Elastik zemine oturan sonlu kirişlerin deneysel incelenmesi

Abstract

ÖZ E T Bu çalışmada Winkler elastik zeminine oturan, sonlu uzunlukta betonarme ve ahşap kirişlerin davranışı incelen miş, orta noktadan etki eden- tekil yükün limit değeri a raştırı lmış tır. Ayrıca kirişin davranışına ve limit yüke, farklı malzeme ve boyutların etkisi incelenmiştir. Mühendislikte Winkler hipotezine dayanarak iyi sonuç lar veren bazı önemli problemler vardır: Gemi kaburgaları, uzay araçları, bina döşemeleri ve köprü tahliyeler indeki ızgara sistemler, bir ve iki doğrultuda sürekli temeller, dönel kabuklar, yatay yük etkisindeki düşey kazıklar vb. Deneysel ağırlıklı bu çalışmanın birinci bölümünde ça lışmanın konusu açıklanmış, konu ile ilgili çalışmalar ve rilerek yapılan varsayımlar belirtilmiş ve çalışmanın amacı açıklanmıştır. İkinci bölüm, konu ile ilgili kuramsal çalışmaya ay rılmıştır, önce problemin bilinen diferansiyel denklemi çı karılmış, daha sonra da bu çalışmada kullanılmak üzere, or tasından tekil yüklü sonlu kirişlerin çözümü yapılmıştır. Bu bölümde ayrıca betonarme ve ahşap kesitlerin yük altın da davranışı hakkında özet bilgi verilmiştir. Üçüncü bölümde, yapılan deneysel çalışmada kullanılan ahşap ve betonarme kiriş ile elastik mesnet yerine geçen lastik takozların mekanik özellikleri saptanmış, deneyin yapılışı ve yapılan ölçümler açıklanmıştır. Deneysel çalış ma, çeşitli boyut ve donatıda 16 betonarme ve 6 ahşap kiriş ile yapılmıştır. Deney düzeni İ.T.Ü. Yapı Laboratuarındaki 20 ton kapasiteli eğilme çerçevesinde oluşturulmuştur. Bu kirişlere orta noktadan tekil yük uygulanırken oluşan çök me ve şekil değiştirme değerleri ölçülmüş, çatlak oluşması izlenmiştir. Dördüncü bölümde kuramsal ve deneysel sonuçların kar şılaştırılması yapılmıştır. Tüm kirişlerin deneyler sonucu elde edilen moment-eğrilik değişimi ve çökme eğrileri in celenmiştir. Ayrıca deneysel sonuçlardan yararlanarak bi rim deformasyon ve gerilme yayılışı ile ilgili şekiller verilmiştir. Betonarme kesit hesaplarında taşıma gücü yön temi, ahşap kesitlerde ise Brochard tarafından kesit için deki gerilme yayılışı için verilen bağıntılar kullanılmış tır. Sonuçlar bölümünde çalışmada elde edilen bazı sonuç lar ve önemli bulgular özetlenmiştir. - ııı -

SUMMARY EXPERIMENTAL STUDY OF FINITE LENGTH BEAMS ON ELASTIC FOUNDATION In this work, the behaviour of reinforced concrete and timber beams of finite length, on elastic Winkler foundation has been analized. The limit value of singular load acting on the beams at the middle point has been ex amined. In addition to this, the effect of using different materials1 dimensions and the effect of different elastic foundation modulus on the behaviour of beam and on limit load have been examined. The problem of beams on elastic foundation has been examined firstly by Winkler. In `Winkler` foundation it is assumed that the pressure of foundation is proportional to the deflection and that the foundation has been composed of independent springs. In engineering applications, there are some important problems which can be handled succesfully by means of Winkler hypothesis : the frames of ships, space vehicles grid systems at plates and bridges, continuous foundations in one or two directions, shell of rotation arid perpendi-' cular piles under the effect of horizontal load are some examples. Mathematical solution of problems of beams on elastic foundation is highly tiresome and difficult, but today's widespread use of computers has greatly eliminated this difficulty. Calculation of reinforced concrete beams ac cording to the elastic theory is being the use, of ultimate strength design is increasinaly becoming wide spread. This method which takes the actual behaviour of reinforced concrete section into consideration has emerged from the idea of taking as starting point ultimate strength against failure. In the first chapter, the problem has been introduced the literature related to the subject and hypothesis have been given and the aim of the work has been explained. In the experimental work related to elastic founda - tion modulus, hypothesis made in obtaining this modulus have been indicated; examination of the variation of elastic foundation modulus under the effect of foundation surface and depth and various tables have been given. In the works which examine the behaviour of finite, semifinite and infinite beams the solution of problem has been made under different loading conditions, and related - iv -tables, and curves have been given. Some information has been given especially on the contents of works analizing the finite beams, Lastly the works examining the beam on elastic and viscoelastic foundation excluding the Winkler foundation have been referred. The second chapter has been assigned to the theoretical works related to the subject. Firstly the differential equation of the problem has been obtained. Then, the solu tion of finite beams loaded in the middle has been obtained to be used in this work. The characteristic beam length is defined as L. İÜ ' where EI is flexual rigidity, b is bC the width of foundation surface width and C is elastic foundation modulus. Eh3, 1/4 L » -^r- at constant uniform rectangular cross section of h height. Foundation reactions depending on characteristic length have been give by Hetergi. By making use of relation and the definition of relative stiffness X. i L Loading of beams has been obtained by dimensionless variables. Then, the equations which give shear strength and bending moment have been obtained by integration. Maxi mum bending moment is at the middle and from here the equa tion for limit load 4M ShX.¦ SinX P = - H L ChX - CosX has been obtained. Then the variation of the value of col lapse has been obtained, by means of Q = C«W. The deflection in the middle of the beam is ChX-* CosX + 2 Wmiddle 2LC ShX + SinX The deflection at the end point is 2P ChX/2 CosX/2 W endpoint ` LC shx + sin x - v -When these two equations have been examined. Wendpoint ` ° ' Wmiddle = ~ for the values of X approaching infinity W_,._. = W endpoint middle ~ CL for the values of X approaching zero. Also the beams on elastic foundation are considered as some beam short beam when X < tt/4 medium length beam when n/4 < X < -rr and long beam when X > u. The materials of the beams used are either reinforced concrete or timber. The work on distribution of stress, unit elongation, variation of in moment curvature and cracking in reinforced concrete beams have been summarized and the main aspects of the method of ultimate strength have been given for the reinforced concrete beams equivalent rectangular distribu tion of stress in the compression zone has been adapted. In the calculation of deflection moment of inertia of cracked beam has been used in each step. Unit deformation distribution of stress has been analized and investigation of deflection, examination of variation of moment curvature of timber beam cross sections have been realized by making use of the definitions given by Brochard. In the third chapter, the mechanical feautures of timber and reinforced concrete beam and rubber wedge used in the experiment, have been determined and information about experimental procedure and measurements has been given. The properties and grading of the materials such as sand, pepple, gravel and cement used in the production of reinforced concrete beam have been determined. The composi tion proportions necessary to obtain C 16 concrete has been given. Mechanical features of the equipment used have been determined by the tensile test. In this way, 8 elements of 10 x 20 cm. cross-section and 240 cm. length and 8 elements of 15 x 30 cm. cross-section and 240 cm. length making a total of 16 elements of reinforced concrete have been produced. Compressive strength, elasticity modulus and unit deformation of each type of beam have been estab lished by using 36 cylindrical samples obtained during the production of these beams. - vx -3 elements of 4.6 x 9.4 cross-section 230 cm. length and 3 elements of 9.3 x 9.5 cross-section and 230 cm. length making of total 6 elements have been used as timber beams. The tensile and compressive strength, the elasticity mo dulus and unit deformation values have been determined by using samples obtained from sections of each beam element. In the experiments, 35 pieces of rubber wedges of 9x9 cm. cross-section and 22 cm. length which are produced in Adapazarı Good Year Rubber Factory have been used as the foundation elements. The elastic foundation modulus of rubber wedges have been determined by the loading experiment. Loading caps of 5 cm., 10 cm., 15 cm., 20 cm., and 30 cm. have been used. Different foundation modulus have been used for each foun dation width, thus obtaining in accordance with the experi mental beam widths. In this way, the effect of using dif ferent modulus dimensions on elastic foundation modulus has been eliminated. Finite beam experiments have been made in the bending frame of Î.T.Ü. Building Laboratory by using experimental elements of timber and reinforced concrete beams and rubber wedges. A rigid foundation has been obtained by means of three elements, of profiles. Rubber wedges made of 7-15 elements have been placed on this foundation. The plate has been placed under the concentrated load acting on the beams on this foundation. A concentrated loading' ât the middle point has been realized by using hydrolic loader. Each loading step has been taken as nearly one eighth of limit load which is found by theoretical solution. In each loading step ; 1. Measurements of shape changes in the central area have been made using a deformeter of 0.0008 mm. sensitivity. 2. Measurements of deflection valves have been made using comparators of 0.001 mm. sensitivity. 3. Cracks have been traced with a crack binoculars. In the fourth section, the results obtained by the ex periments have been given together with theoretical results. Equivalent rectangular distribution of stress in the cross sections of reinforced concrete beams has been as sumed, and by making use of balanced equations. Theoretical cracking moment and limit moment valves have been found. Experimental unit deformation and stress distribution diagrams have been drawn by making use of deformation valves measured by deformeter in the middle section. - vii -Moment valves have been obtained by making use of these valves and mechanical features of beams. M-lC diagrams have been drawn by means of theoretical and experimental moment and curvature valves. Theoretical and experimental M-lC diagrams have been drawn by means of elastic, plastic and limit moment valves and the corresponding curvature valves in timber beams. From the solution of finite beam loaded singularly from the middle point. theoretical deflection valves have been obtained by making use of computer and are given together with the curves of experimental valves. The main results of this study can be summarized as follows : 1. The experimental deflection valves obtained for the reinforced concrete beams have been obtained are close to the theoretical valves given for the middle area. However in the edges, they are different. In the timber beams, valves, very close to the theoretical solutions, have been obtained along the full length of the beam. 2. Since the entire surface of the beams on the founda tion has been supported, the curvature hasn't reached high valves even when the ultimate strength is reached. It can be said that this condition arises from the friction effect of the foundation which hasn't been considered. 3. The character of M-lC diagrams also changes accord ing to the ratio of reinforcement ratio of reinforced concrete beams. C valves are bigger, for low ratio of rein forcement beams and, K valves are smaller for high ratio of reinforcement beams. 4. In the reinforced beams with the longer cross-sec tion values suddenly approach infinity in the middle areas of the beams, whereas in smaller cross-sections this tendency is less pronounced. 5. M-K` diagrams in the timber beams which have been obtained theoretically differ from the experimental diagrams. In the diagram which has been drawn according to the ex perimental results fC values increase linearly. 6. In the experiments made with reinforced concrete beams, cracks have been traced and it has been observed that the vidining of the crack is less. 7. The load and deflection valves which affect the foundation in all areas of the cross -section, have been ob served to remain in the elastic area of load deflection curve which has been obtained during the pre-experiments made with rubber wedges. - vnx -8. The controls have shown that the fracture of beams does not arise from shear, bending or punching effects. 9. The experimental results are in a better agreement to the theoretical results of timber beams of 9x9 cross sections than the 9x4 cross-sections. - ix -