Birinci ve ikinci mertebe limit yükün hesabı için bir yük artımı yöntemi
A load increment method for the calculation of first and second order limit load
Yazar: NURSEL KÜTÜK
Danışman: PROF. DR. ENGİN ORAKDÖĞEN
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / İnşaat Mühendisliği Ana Bilim Dalı / Yapı Mühendisliği Bilim Dalı
Derece: Yüksek Lisans, 2019
Konu: İnşaat Mühendisliği = Civil Engineering

Özet:
Bu çalışmada; düzlem çerçeve sistemlerde birinci ve ikinci mertebe limit yükün hesabı için geliştirilen bir yük artımı yöntemi verilmiştir. Yöntem sistemin özelliklerinden bağımsız olup, Bölüm 2.1'deki varsayımlar kısmındaki koşulları sağlayan bütün düzlem çerçeve sistemlere uygulanabilmektedir. Yöntemin [1]'de verilen yöntemden farkı plastik kesit oluşumlarını izleyen analizlerin, özel bir yazılıma gerek kalmaksızın doğrusal hesap yapabilen yazılımlar kullanılarak yapılabilmesi, oluşan plastik kesitlerdeki akma koşullarının da plastik dönmeye ait eğilme momenti tesir çizgisi yardımı ile ayrık olarak denge denklemlerini de içerecek şekilde yazılabilmesidir.
Yöntemin geliştirilmesinde, malzemenin elastoplastik davranışı ve geometri değişimlerinin denge denklemlerine etkisi gözönüne alınmıştır. Birinci mertebe limit yükün hesabında sadece eğilme momentinin akma koşuluna etkisi gözönüne alınmıştır. İkinci mertebe limit yükün hesabında ise eğilme momenti ile birlikte normal kuvvetin akma koşuluna etkisi de gözönüne alınmıştır.
Verilen yöntemde ikinci mertebe limit yükün hesabı için sistem sabit yükler ve bu yüklere karşı gelen, denge denklemleri ile yeterli yaklaşıklıkta tahmin edilebilen sabit normal kuvvetler altında hesaplandığından, geometri değişimlerinin denge denklemlerine etkisi doğrusallaştırılabilmektedir.
Sistemde ilk adımda henüz plastik mafsal bulunmadığı için, dış yüklerin yük parametresi birim değer alınarak SAP2000 programı yardımı ile analiz edilmiş ve plastik mafsalın oluşabileceği her kritik kesit için araştırma yapılmıştır. Bunun sonucunda ilk plastik kesitin oluştuğu yer saptanarak yük parametresi elde edilmiştir. 
Plastik mafsallardaki veya plastik kesitlerdeki eğilme momentleri, çerçevenin önceki adımda plastik mafsal oluşan noktalarında tanımlanan adi mafsallardaki birim mafsal dönmeleri ve dış yükler için ayrı ayrı yapılan hesaplar sonucunda bulunan eğilme momentlerinin süperpozisyonu ile elde edilmiştir. Birim mafsal dönmelerinden meydana gelen birim dönmelerden oluşan eğilme momenti tesir çizgisi kavramından yararlanılarak elde edilmiştir. Hesaplarda birim mafsal dönmeleri için yani deplasman yüklemesi için hesap yapmak yerine, birim mafsal dönmelerinden oluşan ankastrelik uç kuvvetleri dış yük olarak tanımlanmıştır. Sisteme tanımlanan dış yükler için SAP2000 programı yardımı ile analiz yapılarak plastik mafsalın birim dönmesine karşılık gelen moment diyagramı elde edilmiştir.
Plastik mafsalların oluştuğu kesitlerde akma koşullarını sağlayan denklemler birim dönme yüklemelerinden ve dış yüklerden elde edilen kesit zorlarına bağlı, ayrık olarak yazılmış ve plastik mafsal dönmeleri hesaplanmıştır. Her adımda elde edilen yük parametresi ile akma koşulunu sağlayan dönmelerden oluşan kesit zorlarının birleştirilmesi ile yeni adıma ait kesit zorları diyagramları oluşturulmuştur. Oluşan her plastik mafsal için işlemler tekrarlanarak sistem mekanizma durumuna gelene kadar veya ikinci mertebe limit yük hesabında sistem denklem takımının stabilitesi bozuluncaya kadar işlemlere devam edilmiştir.
Sistemin mekanizma durumuna kadar her adımda bulunan yük parametreleri toplanarak toplam yük parametresi elde edilmiştir. Bu yük parametresine karşılık gelen sistemin yani ikinci mertebe limit yüke ait moment diyagramı elde edilerek işlem sonlandırılır.
Son bölümde verilen sayısal örnekler ile yöntemin birinci ve ikinci mertebe limit yüklerinin hesabının nasıl yapılacağı ayrıntılı olarak açıklanmış ve literatürden örneklerle doğrulanmıştır.

Abstract:
The analysis and design methods of non - linear structures gain much importance due to the fact that these methods yield both more accurate and more economical Solutions. 
The elastic - plastic design methods enable engineers to design more slender structures by taking advantage of the load carrying capac- ity of the material beyond the elastic limit. In this case however, the second - order effects due to the geometrical changes become significant. Therefore, the development of methods to include both the material nonlinearity and the geometric nonlinearity is of great importance.
As the gravity and lateral loads are increased starting from the initial State, plastic sections form at locations where the intemal forces reach the ultimate values defined by the yield condition. After the formation of a plastic section, finite plastic deformations occur while the state of intemal forces remains on the yield curve.
In this thesis, a new load increment method has been given for the calculation of first and second order limit load of plane frames, assuming the plastic hinge concept. The load increment method is applicable to all plane frames under the certain assumptions. The main differences between the methods developed here and the method given in [6] is that the given method doesn't need any special software and the yield conditions can be written seperately and also they include the equilibrium equations indirectly. Moreover, the yield conditions have been directly obtained by revised Muller Breslau principle in terms of plastic hinge rotations [8].
In the method, elasto-plastic behavior of the material and the effect of geometrical changes on equilibrium equations have been considered. The effect of axial force on yield condition is neglected in the first order limit load analysis while the effect of those in second order limit load analysis is considered. The bending moments in any load increment step are obtained by superposing the bending moments due to the external loads in un-hinged frame and due to the unit rotations of plastic hinges developed until the current load increment step. According the Muller Breslau principle, the bending moment diagram due to the unit hinge rotations is equal to the influence line of bending moments due to the unit rotation of the frame. In this thesis, Muller Breslau principle is revised by substituting an element load vector for the unit hinge rotation. The fixed end forces due to the unit hinge rotation are obtained from element stiffness matrices. In SAP2000, fixed end forces can not be defined externally, opposite sign of those have been given as nodal forces. Therefore a correction of bending moments at the element with plastic hinges is necessary.
When the external load parameter reaches the ultimate value the structure collapses as it transforms into the mechanism state, according to the first order theory or it collapses through the loss of stability according to the second order theory. These load parameters will be referred to as the first or second - order limit load, herein. 
In some cases, the structure may be considered as being collapsed due to the excessive deflections and plastic deformations or the rupture of critical sections, as especially is the case for reinforced concrete structures.
The second - order, elastic - plastic analysis methods generally have two practical goals:
1-	to determine the factor of safety of an existing structure using the second - order limit load (or the collapse load) as the basis for comparison, 
2-	to develop design methods which are based on the collapse load criterion.
In both cases, two different approaches can be followed :
1-	The structure is analyzed under the proportionally increasing gravity loads,
2-	The analysis is carried out for constant gravity loads and in¬creasing lateral loads, i.e., for non - proportional loading,
The aim of this study is to develop a method of load increments for the determination of first and second - order limit load of structures subjected to increasing gravity and lateral loads in first order limit load or constant gravity loads and increasing lateral loads in second order limit load. The method accounts for the non - linearities caused by the geometrical changes and the elastic - plastic behavior of material. The effect of the axial force on the yield condition in the second order analysis.
Although the method is developed essentially for non - proportional loading in second order analysis, it is applicable to the analysis of structures subjected to proportional loading as well.
The following assumptions and limitations are imposed throughout the study :
1-	The stress - strain diagram of the material is assumed to be linearly elastic - perfectly plastic
2-	Non - linear bending and axial deformations are assumed to be accumulated at certain sections which are defined as plastic sections, while the other portions of the structure remain elastic. This assumption is the extension of the classical plastic hinge concept.
3-	The effect of the shear force on the yield condition is ignored. Hence, for plane structures, the yield condition is expressed in terms of the bending moment and the axiall force. Further, a piecewise linear approximation is made for the yield curve, i.e., this curve is assumed to be composed of line segments.
4-	The first and the second - order theory is adopted, i.e., the equilibrium equations are formulated for the deformed configuration while the effect of geometrical changes on the compatibility equations are ignored.
5-	Changes in the direction of loads due to the deflections are assumed to be negligible.
6-	For the sake of simplicity, the plane structures loaded in their planes are dealt with.
The structure is composed of prismatic members with constant axial forces. The members which do not satisfy these requirements can be divided into smaller segments for which the above conditions are approximatesly satisfied.
In the application of the conventional elastic - plastic analysis methods for the determination of second - order limit load, several difficulties arise as outlined in the following :
1-	The behavior of the structural system between the formation of two successive plastic sections is not linear due to the second - order effects. Therefore, the load level at which a plastic section forms can onıly be determined either by successive approximations or by sufficiently small load increments.
2-	The behavior of the structural system changes due to the formation of plastic sections. Therefore, at each step of the analysis, the stiffness and loading matrices of the structure should be revised to account for the last plastic section. Moreover, the revised set of equations must be solved again to determine the unknowns and the intemal forces.
3-	As the loads are increased, the intemal forces at the plastic sections continuously change and must be controlled to satisfy the yield condition.
In the proposed method, the structure is analyzed under factored constant gravity loads and increasing lateral loads, i.e., the non - proportional loading is adopted.
This approach is considered more realistic as compared with the analysis for proportional loading. Because, the probability of change of lateral loads whieh generally represent the wind and earthquake effects is more than that of the gravity loads which include the un- altered dead loads.
In second order analysis, limit load calculations are performed for stationary dead loads and monotonically increasing lateral forces depending on a load parameter. In the calculation of second order limit load, the effect of axial forces on yield condition is considered together with the bending moment. In the analysis, equilibrium equations are nonlinear as the axial forces are changing depending on load increments. As, the dead loads are stationary and the sum of the axial forces at the story remains unchanged, the axial forces are assumed to be stationary at any load increment. Thus the behavior of the frame is assumed to be linear between the developing plastic hinges. On the other hand, the yield conditions can also be idealized by line segments. Thus, the load increment method can be applicable to the second order limit load analysis under the same assumptions which are valid for first order limit load analysis.
The load increment procedure is ended when the frame transforms into total or partly mechanism in first order limit load analysis or the determinant of global stiffness matrix has a negative value in second order limit load analyse.
Five illustrative examples from the literature are given for explanation and application of the proposed load increment method.