
frequency