Süreksiz mesnetli plakaların hesabı
The Computation of plates mith discontinious support
- Tez No: 39647
- Danışmanlar: DOÇ.DR. MELİKE ALTAN
- 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: 51
Özet
ÖZET Günümüzde birçok yapılarda özellikle konut ve büro tipi yapılarda kullanılan süreksiz mesnetli plakların kesit tesirlerinin hesabında, süreksiz mesnetli kısım sürekli kabul edilerek hesap yapılmakta, süreksizlik bölgesinde açıklığı süreksizlik boyu kadar olan bir gizli kiriş düşünülmektedir. Gizli kirişe sık boyuna donatı ve etriye yerleştiril mektedir. Yaklaşık çözümlerin doğruluk derecesini araştırmak ve gerçekçi bir hesap ve donatı düzeni önermek amacıyla ; iki tip mesnetlerime şekline sahip plakların kesit te sirleri, bir kenarının belli bir uzunlukta süreksiz mesnetlenmesi durumunda plak kenar oranlarına ve süreksizlik boyunun süreksizliğe paralel plak boyuna oranına bağlı ola rak Sonlu Elemanlar Yöntemi ile hesaplanmış, diyagramlar ve tablolar verilmiştir. Sekiz bölümden oluşan bu çalışmanın birinci bölümünde çalışmanın genel kapsamı, ikinci bölümde ince plak teorisinden faydalanılarak plak denkleminin çıkarıl ması ve sınır koşulları, üçüncü bölümde ise sonlu elemanlar yönteminin genel tanım lan yapılmış, dikdörtgen plak sonlu elemanın rijitlik matrisinin çıkarılması kısaca gös terilmiştir. Dördüncü bölümde, uygulamada sık rastlanılan mesnetlerime ve kenar oranlarına sahip süreksiz mesnetli plak tiplerinin seçimi yapılmıışur. Beşinci bölümde, daha evvelce geliştirilen bir bilgisayar programı yardımı ile plak çözümlerine ait veri girişleri ve boyutsuz sonuçlara ait formülasyon verilmiştir. Altıncı bölümde, elde edi len boyutsuz sonuçlara ( seçilen plak tipleri için ) ait maksimum ve minimum moment değerleri, moment diyagramları ve tablolar ile sunulmuştur. Yedinci bölümde, yakla şık yöntem kısaca özetlenmiş ve herbir tip için sayısal uygulamalar yapılarak ; sonuçlar tablolarda karşılaştırılmıştır. Son bölümde ise bütün bu çalışmadan elde edilen sonuçlar değerlendirilmiş ve öneriler sunulmuştur. Süreksiz mesnetli plakların sonlu elemanlar metoduyla yapılan hesaplamalar sonucu, plağın yüzeysel taşıyıcılık özelliği gösterdiği ve süreksiz mesnedi sürekli var sayarak yapılan hesaplamaların da plağın gerçek taşıma şeklini tam yansıtmadığı görül müştür. Plağın gerçek taşıma şekli ise yüzeysel taşıyıcılık özelliğinden dolayı üzerin deki yükü x ve y yönündeki şeritlere Ly / Lx ve L' / Lx oranına bağlı olarak dağı tıp, taşımaktadır. Süreksiz mesnetli plak, sürekli mesnetli plak varsayılarak hesapla nan momentlere göre plağa donatı konlursa ; plağın ortasından gizli kirişe kadar olan kısmının x yönündeki donatısı yetersiz kalacaktır. Buna karşın gizli kirişe gereğinden fazla donat konulacaktır. Sonlu elemanlar yöntemi ile hesaplanan tablolar ve öneri len donatı düzeni yardımı ile mesnedinde kısmi süreksizlik olan plakların daha gerçek çi, güvenli ve ekonomik bir şekilde hesabı ve donatılması mümkün olacaktır.
Özet (Çeviri)
THE COMPUTATION OF PLATES WITH DISCONTINIOUS SUPPORT SUMMARY Usually, in most of the structures such as residence buildings and office buildings, the support of the plates is partially discontinious. Until now, the com putation of the plates with discontinious support has been made by assuming the discontinious support as the continious support. It has been thought that there is a hidden beam along the discontiniousness, length of which is equal to disconti- niousness 's length and thickness is equal to the plate 's thickness. The reinforcing bars of the plate has been placed concentratedly and with strirrups. This thesis' aim is to investigate the accuracy of these approximate compu tations. The“ computations of the plates with discontinious support have been made' for different support conditions ( Simply supported edge, built in edge and free edge ) and different edge length ratios ( Ly / I^ and L1 / L^ ) with the finite element method which has been supplied by a computer programme. The Equation (5.1) and Equation ( 5.2 ) have been formulated for getting the results unitless. ^ L* ^ Ly -/- Lx Fig. 1 Plate with discontinious support This study consists of eighth chapters. In the first chapter, the general view of this study, in the second chapter the evaluation of the thin plate differantial equation and the boundary conditions, in the third chapter the summary of the finite element method and the evaluation of the matrices for the plate bending elements have been presented shortly. In the fourth chapter, the selection of the plates with discontinious support is made which are encountered frequently in application and have different boundary conditions ( Simply supported edge, built in edge and free edge ), edge ratioes ( Ly / 1^ and L7 1^ ) with single span and more than single span. In the fifth chapter, input data for the finite element computer programme and the formulas for obtaining unitless values have been presented. In the sixth chapter, VIthe obtained unitless results are presented in tables 6-1 - 6-25. These results have been choosen as maximum span moments and mini-mum support moment values. These results can be used for different plate thickness and different concrete class. In the seventh chapter, the approximate method ( The hidden beam method ) has been sum marized shortly, numerical examples have been given for every type plate and the results are compared for each computation method in tables. Finally, in the last chapter, the results obtained from this study have been evaluated and recommenda tions have been presented. Approximate ( Hidden Beam ) Computation of the Plates with Discontinious Support : The plate with discontinious support has beams in plate, thickness of which is equal to the thickness of the plate. If the support of the plate is discontinious, the com-putation of the hidden beam should be made for the discontinious support. The way for carrying the load of plates with discontinious support varies according to the length of discontinious support and the thickness of the plate. This effect is written related to L and d. Figures concerning approximate computation method are in the seventh chapter. 1 ) For L / d < 7, the constructive reinforcing bars are placed to the hidden beam of plates with discontinious support. 2 ) For 7 15, the computation of the plate is made with the thin plate theory. Approximate ( Hidden Beam ) Computation Method ( 7 < L / d Lp2 : Edge lengths of the neighbouring plates perpendicular to the discontinious support. q=-^+^! (2) 4 2 2 K ' Moments for the unit width : For plates with single span -qL2/12 qL Support moment MK = - - = - -*- ( 3 ) vv ”0,125 L 1,5 v ' _ 4.. qL2/24 qL... Span moment Mrf =nQ25L =:y ( 4 ) For plates with more than single span : Support moment ML = - - - - = - - ( 5 ) vv“ 0,25L 3 v J qL2/24 qL,,x Span moment Mrf = = - - ( 6 ) v rf 0,50L 12 v ' Tables prepared for moments of the plates with continious support are used for the other moments of the plates with discontinious support except the hidden beam' s moments. IXThe thin plate is defined as a three dimensional body of constant or variable thickness h, the midpoints of which are on a plane ; the middle plane. In addition h is very small compared to the other dimensions of the plate. Another property is that loads are perpendicular to the middle plane. Thin plate theory, differantial equations and boundary conditions are not used in the calculations of the plates with discon- tinious support since the difficulty of solving the difTerantial equation and determining the boundary conditions along the discontiniousness for colleaques who work in practice. The finite element method is used for the calculation of the plates with dis- continious support instead of the thin plate theory. Thin plate theory has been sum marized shortly in the second chapter. The Finite Element Method : The finite element method is a relatively new and highly useful method for stress analysis of structural continua. The method relies strongly on the matrix formu lation of structural analysis. The finite element method allows the structural conti nuum to be replaced by a fictious system consisting of discrete elements of finite dimensions. The system is normally analysed by means of the displacement method which is already well known due to its extensive use in frame analysis: In a matrix analysis of frames and trusses, the standart approach is to divide the structure into a finite number of elements connected at joints or nodal points. The stiffness or flexibility properties of each individual element are then established by an element analysis resulting in a stiffness of flexibility matrix for the element. In the following discussion, a displacement method of analysis will be presupposed. Thus, the ele ment analysis comprises the development of an element stiffness matrix. The stiffness matrices for plate bending elements : Only plane stress problems have been discussed so far. Another major field of problems concerns the bending of elastic plates. The plate bending problem can be treated in the same manner as the plane stress problem, namely by subdividing the plate into finite elements and performing a matrix stiffness analysis of the resulting idealised structural system. In case of the bending of a plate with unstrained middle surface, the straining of the plate is completely defined by the lateral deflection. General approach to the element stiffness anlysis ; in the thoery of plate bending curvatures of the plate are expressed by the second order derivatives of the deflection, which may be assembled in a vector of curvatures. [c]= W w. yy. (7) W, xyjThe factor 2 in the last element has been introduced merely because it is convenient when expressing the internal work. The two bending moments and the twisting moment are also assembled in a vector. H= M. K] (8) Positive directions of bending moments are shown in Fig.2, which also shows positive directions for shear forces, external load, and deflection. Fig. 2 Moments and shear forces of the plate with their positive directions under load P Since the deflection is taken to be possitive downwards, negative curvatures, as defined by eq. ( 7 ), correspond to possitive moments. In the case of an elastic isotropic material, the bending moments are related to the curvatures as follows ; [Mj = -[D][[W”]-v[W“]] [My] = -[D][[Wyy]-,[Wj] (9) where, Kh-Mo-,)^] D = Eh3 12(1 -V) (10) using the notation of eqs. ( 7 ) and ( 8 ), eq. ( 9 ) may be written, H[m] = -[D]*[C] (11) where. [D] -D (12) is a stiffness matrix expressing the bending stiffness of the isotropic plate. Eq. (11) also applies to an arbitrary orthotropic elastic plate, provided the appropriate stiffness matrix [ D ] is used. The evaluation of stiffness matrices for plate bending elements follows the same general procedure as already described for elements in plane stress analysis. The procedure may be summarized as follows : a ) For a choosen element shape, nodal point system, nodal point parame ters, all element nodal parameters are assembled in a vector [ V ]. The deflection [ W ] is described by assumed deflection functions ( usually polynomials ) as, [W] = [«]T»[q] (13) The vector contains the assumed displacement functions and the vector [ q ] contains a set of generalized displacements. Alternatively the deflection may be expressed as, [ W ] = [ Bco ]T * [ V ] (14) where, Bo> are interpolation functions. The matrix [ B ] depends on the choice of the displacement functions [ w ]. In simple cases, [ Bo ] may be written down on inspection. In any case a relation [V] = [G]T*[q] (15) can be established by introducing the nodal coordinates into eq. ( 13 ) and in the appropriate derivatives of this expression. The matrix [ B ] may then be found by inversion, [B] = [G]”1 (16) xub ) By differentiation of [ W ] the curvature vector [ C ] ( see eq. ( 7 ) ) is obtained in one of the two alternative forms : where, from eq. ( 13 ) [C]=[Pq]*[q] from eq. ( 15 ) [C]=[P].[V] (17) (18) [Pa]' [w«r K]T (19) [P] = [Pa]*[B]T (20) c ) The moment vector [ m ] ( see eq. ( 8 ) ) is found from eq. (11) by use of eq. ( 17 ), yielding [m]=-[D]*[Pq]*[q] or by use of eq. ( 12 ), yielding [m]=-[D]*[P]*[V] (21) (22) d ) Stress resultants [ Q ] or [ S ] are found from the principle of virtual work. In the case of plate bending, the internal virtual work intensity may be expres sed as the sum of the products of the moment components and the corresponding curvature components. Thus, taking virtual displacements equal to actual displace ments [V]T *[S] =[qf *[Q] = -j [cf *[m] *dA A (23) where A denotes the area of the element. e ) Substituting eqs. ( 17 ) or ( 18 ) and ( 21 ) or ( 22 ) into eq. ( 23 ) gives the element stiffiiess relation in the form [Q] = [kq]*[q] (24) xmor in the form [S]=[k]*[v] where, [K]= jMT*M*[p,]*dA A (25) (26) [k]= |[P]T*[D]*[Pq]*dA A (27) by use of eq. ( 20 ) one finds M=[B]*[kq]*[B]T (28) f ) For the final assembly of the element stiffness matrices [ k[ ] to form the structure stiffness matrix [ K ], the nodal parameters should be arranged in the form ( 29 ) where the parameters related to element node number 1 are listed first, node number 2 next and so on. Depending on the arrangment of nodal parameters in the element analysis a rearrangment may be necessary. The corresponding rearrangment of the stiffness matrix [ k ] consists of interchanging rows and columns respectively [V] = V, (29) Results and Recommendations : For Ly / Lx > 1,5, the discontiniousness of the plate' s support does not effect much the way of the plate's carrying load. Although the value of L7 Lx has increased, the span moments Mj^ and Mym do not change much. Moments of the plate with discontinious support are approximately equal to the moments of the plate with conti-nious support. For Ly / 1^ < 1,5, the way of the plate1 s carrying load changes depending on the increase of the L'/Lx ratio from y direction to the x direction. It is concluded that the computation assuming the discontinious support as the continious support does not represent the way of plate' s carrying load. Plates XIVactually carry the loads by distributing them to the strips in the x and y directions, related to the Ly/Lx ^d L'/Lx ratioes. If the reinforcing bars of the plate with discontinious support is placed according to the approximate computation method, the amount of the reinforcing bars placed between the middle of the plane and the discontinious support in the x direction will not be sufficient. Meanwhile, the amount of the reinforcing bars placed to the hidden beam will be more than required. As a result, the approximate ( Hidden Beam ) method is not safe for the direction of the discontinious support and not economic for y direction of the plate with discontinious support. For plate with single span : if M^-m > Mj^, the additional reinforcing bars corresponding to the difference ( Mx^ - M}aa ) should be placed with Ly / 2 width from the discontinious support and ( D + 2Lj, ) length to the region ( L' + 2L5, Ly / 2 ) in the x direction. L^ is the reinforcing bar's anchorage length. For plates with more than single span : if M^^y > Mxm, the additional reinforcing bars correspon ding to the difference ( Mx^ - Mj^ ) should be placed with Ly width and ( L' + 2Lb ) length to the region ( L' + 2Lb, Ly ) in the x direction. If My^ > Mym, the additional reinforcing bars corresponding to the difference ( My^ - Mym ) should be placed ( Ly + 2L5 ) length and L' width to the region ( L', Ly + 2L5 ) in the y direction. The computation and placing of the reinforcing bars of the plates with discontinious support will be actual, economic and safe with the help of tables given and arrangment of reinforcing bars recommended. XV
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