Betonarme yapı sistemlerinde ikinci mertebe limit yükün ve göçme güvenliğinin belirlenmesi için bir yük arıtımı yöntemi

Abstract

Bu çalışmada, betonarme yapı sistemlerinde ikinci mertebe limit yükün hesabı ve göçme güvenliğinin belirlenmesi için bir yük artımı yöntemi geliştirilmiştir. Altı bölüm halinde sunulan çalışmanın birinci bölümünde konunun tanıtılması, konu ile ilgili çalışmaların gözden geçirilmesi, çalışmanın amacı ve kapsamı yer almaktadır. İkinci bölüm, betonarme çubukların elastoplastik davranışına ayrılmıştır. Bu bölümde ilk olarak, çeşitli iç kuvvetler etkisindeki betonarme çubuk elemanların iç kuvvet- şekil değiştirme bağıntılarının ve taşıma güçlerinin tayini hakkında bilgi verilmiştir. Daha sonra, bu çalışma kapsamı içinde, iç kuvvet-şekildeğiştirme bağıntılarının ve bileşik iç kuvvet durumuna ait taşıma güçlerini ifade eden akma koşullarının nasıl idealleştirilebileceği açıklanmıştır. Üçüncü bölümde, betonarme uzay çubuk sistemlerde ikinci mertebe limit yükün hesabı ve göçme güvenliğinin belirlenmesi amacıyla geliştirilen yük artımı yönteminin dayandığı varsayımlar, yöntemin esasları, formülasyonu ve yöntemin uygulanmasında izlenen yol açıklanmıştır. Yöntemde, düşey işletme yüklerinin bu yükler için öngörülen güvenlik katsayısı ile çarpımından oluşan belirli değerleri altında, aralarındaki oran sabit kalacak şekilde artan yatay yüklere göre hesap yapılmaktadır. Düşey yükler belirli olduğundan, büyük ölçüde denge denklemlerine bağlı olan normal kuvvetler başlangıçta kolaylıkla tahmin edilebilmekte; böylece geometri değişimlerinin denge denklemlerine etkisi lineerleştirilmektedir. Bileşik iç kuvvet durumuna ait akma koşullan lineer bölgelerden oluşacak şekilde idealleştirildiklerinden, iç kuvvet durumunun akma yüzeyi üzerinde kaldığım ifade eden akma koşulu denklemleri de lineer denklemlere dönüşmektedir. Böylece her plastik kesitin meydana geldiği yük parametresi ardışık yaklaşıma gerek kalmadan doğrudan doğruya hesaplanmaktadır. Bu bölümde ayrıca, döşemeleri kendi düzlemleri içinde sonsuz rijit olan çok katlı yapılarda ve büyük yapı sistemlerinde, yöntemin etkin kullanımım sağlayan bir matematik formülasyon önerilmiştir. Dördüncü bölümde, yöntemin sayısal uygulamaları için hazırlanan ve FORTRAN dilinde kodlanan bilgisayar programlan hakkında bilgi verilmektedir. Beşinci bölüm, geliştirilen bilgisayar programlarından yararlanılarak çözülen örneklere ve bu örneklerin sonuçlarının tartışılmasına ayrılmıştır. Altıncı bölümde, bu çalışmada elde edilen sonuçlar açıklanmıştır. Çalışmanın Ek A bölümünde çubuk rijitlik ve dönüştürme matrisleri, Ek B bölümünde ise plastik şekil değiştirme bileşenlerinin birim değerlerinden oluşan uç kuvvetleri matrisleri verilmektedir.

The use of elastic-plastic analysis and design methods, which consider the non-linear behavior of reinforced concrete as well as the non-linearity caused by geometrical changes, may result in both more realistic and more economical solutions. Furthermore, the collapse safety of existing reinforced concrete structures can be examined accurately by the use of these methods. In this study a method of load increments is developed for the determination of second-order limit load and collapse safety of reinforced concrete framed structures subjected to factored constant gravity loads and proportionally increasing lateral loads. The thesis consists of six chapters. In the first chapter, after introducing the subject, the results of a literature survey is given and the scope and objectives of the study are explained. The aim of this study is to develop a load increments method for the non-linear analysis of reinforced concrete structural systems. The non-linearities due to both elastic-plastic behavior of reinforced concrete and geometrical changes are considered. The method is independent of the characteristics of the structural system and can be applied to all reinforced concrete space structures under certain assumptions and limitations made throughout the study. The second chapter outlines to the detailed investigation of the non-linear behavior of reinforced concrete members. The investigation covers the real internal force- deformation relationships, the yield (failure) conditions and the idealization of non linear behavior. This investigation is based on three basic assumptions made for reinforced concrete, such as a- plane sections remain plane after bending, b- full bond exists between concrete and reinforcing steel, c- tensile strength of concrete is negligible after cracking. XVll The non-linear behavior of reinforced concrete frame elements under biaxial bending combined with axial force is idealized by ideal elastic-plastic internal force- deformation relationship. This idealization corresponds to the plastic section hypothesis. When the state of internal forces at a critical section reaches the ultimate value defined by the yield (failure) condition, plastic deformation occurs. The plastic deformations are limited to the rotational capacity. The rotational capacity may be expressed in terms of the length of plastic region and the ultimate plastic curvature. In this study, the yield surface for reinforced concrete elements subjected to biaxial bending combined with axial force, is approximated as being composed of 24 planes. Similarly, the ideal elastic-plastic behavior is also assumed for reinforced concrete sections subjected to torsional moment. In the third chapter, the assumptions, the basic principles and mathematical formulation of the method are presented and the proposed analysis procedure is explained. The following assumptions and limitations are imposed throughout the study. a-The internal force-deformation relationships for reinforced concrete frame elements under both biaxial bending combined with axial force and torsional moment are assumed to be ideal elastic-plastic. b-Non-linear deformations are assumed to be accumulated at plastic sections while the remaining part of structure behaves linearly elastic. This assumption is the extension of classical plastic hinge hypothesis which is limited to planar elements subjected to simple bending. c- Yield (failure) conditions may be expressed in terms of bending moments, axial force and torsional moment. The effect of shear forces on the yield conditions is neglected. Furthermore, it is considered that the yield conditions for biaxial bending with axial force and torsional moment may be expressed independently. The non-linear yield condition for biaxial bending combined with axial force can be approximated by linear regions. d-The plastic deformation vector for the case of biaxial bending combined with axial force is assumed to be normal to the yield surface. e-The second order theory may be applied to the analysis of slender structures with high axial forces. In the second order theory, the equilibrium equations are formulated for the deformed configuration while the effect of geometrical changes on the compatibility equations is ignored. XVUl f-The second-order effects caused by torsional displacements are not taken into account. Hence, only the second-order effects due to the geometrical changes within the principal planes of members are considered. g-Changes in the direction of loads due to deflections are assumed to be negligible. h-The structure is composed of straight prismatic members with constant axial forces. The members which do not meet these requirements can be divided into smaller straight and prismatic segments with constant axial forces. i-Distributed loads may be approximated by sufficient number of statically equivalent concentrated loads. In the load increments method developed in this study, the structure is analyzed under factored constant gravity loads and proportionally increasing lateral loads. Thus, at the end of the analysis, the factor of safety against lateral earthquake or wind loads is determined under the anticipated safety factor for gravity loads. When the gravity loads are known, the member axial forces can be easily estimated through equilibrium equations. Thus, the second-order effects are linearized by calculating the elements of stiffness and loading matrices for the estimated axial forces. If the member axial forces obtained at the end of the analysis are quite different that the estimated values, the analysis is repeated. However, in most practical cases such a repetition is not required. In the proposed method, the structure is analyzed for successive lateral load increments. At the end of each load increment, the state of internal forces at a certain critical section reaches the limit state defined by the yield condition, i.e., a plastic section forms. Since the yield vector is assumed to be normal to the yield curve, the plastic deformation components may be represented by a single plastic deformation parameter which is introduced as a new unknown for the next load increment. Besides, an equation is added to the system of equations to express the incremental yield condition. This equation is linear, because the yield surface is approximated to be composed of linear regions. Since the system of equations corresponding to the previous load increment has already been solved, the solution for the current load increment is obtained by the elimination of the new unknown. In the second-order elastic-plastic theory, the structure generally collapses at the second-order limit load due to the lack of stability. This situation is checked by testing the determinant value of the extended system of equations. If the magnitude of determinant is less than or equal to zero, then the second-order limit load is reached. Hence, the computational procedure is terminated. XIX In some cases, the structure may be considered as being collapsed due to large deflections, excessive plastic rotations, large cracks or failure of critical sections. The analysis method developed herein allows for the detection of collapse load caused by these reasons. The generalized yield condition for a reinforced concrete space frame element may be written as, K(Mx,My,Mz,N,Tx,Tz) = 0 where K is a non-linear function of internal forces, Mx and Mz are bending moments, My is torsional moment, N is axial force and Tx, Tz are shear forces. Neglecting the effect of shear forces, this condition becomes K(Mx,My,Mz,N) = 0 Considering that the total longitudinal reinforcement may be divided into two parts, such as flexural and torsional reinforcements, the interaction between flexure and torsion can be ignored. Thus, the yield conditions become K1(Mx,Mz,N) = 0 and K2(My) = My-Myp=0 where Kj is a non-linear function of bending moments and axial force and Myp is the torsional plastic moment. Under increasing loads, when internal forces reach the ultimate values defined by the above yield conditions, plastic deformations occur. The state of internal force is not allowed to violate the yield conditions. For the case of biaxial bending combined with axial force, this property is stated as dKi = ^J-dMx + ^-dMz +^-dN = 0 ' 3MX * dMz 2 dN TIK tiK t$K where - - -,^r-L,-r- '- denote the partial derivations of function Kj(Mx,Mz,N) dMx dMz dN with respect to Mx, Mz andN. XX The plastic deformations which develop at plastic sections are defined by the yield vector d(x,z,A) in which <\>x, §z and N are the plastic deformation components in the directions of Mx, Mz and A, respectively. Since the yield vector is assumed to be normal to the yield surface, the plastic deformation components may be expressed in terms of a single parameter, as in the following,, oK,,, oK,.,oK, *?-?»£ ?*?-?*£ -A=f» The parameter 0 is called as plastic deformation parameter. In order to linearize the structural behavior the yield surface is approximated by K,(MX,MX,N) = A1Mx + A2Mz + A3N+B = 0 where A,,A2,A3 and B are constants which depend on the material and cross-sectional characteristic and the amount and distribution reinforcement. According to this approximation, the plastic deformation components become *.-*%-#.. *.-!£-«.. *-?§-". At each step of the proposed method, a structural system with several plastic sections is analyzed for a lateral load increment. In the mathematical formulation of the method, two groups of unknowns are considered, such as a- nodal displacement components, b- plastic deformation parameters at plastic sections. The equations are also considered in two groups. a-The equilibrium equations of nodes in the directions of nodal displacement components. These equations may be written in matrix form as where, XXI [Sjj] : system stiffness matrix obtained by omitting the plastic sections, \d\ : unknown nodal displacements vector, [S^] : a matrix which represents the effects of imposed unit plastic deformations on the equilibrium equations, [$] : unknown plastic deformation parameters vector, [P"] : vector of fixed-end forces due to member external loads,