Yüzeyinden harmonik zorlama etkisindeki boşluklu yarım düzlem problemi
Yüzeyinden harmonik zorlama etkisindeki boşluklu yarım düzlem problemi
Dosyalar
Tarih
1996
Yazarlar
Ünal, Fatih Han
Süreli Yayın başlığı
Süreli Yayın ISSN
Cilt Başlığı
Yayınevi
Fen Bilimleri Enstitüsü
Özet
Elastik bir ortam içindeki yapıların dinamik etkiler altındaki davranışı geçmişte pek çok araştırmaya konu olmuştur. Yeraltı tünelleri ve boru hatları, yeraltı santralla- n, su altı boru hatları bu tip araştırmaların temel problemleri olmuştur. Sonsuz ortam ile ilgili çalışma çokluğuna karşılık yan sonsuz ortamdaki yapı ların dinamik davranışı ile ilgili çalışmalara yalan zamanlarda rastlanmaktadır. Yan sonsuz ortam kabulünün yeraltı yapılan için sonsuz ortam kabulüne göre daha uygun olmasına karşılık matematik yapısı bilhassa sınır koşullarının sağlatılmasının güçlüğün den dolayı daha ağırdır.. Bu çalışmada ise elastik bir yarım uzay içinde bulunan ve serbest yüzeyden harmonik basınç etkisindeki silindirik boşluğun ve yarım uzayın davranışı analitik olarak incelenmiştir. Birinci bölümde, konumuz ile ilgili ve literatürde bulunan çalışmalardan ve bu çalışmalarda kullanılan yöntemlerden kısaca bahsedilmiştir. Elastisite problemlerinin genel özelliklerinden bahsedilirken koordinat takımının ve potansiyel fonksiyonlarının önemi üzerinde durulmuştur. Bu çalışma genel ifadelerle tanıtılarak çözüm yöntemin den kısaca bahsedilmiştir. ikinci bölümde çözüm için kullanılan temel denklemlerin elde edilişi açıklan mıştır. Burada denge denklemleri, sınır koşullan, Hook yasası, Lame sabitleri ve şekil değiştirme yer değiştirme bağmtılan yazılarak Navier denklemleri elde edilmiştir. Üçüncü bölümde analitik çözüm metodu anlatılmıştır. Burada Navier denk lemlerinin çözümü için gerekli formülasyonlar gösterilmiştir.
Dynamical analysis of the underground structures has been subject of a lot of studies.- These problems are of interest in earthquake engineering and in other areas of engineering where it is important to know the dynamic response of the shell struc tures. Examples are buried pipelines tunnels, etc. Previous works like these were done by Datta [9, 10] and his colleagues for the case of harmonic waves incoming from infinity. In these works, some solution techniques are used as like the method of matched asymptotic expansions, the method of successive reflections, hybrid finite element and eigenfunction expansion technique, series of Bessel-trigonometric functions, and etc. Balendra [12, 13] and his colleagues dealt with dynamic response of buildings due to trains in underground tunnels. In these works, the displacements fields are formulated by the method of wave function expansion and the boundary conditions are satisfied only at a finite number of points along the traction free surface, the tunnel-soil interface and the soil-foundation interface. The tunnel and the foundation have been assumed as being rigid. This work discusses the dynamic response of a homogeneous, isotropic and perfectly elastic half-space which contains an elastic circular cylindrical cavity of infinite length at the and of a finite depth below the plane boundary and parallel to it. Loading is an harmonic pressure on the surface. 2. STATEMENT OF THE PROBLEM AND ANALYTICAL SOLUTION Since the geometry, the material properties and surface pressure are independent of the third coordinate, the plane strain case is considered. A section of the medium by a plane normal to the axis of the circular cylindrical cavity is represented in Fig. 1. The y-axis is taken parallel to the free surface and the x-axis perpendicular to it. The z-axis are assumed to be drawn perpendicular to this plane section. It is convenient to use polar coordinate systems as r and 9 for reason of the geometry. Loading on the surface of layer by a harmonic pressure is as follows ax =a0Cos(cot) (1) where cr0 is the amplitude of pressure, w the frequency and t the time. The displacement vector U in the soil medium must satisfy the Navier equations (2+//)VV-C7 + //V2f/-p d2 U ât2 = 0 (2) ax =c 0Cos(wt) x UUUI UUIU Figure 1 Geometry of the problem arid coordinate used Where X and// are the Lame constants for the half space with a Poison's ratio v and density p. It is possible to obtain a simpler set of equations by intro ducing the scalar and vector potentials » n - n n ^-^-UK^ + K-Y^k.r)^ Yn(^r)-fYtt+x(kxr) + Cos n0 (17) VUl n=0 + B" +C. +£> f 2 ~\ >. ~j^ + 0.5 {k2f Jn(k2r) - £- Jfl+1(/r2r) r f 2 _ \ - ?-^ + 0.5 (k2)2 Y"(k2r) - -f- Y"+1(k2r) k r J ı ' Sin n9 (18) ax, a and r are computed by the following equations ax = ^-Z- + Z~t- Cos 29 - t" Sin 29 a, = r * + r * Cos 20 + Tr0 Sin 29 xxy = °r a& Cos 29 + xra Sin 29 Â* ?re (19) (20) (21) 3. BOUNDARY CONDITIONS The boundary conditions at the surface of the cylindrical cavity ( first case ) are °"r = ° > xrB - ° at r^a (22a) and for the rijit cylinder ( second case ) Ur = 0, Ue-0 at r=a (22b) Furthermore, on the surface of half space, x=h, In first and second case we have ax = a0Cos(wt), x = 0 at r = hi Cos 9 (23) IX Because, difficulty of using infinite series, we had to truncate the series in a finite number N. The boundary conditions at the surface of the cavity are satisfied exactly. The boundary conditions at the surface of half space are satisfied approxi mately as follows : J= J [fa ~ °"o)2 + fa y) \dy=m minimum (24) d A. "(J) = 1 0 L / v dax âzxy dy = 0, n = 0,1,2,. (25) dB. "(J) A fa - ob) âax d t dB, +Txy âB, xy dy = 0, n = 0,1,2,. (26) After this, For first and second case, the unknown integration constants are calculated using all of the boundary conditions (22a), (22b) and (23). Choosing definite values for the circular cylindrical cavity geometry, the material properties, the depth of the circular cylindrical cavity axes from surface, the displacements of some special points have been investigated with respect to forcing frequency ca. In these calculation, Mathematica program is used.
Dynamical analysis of the underground structures has been subject of a lot of studies.- These problems are of interest in earthquake engineering and in other areas of engineering where it is important to know the dynamic response of the shell struc tures. Examples are buried pipelines tunnels, etc. Previous works like these were done by Datta [9, 10] and his colleagues for the case of harmonic waves incoming from infinity. In these works, some solution techniques are used as like the method of matched asymptotic expansions, the method of successive reflections, hybrid finite element and eigenfunction expansion technique, series of Bessel-trigonometric functions, and etc. Balendra [12, 13] and his colleagues dealt with dynamic response of buildings due to trains in underground tunnels. In these works, the displacements fields are formulated by the method of wave function expansion and the boundary conditions are satisfied only at a finite number of points along the traction free surface, the tunnel-soil interface and the soil-foundation interface. The tunnel and the foundation have been assumed as being rigid. This work discusses the dynamic response of a homogeneous, isotropic and perfectly elastic half-space which contains an elastic circular cylindrical cavity of infinite length at the and of a finite depth below the plane boundary and parallel to it. Loading is an harmonic pressure on the surface. 2. STATEMENT OF THE PROBLEM AND ANALYTICAL SOLUTION Since the geometry, the material properties and surface pressure are independent of the third coordinate, the plane strain case is considered. A section of the medium by a plane normal to the axis of the circular cylindrical cavity is represented in Fig. 1. The y-axis is taken parallel to the free surface and the x-axis perpendicular to it. The z-axis are assumed to be drawn perpendicular to this plane section. It is convenient to use polar coordinate systems as r and 9 for reason of the geometry. Loading on the surface of layer by a harmonic pressure is as follows ax =a0Cos(cot) (1) where cr0 is the amplitude of pressure, w the frequency and t the time. The displacement vector U in the soil medium must satisfy the Navier equations (2+//)VV-C7 + //V2f/-p d2 U ât2 = 0 (2) ax =c 0Cos(wt) x UUUI UUIU Figure 1 Geometry of the problem arid coordinate used Where X and// are the Lame constants for the half space with a Poison's ratio v and density p. It is possible to obtain a simpler set of equations by intro ducing the scalar and vector potentials » n - n n ^-^-UK^ + K-Y^k.r)^ Yn(^r)-fYtt+x(kxr) + Cos n0 (17) VUl n=0 + B" +C. +£> f 2 ~\ >. ~j^ + 0.5 {k2f Jn(k2r) - £- Jfl+1(/r2r) r f 2 _ \ - ?-^ + 0.5 (k2)2 Y"(k2r) - -f- Y"+1(k2r) k r J ı ' Sin n9 (18) ax, a and r are computed by the following equations ax = ^-Z- + Z~t- Cos 29 - t" Sin 29 a, = r * + r * Cos 20 + Tr0 Sin 29 xxy = °r a& Cos 29 + xra Sin 29 Â* ?re (19) (20) (21) 3. BOUNDARY CONDITIONS The boundary conditions at the surface of the cylindrical cavity ( first case ) are °"r = ° > xrB - ° at r^a (22a) and for the rijit cylinder ( second case ) Ur = 0, Ue-0 at r=a (22b) Furthermore, on the surface of half space, x=h, In first and second case we have ax = a0Cos(wt), x = 0 at r = hi Cos 9 (23) IX Because, difficulty of using infinite series, we had to truncate the series in a finite number N. The boundary conditions at the surface of the cavity are satisfied exactly. The boundary conditions at the surface of half space are satisfied approxi mately as follows : J= J [fa ~ °"o)2 + fa y) \dy=m minimum (24) d A. "(J) = 1 0 L / v dax âzxy dy = 0, n = 0,1,2,. (25) dB. "(J) A fa - ob) âax d t dB, +Txy âB, xy dy = 0, n = 0,1,2,. (26) After this, For first and second case, the unknown integration constants are calculated using all of the boundary conditions (22a), (22b) and (23). Choosing definite values for the circular cylindrical cavity geometry, the material properties, the depth of the circular cylindrical cavity axes from surface, the displacements of some special points have been investigated with respect to forcing frequency ca. In these calculation, Mathematica program is used.
Açıklama
Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1996
Anahtar kelimeler
Dinamik davranış,
Harmonikler,
Uzay,
Dynamic behavior,
Harmonics,
Space