Gövdesi boşluklu çelik I-kirişlerin incelenmesi

Abstract

Bu çalışmada, gövdesi boşluktu çelik I-kirişlerin plastik kurama göre hesabında yararlanılabilecek yöntemler üzerinde durulmuştur. Çalışma altı ana bölümden oluşmaktadır. Birinci bölümde, ele alınan problem tanımlanmış, bugüne kadar yapılan çalışmalardan ulaşılabilenler, varsayımlar ve bulgularla özetlenmiştir. Deneysel çalışmalarda ele alınan kiriş tipleri ve deney sonuçlarına etki eden faktörler üzerinde durulmuştur. Son olarak bu çalışmanın amacı ve kapsamı verilmiştir. İkinci bölümde, dışmerkez gövde boşluktu çelik I-kirişlerin elastik ve plastik kurama göre hesabında karşılaşılan önemli problemlerden biri olan, kesme kuvvetlerinin başlıklara dağılımı problemi incelenmiştir. Birçok çalışmada problem basite indirgenerek, kesme kuvvetinin yalnızca üst başlık tarafından taşındığının varsayılmasına karşın, burada problem daha gerçekçi bir yaklaşımla ele alınarak analitik olarak çözülmüştür. Çözümde, kirişin düşey olarak yüklenmesi durumunda üst ve alt başlıkların aynı yer değiştirmeyi yapması koşulundan (uygunluk koşutu) yararlanılmıştır. Üçüncü bölümde, dışmerkez gövde boşluklu çelik I-kirişlerde göçme incelemesi plastisitenin alt ve üst sınır teoremleri yardımıyla yapılmıştır. Önce plastisitenin temel kavramları üzerinde durulmuş, sonra göçme hesabında yararlanmak üzere tek simetri eksenli çelik I ya da T-kesitlerin bileşik eğilmesinde akma eğrileri boyutsuz kesit parametreleri cinsinden elde edilmiştir. Gerçek akma eğrilerinin yanında, göçme yükü hesaplarında oldukça kolaylık sağlayan tek simetri eksenli I-kesitler için sandviç kesit modeli geliştirilmiştir. Bu yoldan bir gövde boşluklu I-kirişte alt ve üst sınır teoremlerine göre göçme yükleri hesaplanmış ve ilgili bağıntılar verilmiştir. Göçme yükünün kirişle ilgili parametrelere göre değişimi eğrilerle verilmiştir. Dördüncü bölümde, dışmerkez boşluklu çelik I-kirişlerin boşluk bölgesinin taşıma gücü diğer bir yoldan incelenmiştir. Bu yöntemde sekonder eğilme momentlerinin etkisi dikkate alınmış, göçmeye boşluk bölgesinde alt ve üst başlık elemanlarının uçlarında oluşan toplam dört adet plastik mafsalla ulaşıldığı varsayılmıştır. Burada, önce boşluklu kesitin eğilme momenti ve kesme kuvveti taşıma gücü hesaplanmakta, daha sonra bir etkileşim diyagramı ile boşluktu kesitin bileşik kesit zorları altında güvenli olup olmadığı kontrol edilmektedir. Beşinci bölümde, bu çalışmada geliştirilen bağıntıların uygulanılışını göstermek ve mevcut literatürdeki kuramsal ve deneysel sonuçlarla karşılaştırmak amacıyla sayısal örnekler üzerinde durulmuştur. Altıncı bölümde, bu çalışmada elde edilen bulgular özetlenmekte, konuyla ilgili ileride yapılabilecek çalışmalardan söz edilmektedir.

An important problem m the design of high-rıse buıldings is depth restrıctıons due to economical, architectural and functional reasons. Since, steel I-shaped beams vvith web openings are extensıvely used in such buildings to allow the passage of utılıty ducts and piping, the height of the floors, which greatly affects the construction cost can sıgnifıcantly be reduced. For small spans this may not be possible, because the beam will be too shailow for the size of service. Beams with web openings are frequently used as Castella beams having a greater strength and rigidity than beams with equal sectional area and without openings. Another use of open-web beams can be defined as beams with arbitrarily placed openings. While considerable material saving can be obtained in Castella beams, the second case may lead to sıgnıficant loss in strength and rigidity at the opening sections. On the other hand, structural behaviour of such beams is rather different from beams without openings. Numerous theoretical and experimental investigations have been made to predict the elastic and post-elastic strength of such beams. in steel I-beams vvith web openings, shear produced along an opening modüle dominates the overall beam behaviour. These shear forces create secondary bending moments at the ends of the T-shaped elements of the bottom and upper chords. Öne of the aims of the present study is to develop a rapid and simple method to take secondary bending moments, vvhich is generally called the Vierendeel effect, into account in the plastic design of I-beams at unreinforced and reinforced, concentric ör eccentric web openings. in spite of several advantages of opening-up a beam, these openings reduce both shear and moment capacities of the beam at opening sections. The proposed methods mainly assume that the material is rigid-perfectly plastic vvıthout strain-hardening and contraflexure moment point occurs in the middie of the opening length. Since the determination of secondary bending moments depends on the shear produced along an opening, interaction between moment and shear has been controlled by von Mises yield criterion. On the other hand, interaction betvveen primary bending moment and shear forces has also been taken into account at sections with openings by a piecewise linear approach. Therefore, not only the effects from primary bending of overall system, but also the effect of secondary bending moments are taken in the calculations. Outstanding examples are solved to show the importance of secondary bending moments on the ultimate strength of I-beams at opening sections. The results obtained by the proposed methods are compared vvith available experimental ones. Although the opening up of a rolled beam increases its plastic section modulus and the rigidity of the beam, a considerable reduction in strength at opening sections due to secondary bending effects, can be expected. The ultimate strength of an opening segment must be calculated accurately to determine vvhether reinforcement is necessary. xiii In the first chapter, some information is given about the previous studies on the elastic and plastic analyses of I-beams with web openings. For each study, the method used during the analysis, assumptions and results are evaluated. Finally, the scope and the aim of the thesis are given. Since I-shaped beams with web openings are frequently used in steel or steel- concrete composite framed construction to obtain a considerable material saving, this very wide use of such beams requires more analytical and experimental investigations in order to provide a faster and reliable design procedure. Clawson and Darwin [14] presented an experimental investigation on the behaviour of composite beams with rectangular web openings. They have developed moment-shear interaction curves for a limited amount of particular beam sizes and opening configurations. Further, beams with high moment-shear ratios fail by general yielding in the steel below the neutral axis and crushing in the concrete. Beams with medium to low moment-shear ratios fail by the formation of plastic hinges in the steel below the opening, accompanied by a diagonal tension failure in the concrete slab. Another important result is that the test results indicate that the concrete slab significantly contributes to the overall shear strength of composite beams at web openings. Other studies have both analytical and experimental contribution to the present problem with some varying approximations. Although, it is possible to obtain rather exact formulae for the ultimate design of open-web I-beams [25], these formulae require cumbersome calculations. In this sense, this thesis describes specific design approaches for the post- elastic analysis of steel I-beams with web openings; secondary bending effects are considered under some simplified assumptions that lead us faster and sufficient results. Çelik and Özgen [36] developed some expressions for the plastic design of Castella beams by taking the secondary bending moments into consideration. Furthermore, the optimum sectional properties of I-beams with web openings have been investigated to get the maximum sectional values for moment of inertia, plastic and elastic section moduli for a given amount of material [51]. While many studies include the sectional calculations only, the effect of concentrated loads on the plastic behaviour of such beams is investigated for the different positions of concentrated loads on the beam [52]. Some design and construction principles are recommended based on the attained numerical results. Cho, on the other hand, investigated the slab behaviour in composite beams at web opening regions using the truss analogy and obtained satisfactory results with respect to the experimental ones [27]. In the second chapter, determination of shear force distribution in steel I-beams with eccentric web openings is investigated. The problem is reduced to the determination of shear deflection coefficients (ks) of each flange members at the opening segment. As well known, the shear deflection coefficient of any section is defined as k.-4((^M ^7 Here A is the total area of the section, Ix is the moment of inertia, Qx is the area moment of each segment according to the gravity center of the section and t is the thickness of each segment. Thus, the shear forces created by the unit displacements xiv of the flanges are calculated and shear force distribution is obtained by equating the displacements of top and bottom elements. This procedure gives the following expressions: where V,, V b :top and bottom shear forces, a0 : opening length, /,, Ih : moment of inertia of top and bottom flanges, A,,Ah :areas of top and bottom sections. In the third chapter, collapse loads for steel I-beams with eccentric and concentric web openings under the effect of concentrated loads are determined by lower and upper bound theorems of limit analysis. The load may either be located on the top chord or far from the opening region. The most probable collapse mechanism is obtained as four-hinge mechanism at an opening module. The first part of this chapter mainly deals with the fundamental principles of structural plasticity. Although, there are many studies on the yield conditions of symmetrical steel I- sections, bending moment(A/)+axial force(A0 yield curves for monosymmetncal I- sections and for T-sections have not widely been examined. The derivation of a yield surface for steel structural members is a part of an investigation of post elastic behaviour of steel. Herein, this behaviour has been approximated by assuming the material to be rigid-plastic. Analytical expressions for the yield conditions of bar elements subjected to combined axial force and bending moment are obtained in a nondimensional form which provides simplicity in design. Since exact yield surfaces obtained analytically or experimentally are generally nonlinear, yield surface linearization has been carried out to provide simplicity in the plastic design. On the other hand, a piecewise yield condition is very useful from the mathematical point of view. Indeed, if the yield surface consists of plane faces, as long as the stress point remains on a given plane, the stress vector at yield retains the same direction normal to that plane and the yield mechanism does not change. In more complex problems, the determination of exact yield surface is complicated and needs cumbersome calculations. In many practical cases, the effect of shear force is very small. On the other hand, there are many ways to consider the effect of shear on the plastification moment of a section; this is done here by reducing the thickness of the web as given in the Turkish Code "Rules For Plastic Design of Steel Structures" (TS4561). In the fourth chapter, an alternative design method based on the plastic theory is proposed. The method may be adopted to unreinforced, either eccentric or concentric web openings. The opening is considered under the effect of a positive bending xv moment M and a shear force V at the centerline of the opening modüle. Here, the parts of the beam above and belovv the opening will be named as top and bottom tees respectively. No axial force acting on the section is considered. Secondary bending moments acting on the top and bottom tees are represented by M, and Mh vvhile the shearing forces are taken as V', and Vb, As in many analyticai studies on open-web beams, it is assumed that the point of contraflexure is located at the center of the opening on both of the top and bottom tee elements, unless the opening is not subjected to other external loads. Some experimental results given by Clavvson and Darvvin [1] indicate that the assumptton involving the contraflexure moment point may not be valid near the ultımate load. Basıc design procedures involve determination of maxımum nominal flexural capacıty, maxımum nominal shear capacity and interaction of flexure and shear. The maximum moment capacity Mp can easily be calculated by the well-known conventional methods taking ınto account the loss of material within the vveb, The maximum nominal shear capacity is based on a four-hinge mechanism of the opening modüle. Maximum shear capacity of the beam at opening sectıons ıs composed of the individual shear strengths of both top and bottom tees. If the tees are considered as geometrıcally equal, they contribute to the total shear strength with the same amount Next, the problem is reduced to the determination of shear strengths of the bottom (V ^ ) and the top (Vpl) elements. There are several methods to obtain the shear strength of any cross section. in the present study, in order to provide sımplicity and rapid calculation, since the fîange thicknesses of each stub are small in practice relative to the stub depths, the contribution of the flanges to the secondary bending moments is also small and can be neglected. in addition, the normal and shear stresses in the vveb of a stub are taken as uniform through the stub depth ignoring local equilibrium. When normal and shear stresses present at a section, as vvell known, these stresses are limited by the von Mises yield criterion. Finally, the reduced shear strength of top and bottom tee members due to secondary bending effects are attained as follovvs: TP<=VP^*. V^T^ t b where s, =3(s, Ia0 )2 and eb =3(sh Ia0 )2 are the coefficients depending on the sectional and opening properties. The variation of the shear strength reduction at opening sections due to the secondary bending moments is nonlinear vvith respect to the dimensionless ratio of a/j. A yield surface for bending moment and shear force at an opening section is handled using a second order parabola interaction curve vvhich relates the design moment and shear strengths Md and Vd vvith maximum moment and shear capacities Mp and V'p. Although this approxımatıon represents the vveak interaction betvveen flexure and shear and provides a good agreement vvith available test results given by Darvvin and Donahey [5], it is further assumed here that this interaction can be taken by means of a bilinear curve vvhose intersection points on the second order parabola. These linear interaction curves provide a faster and simple proportioning in the design of open-vveb I-beams. On the other hand, a different interaction curve for bending moment and shear force is given in [10]. xvı On the other hand, a brief evaluation of the existing codes on Steel I-Beams with Web Openings is given in this chapter. A new proposed specification is introduced. This specification covers the design of composite and noncomposite beams with web openings. It follows load and resistance factor design (LRFD) philosophy. Basic design procedures involve determination of maximum nominal flexural capacity, maximum nominal shear capacity and interaction of flexure and shear. The maximum nominal flexural capacity is calculated using standart strength procedures for both composite and noncomposite sections. The maximum nominal shear capacity is based on a simplified four-hinge mechanism, with one hinge at each corner of the opening. The general design approach is identical for composite and noncomposite members with or without opening reinforcement. The design procedures apply only to compact sections. Additional criteria are applied to ensure ductile behaviour and performance in accordance with the design expressions. The design procedure is valid for rectangular and circular openings. Criteria for placement of concentrated loads and positioning of openings are included. In the fifth chapter, a number of examples are carried out to understand the use of developed design formulae. Furthermore, the results of different methods of analyses are compared with the ones obtained by linear Finite Element solution and existing experiments in literature. Both of the proposed design methods give conservative results with respect to the experimental ones. In the last chapter, the conclusions of this thesis are outlined. The following results may be deduced from this study: o Web openings in steel I-beams highly reduce the strength of such beams due to secondary bending moments. o Interaction between primary moment and shear at opening sections is weak. In other words, the moment capacity at an opening section is relatively unaffected by the shear force, until the shear approaches the shear capacity of the section. o The Moment/shear force ratio at an opening section has a great effect on the mode of beam failure. Therefore, the present method is valid especially for relatively small M/V ratios. o If the opening dimensions are large enough with respect to beam dimensions, the strengthening of the vicinity of openings is strongly recommended to increase the shear and moment capacities as well as the rigidity of such beams. o The proposed design procedure provides rapid and practical solutions for the structural engineers. o The best position for an opening depends on the loading type and boundary conditions of the beam. Generally, noncomposite I-beams can accept openings in their webs fairly easily if the opening is at mid-span. xvn Near the support, where the shear is high the same size opening may require an unreasonable amount of stiffening or lesser opening. The methods proposed here may easily be extended to the plastic analysis of steel-concrete composite and cold formed steel beams with web openings.