İki parametreli elastik zemine oturan dairesel plak problemleri
The Circular plates problems on the two-parameter elastic foundation
- Tez No: 46280
- Danışmanlar: PROF.DR. HASAN ENGİN
- Tez Türü: Yüksek Lisans
- Konular: İnşaat Mühendisliği, Civil Engineering
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1995
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 45
Özet
ÖZET Bu çalışmamda iki parametreli, tek tabakalı elastik bir zemine oturan, ağırlıksız dairesel bir plağın tekil, düzgün yayılı ve şerit düşey yükler etkisindeki davranışı incelenmiştir. Birinci bölümde, konu ile ilgili genel tanımlar ve açıklamalar yapılmıştır. Konu ile ilgili daha önce yapılmış çalışmalarla, yapılan çalışmanın amaç ve kapsamı verilmiştir. İkinci bölümde, elastik zemin üzerine oturan dairesel plak ile yüzeyin elastik eğrisi dönel simetrik dış yük için çözülmüştür. Üçüncü bölümde ise ilk olarak, zeminin iki yönde de çalıştığı kabul edilerek ortasından tekil yük ile yüklü ince dairesel plak problemi ele alınmıştır. Çözüm sonunda plak ve yüzeye ait çökme, kesme kuvveti ve radyal eğilme momenti değişimleri incelenmiştir. Daha sonra iki parametreli Pasternak zemininin yalmz basınç aktardığı göz önüne alınarak plağın merkezinden tekil yük, belirli bir bölgesinin düzgün yayılı yük ve şerit yük ile yüklenmesi halinde plağın tam batma ve batmama halleri incelenmiş, tam batmama halinde ayrılma noktasının yeri belirlenmiştir. Sonuç bölümünde ayrılma noktasının plak yarıçapı ve zemin parametreleri ile değişimi yarıçap boyunca yerdeğiştirme ve kesit tesirlerinin değişimi grafiklerle gösterilmiştir. VII
Özet (Çeviri)
SUMMARY THE CIRCULAR PLATES PROBLEMS ON THE TWO-PARAMETER ELASTIC FOUNDATION In recent years the development of solid-propellant rocket motors, the increased use of soft filaments in aerospace structures, and the building activities in the cold regions intensified the need for solutions of various problems of beams, plates and shells continuously supported by elastic or viscoelastic media. The usual approach in formulating these problems are based on the inclusion of the foundation reaction into the corresponding differential equation of the beam, plate or shell. The foundation is very often a rather complex medium; i.e., a rubber like fuel binder, snow or granular soil. But since of interest here is the response of the foundation at the contact area and not the stresses or displacements inside the foundation material, the problem reduces to finding a relatively simple mathematical expression which should describe the response of the foundation at the contact area with a reasonable degree of accuracy. The problem of beams on elastic foundation has been examined firstly by Winkler in 1 867. According to this hypothesis, the deflection at every point of the foundation is proportional to the pressure applied at that point and independent of pressures acting at near by points of the foundation. This assumption is equivalent to considering the foundation to be composed of independent elastic springs. In Winkler foundation it is assumed the deflection a>, of the soil medium at any point on the surface is directly proportional to the stress p, applied at that point and independent of stresses applied at the locations i.e. p(x,y)=k /// tttt t it tttt it 1 1 (a) (b) Figure 1. The Corresponding Deformations of the Foundation Surface Figure 2. The Displacement of the Foundation Surface In engineering applications, there are some important problems which can be handled successfully by means of Winkler hypothesis: the frames of ships, space vehicle, grid systems at plates and bridges, continuous foundations in one two directions, rotationally shell and perpendicular piles under the effect of the horizontal load. In Winkler model there is only one parameter k characterising the foundation but some researchers improved two-parameter models including shear stress to make the foundation to be represented similar to the real foundation as well as possible. Some of these models are given as follows; 1) Filonenko-Borodich Model 2) Hetenyi Model 3) Pasternak Model 4) Vlasov Model 5) Reissner Model In Pasternak model, a shear layer is assumed which is on the Winkler springs. Pasternak considers the existence of shear interactions between the spring elements. Assuming a shear layer to be homogeneous and isotropic in the plane (x,y), the pressure is: p(x,y) = k ö(x,y)-GV2 o(x,y) IXT 4- /// /////// 1// n ttı/nıtf it t, Figure 3. Pasternak Foundation Model The second term on the right-hand side of equation is the effect of the shear interactions of the vertical elements. Here ca is the deflection of surface and G is the shear modulus of layer V2 is the Laplace operator in x and y. In the second chapter, problem of axisymmetrical deformation of circular plates on elastic single-layer foundation has been considered. The external load has been applied symmetrical relative to the plate center, so that the plate is subjected to an axisymmetrical deformation. Polar coordinates (9,r) have been used. The origin of coordinates being placed at the plate center, and the distance from the center to a given point denoted by r. The differential equation of bending of a plate resting on an elastic single-layer foundation is in polar coordinates: DV2V2co(r,0)-GV2co+kco=P(r,0) D=- EhJ 12(1 -v2) where P=external load D=flexural rigidity of plate h=thickness E=elastic modulus of plate v=Poisson's ratio of material of plate By virtue of the axial symmetry, the plate deflections (r) are independent of the polar angle 8, then the governed equation of the plate is: DV2V2co(r)-GV2o(r)+kcu(r)=P(r) Laplacian V2 is: 72_ 1 d + - dr2 rdr Let us replace the coordinate p by a new dimensionless independent variabler where L stands for a characteristic length defined as follows -f Vp the differential operator ^2 d2 Id V =+p dp2 Pdp The governed equation of the plate is by the dimensionless coordinate V2V2© -2r0V2CD +co =- 8(p) This equation differs from the equation of bending a circular plate on an elastic Winkler foundation by the term; -2r0Vpco through which allowance is made for the work done by the shearing stresses acting in the single-layer foundation. The differential equation of the plate have been solved according to Bessel and Hankel functions, then constants of integration have been calculated from boundary and continuity conditions. In the last operation, bending moments Mr,Me and shearing forces Qr can be calculated as shown below: ^ d“co vdco ^”-,-> 1-vdco Mr = -D(- - + ~- ) = -D( V;co- ) r dr2 rdr r r dr; ^ d2co 1 dö) ^, t l-vdcox Mq = -D(v - T + - - ) = -D(vV2oo +- ) ^ dr2 r dr Vr r dr; _ _ d d*-© 1 de> _ d,"-> v As is mentioned in chapter three, behaviour of a weightless thin circular plate that subjected to a concentrated load at the origin is investigated. Firstly, we assumed that the Pasternak foundation reacts in compression and also tension. At the end of the solution, the changes of vertical displacements, radial bending moments and shear forces via the radius of circular plate are given in the figures. The effect of k (the foundation stiffness), in the displacement, shear force and bending moment is shown in related figures. xiThen we assumed that the foundation reacts in compression only. In this chapter, some examples have been considered. Concentrated load at the origin, uniformly distributed load in some coaxial region and the uniformly distributed load on the cicumference with radius a. In this case, the plate and foundation is contact in some region. Formulation of problem is carried out in two region. Solution in contact area is obtained using Bessel functions with complex argument. In the lift-off region, solution is calculated with analytical functions. Using the boundary and continuity condition, some expressions obtained for the integration constants and the lift-off point. These expressions are linear in the integration constants and non-linear in the lift-off point. Eliminating the integration constants, a tedius function occured for the lift-off point. The root of this function which represents the lift-off point is calculated using the Newton-Raphson method, numerically. In the end of the solution, related to the loading case, the variations of lift-off point are calculated with respect to the radius of the plate and the foundation stiffness, numerically. Xll
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