Available online at www.sciencedirect.com ScienceDirect Procedia Engineering 172 (2017) 883 – 890 Modern Building Materials, Structures and Techniques, MBMST 2016 Research in redistribution of bending moments in the beams of reinforced concrete early loaded Dolny Piotra,*, Kamiński Krzysztofa Warsaw University of Technology, Wydział Budownictwa, Mechaniki i Petrochemii, ul Łukasiewicza 17, 09-400 Płock, Poland a Abstract This paper presents results of a study of a reinforced concrete beam, simulating a double-span beam, early loaded The beam was designed with limited redistribution of bending moments from the support to the span The tested element was loaded at the operational load level, after the concrete had reached 80% of its predicted compressive strength In a two-month test it was observed that the distribution of bending moments changed in the direction opposite to what was expected, i.e from the span to the support Moment redistribution in the tested beam leads to a reduction of security level in the middle support section of the beam © 2017 Published by Elsevier Ltd This is an open access article under the CC BY-NC-ND license © 2016 The Authors Published by Elsevier Ltd (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the organizing committee of MBMST 2016 Peer-review under responsibility of the organizing committee of MBMST 2016 Keywords: redistribution of moments; reinforced concrete beam; the distribution of internal forces; plastic deformation Introduction Materials forming concrete structures demonstrate nonlinear elastic stress-strain relationship To simplify calculation, this nonlinear relationship is often ignored in design of concrete structure (linear elastic method) More detailed calculations would require examination of a nonlinear activity of material (nonlinear and plastic methods), or would need checking the possibility of plastic hinge (linear elastic method with limited redistribution) All these methods lead to changes in the distribution of internal forces in the particular section with reference to calculations of a linear elastic analysis method The differences between those forces are called the redistribution of internal forces Distribution of bending moments is understood by different authors, as a change in the distribution of internal forces: x after a plastic hinge has been formed at Ultimate Limit State of construction, cf.: [3,8], x after flexural stiffness along cracked sections has been changed - at Ultimate Limit State and at an operational phase of construction, cf.: [6] 1877-7058 © 2017 Published by Elsevier Ltd This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the organizing committee of MBMST 2016 doi:10.1016/j.proeng.2017.02.096 884 Dolny Piotr and Kamiński Krzysztof / Procedia Engineering 172 (2017) 883 – 890 Linear elastic analysis with limited redistribution is the simplest method of approximation of linear elastic analysis forces to their actual values, taking into account plastic properties of concrete Bending moments calculated in linear static analysis are redistributed in a limited range from more loaded sections to less loaded sections This calculation is not complicated, but difficult to verify, which makes it controversial As stated by [7], linear elastic analysis with redistribution, also called plastic balancing of moment method [10] is performed by the same assumptions as the analysis of linear elastic method throughout the analyzed system excluding joints which provide redistribution moments Modification of moments in the selected joints causes a linear change in moment distribution in the remaining construction due to the necessity of maintaining an equilibrium level with respect to the applied load Application of the linear elastic method of redistribution of bending moments is only permitted after having checked the ultimate limit state in continuous beams, in which the relation between adjacent spans length is less than in the beams in nonsway frames and elements which are subject to bending, cf [5], such as one-way slabs which are regarded as multi-span beams This kind of analysis assumes a number of simplifications, authorized by [11] Fig Redistribution of bending moments in the beam continuous Knauff (2006) Supports not restrict freedom of rotation in continuous beams and slabs, and the system of loads on not transmitted elements is assumed as if for free supported elements, as long as the span of two adjacent spans is not greater than 30% When the element is loaded with evenly distributed load, it is recommended to use only two patterns of its distribution, either live loads are applied to two adjacent spans, or alternative spans, both combinations with dead loads on all spans The effect of shrinkage and the influence of temperature are generally ignored if dilatation intervals are not more than 30m Redistribution of moments is an important phenomenon and should be taken into account when designing multispan reinforced concrete beams Requirements for the safe creation of plastic hinge occur in various forms: to provide reinforcement steel of adequate ductility, to check the maximum angle of rotation of plastic hinge, or to check relative compression zone height None of the methods used for structural analysis allows for even a single plastic hinge Other methods of calculation imply diversification of safety factors and the use of plastic reserve in the statically indeterminate constructions, cf [5] [11] provides conditions under which this time-consuming determination of rotation angles of critical sections can be skipped, namely: the value of the redistribution coefficient δ be not less than (1) for concrete fck İ 50MPa, and not less than (2) for concrete fck >50MPa ߜ ͲǡͶͶ ͳǡʹͷ ௫ (1) ௗ ߜ Ͳǡͷ ͳǡ͵͵͵ ௫ ௗ Nomenclature δ x/d the ratio of redistributed bending moment and linear elastic bending moment the relative height of compression zone (2) 885 Dolny Piotr and Kamiński Krzysztof / Procedia Engineering 172 (2017) 883 – 890 [8] point out to two possible directions of redistribution of bending moments in the phase of structural damage, from the support to the span and from the span to the support The direction of redistribution depends on the order of scratches of the support and span sections, and the relation of their capacity (3,4) (Fig 2) The direction of redistribution is determined by the inequality (3) and (4) Fig Examples of redistribution of moments in the beam statically indeterminate under first load [8] ȁ ȁ ெ ெ ெ ெ ȁ൏ȁ ெ ெ ȁ ȁ ൏ ͳǡ (3) (4) Nomenclature MfI MfII MelI MelII span cracking bending moment support cracking bending moment nonlinear elastic span bending moment nonlinear elastic support bending moment Both directions of redistribution of forces is also described by [6], who analyzed the impact of changes in stiffness and deformability of joints on the redistribution of internal forces in a multi-frame structure In his example, he obtained a change of bending moments at the beams within from -20% to +13% using elastic links between beams and fixed shields (simulating semi-stiff joints) [1,2] obtained similar results, studying concrete beams reinforced with composite rods and strengthened by carbon fibre laminates It can be concluded from the referred literature that the distribution of internal forces in reinforced concrete structures may differ from the distribution obtained by linear elastic analysis Change in stiffness of the sections caused by scratches, creep of concrete, or plastic deformation of the reinforcing steel may result in modification of distribution of forces Research program 2.1 Materials The beam for the experiment was made from materials available locally in the Plock proximity: aggregates from Mława, sand from the Vistula River, CEM II/B-V and reinforcing steel – C class ductility, available from market stores In order to determine the progression of compressive strength of concrete, two series of cubic samples, 15 cm side, were made They were formed into steel forms, cured for days, removed from the forms, at the same time as the beam The samples of the first series were then cured in water at 20°C ±2°C - according to the requirements of the EN 206-1 The other series of samples were stored together with the beam, in the air-dry conditions, according to the PN- 886 Dolny Piotr and Kamiński Krzysztof / Procedia Engineering 172 (2017) 883 – 890 B 06250:1988 standard The samples stored in water were used to determine the class of concrete, according to EN 206-1 standard The class of concrete was C30/37 The test samples were cut from the same rods as the main reinforcement of the test beam Yield stress was determined as the upper limit of the yield Modulus of elasticity of steel was determined by approximation by line form a strain-stress (ε-σ) graph Summary of the results is provided in Table Table Properties of reinforcing steel Type of reinforcement Top Bottom The average modulus of elasticity (GPa) The average tensile strength (MPa) Ф 16 mm 229,6 557,2 Ф 12 mm 218,7 531,0 Ф 16 mm 283,7 542,3 Ф 10 mm 235,0 555,7 The modulus of elasticity of concrete was tested on cylindrical specimens of 30 cm height and a base diameter of 15 cm The samples were loaded to achieve a tension corresponding to the percentage of 28-day strength of concrete The modulus of elasticity of concrete, in a specified load range, was calculated from the values of stress and strain readings based on [9] standard The results are summarized in (Table 2) Table Modulus of elasticity of concrete Age of concrete (days) Stress range (%fcm,28days) Average modulus of elasticity of concrete (GPa) 14 0-36 26.6 0-54 24.1 0-40 28.4 0-60 26.1 0-40 28.0 28 150 2.2 Test stand The tested element was designed to simulate a half of a double span beam, statically indeterminate, of a span length equal to 3.80 m, loaded with two concentrated forces at distances of 1.05 m and 1.95 m measured from the roller support of the beam Concentrated forces were applied by using a rigid beam Only total force measurement was taken Further, the active load was placed on the overhang to measure a support bending moment With the help of a computer program, the force applied was adjusted to the overhang in order to compensate for rotation of the beam on the B support Additionally, a measurement of the reaction on the supports A and B was made in order to control the process in the beam, and to make possible the calculation of the internal forces in the tested element A schematic view of the beam and reinforcement bars is shown in Fig (in mm), whereas Fig presents a view of the beam on the test stand The test area was separated from the rest of the laboratory in order to minimize environmental changes around the tested element Changes in temperature and humidity were measured Dynamometers 1-4 (Fig 3) measured the forces of a load on the beam and support reactions, to the nearest 0.1 kN Sensors 6-17 (strain gauges) measured the vertical deformation of the longitudinal axis of the test element with an accuracy of 0.001 mm The readings were recorded by computer at fixed intervals, analyzed and filtered in accordance with the test program 887 Dolny Piotr and Kamiński Krzysztof / Procedia Engineering 172 (2017) 883 – 890 B A 75 1053 900 1503 75 598 150 150 455 15 10 450 11 450 13 12 450 450 14 450 505 16 17 1282 1o16 2o6 2o12 2o10 1o16 Fig Schematic view of reinforcement of beam Fig Test stand view 2.3 Distribution of internal forces Distribution of bending moments obtained from linear elastic method loaded by dead and live loads (two strengths of 32,7 kN each) are shown in (Fig 5a) The calculations took into account the weight of component and accessories (traverse, sensors) In further calculations, the resulting bending moments underwent redistribution by a δ=0,90 factor The results are shown in (Figure 5b) The span moment was calculated by loading the freely supported beam, with concentrated forces and the moment, was calculated from the limited redistribution of bending moment 888 Dolny Piotr and Kamiński Krzysztof / Procedia Engineering 172 (2017) 883 – 890 linear elastic analysis mathod linear elastic analysis mathod with limited redistribution (d=0,90) 30,1 31,2 38,6 34,8 27,1 29,0 Fig Bending moments in the beam: a the analysis of linear elastic method b the linear elastic analysis with limited redistribution 2.4 Bearing capacity design Sections of reinforced concrete were designed for bending moments calculated with redistribution in accordance with the graph shown in (Fig 5b) Rods and cross sections, characteristic of the beam, are shown in (Fig 6b) Support reinforcement was composed of two rods, 12 mm in diameter, and additional rod of 16 mm diameter with the total cross-sectional area of 4.27 cm2 Reinforcement of the span was two rods, 10 mm in diameter, and one rod of 16 mm in diameter with the total cross-sectional area equal to 3.37 cm2 All rods were made of steel class AIII-N, ductility C according to [11] The calculations were performed for factored values (in accordance with [11]) of the strength of materials and the average values obtained from tests for individual materials Calculated load capacity was equal to 35.0 kNm on support B, and 31.7 kNm at a span The proportion of average strength in designed cross-sections was equal to (5): ߕൌ ெೃǡೞೠೝ ெೃǡೞೌ ൌ ଷହǤ ଷଵǤ ൌ ͳǤͳͲ (5) 2.5 Safety factors at the time of destruction During the experiment, the total span moments varied in the range of 25.3-24.8 kNm and support moments - in the range of 35,5-37,5 kNm The strain of sections during the beam test is summarized in Table Total load capacity was calculated by using average values of material properties obtained in laboratory tests (Table and 2, Fig 3) The proportion of the bearing capacity of cross-sections, at the time of destruction, was almost equal to the designed ratio of the load (6): ߕൌ ெೃǡೞೠೝ ெೃǡೞೌ ൌ ସଷǤ ଷ଼Ǥ ൌ ͳǤͳ͵ (6) Table The strain of sections during the beam test Span Support The strain of section in relations to factored strength 79,8-78,2% 101,4-107,1% The strain of section in relations to average strength 65,3-64,1% 81,4-86,0% Research methodology The beam was loaded for over hour up to operational load level of the simulated element The experiment program assumed simulation tests, in the scale of 1:1, a half of the double span beam, with a span of 3.8 m span, and a crosssection of 15x25 cm, early-stage load - from obtaining the minimum concrete compressive strength of at least 80% of its average, 28-day compressive strength, for several months, under a load at operational level The tested element was designed in accordance with [11], the current state of technical knowledge and with assumption of the possible redistribution of internal forces from the support to the span, ie the direction of Dolny Piotr and Kamiński Krzysztof / Procedia Engineering 172 (2017) 883 – 890 redistribution thoroughly tested and recommended, cf [4] The degree of reinforcement of the span equalled 1.13%, and the central support of modelled beam reached 1.28% The beam was slowly loaded to meet two parameters: x to reach designed bending moment value on span, x to make even the rotation of the supporting section B to simulate fixed support – equal load of both spans of simulated double span beam During the test, all loading forces and supporting reactions were measured Also, a vertical deflection of the beam was measured Results 4.1 The course of study The test began on day 14 after the beam and the samples were concreted The beam was placed on the testing stand, and the strain sensors were placed on it The beam was loaded gradually until the bending moment of value 25.3 kNm on span was reached, and the balance (lack) of rotation on the B support was achieved Charging of element longitudinal deformation and internal forces was observed The rotation balance over B support was controlled The measurement results showed correct deflection of the beam and the balance of internal forces was not disturbed During the test, the subsidence of both passive supports of the beam was controlled The observed differences in subsidence of supports were extremely small (0.3 mm) and did not change after the load at the designed level was settled 4.2 Redistribution of bending moments The redistribution of bending moments is presented as a relation of the supporting MB and the span M1 moment on the chart (Fig 6) Redistribution from the first moments of the load occurs in the opposite direction than the phenomenon described in the standard and literature It can also be observed that there is a distinct increase of support bending moment MB (from a value of 33.3 kNm on June 24 to 35.3 kNm on Aug 25.) and a slight decrease of span bending moment M1 (from a value of 25.3 kNm on June 24 to a value of 24.8 kNm on Aug 25) As a result, the observed redistribution bending moment occurs in the direction from the span to the support, changing the ratio of the bending moments from 1.316 on June 24 to 1.423 on Aug 25 Summary It was observed that the direction of redistribution of bending moments in a beam, early loaded, was opposite to the Eurocode standard specifications Distribution of bending moments in the beam, statically indeterminate, was reverse in relation to the distribution obtained in the analysis The disproportion between the projected strain of cross sections of the beam and the results of the experiment kept growing consistently, with each day of the test 889 890 Dolny Piotr and Kamiński Krzysztof / Procedia Engineering 172 (2017) 883 – 890 Fig Redistribution of bending moments in the beam – the relation of the supporting moment (MB) to span moment (M1) during the test Assuming the degree of redistribution lower than 1, while designing RC elements, although in accordance with numerous examples available in the literature, eg [5], it may lead to a reduction in safety factors, below the level allowed by the designer, particularly in the sections of the support In the tested element, the redistribution of bending moment reduced the safety factor of approximately 5.7% References [1] M.A Aiello, L Valente, A Rizzo, Moment redistribution in continuous reinforced concrete beams strengthened with carbon-fiber-reinforced polymer laminates, Mechanics of Composite Materials 43(5) (2007) 453-466 [2] Ilker Fatih Karaa, Ashraf F Ashourb, Moment redistribution in continuous FRP reinforced concrete beams, Construction and Building Materials 49 (2013) 939-948 [3] M Jędrzejczak, M Knauff, Redystrybucja momentów zginających w żelbetowych belkach ciągłych - zasady polskiej normy na tle Eurokodu, Inżynieria i Budownictwo 58(8) (2002) 428-430 [4] M Kamiński i in., Podstawy projektowania konstrukcji żelbetowych według Eurokodu 2, Wydawnictwo Naukowe PWN, Warszawa – Wrocław, 1996 [5] M Knauf i in., Podstawy projektowania konstrukcji żelbetowych i sprężonych według Eurokodu 2, Dolnośląskie Wydawnictwo Edukacyjne, Wrocław, 2006 [6] J Malesza, Wpływ zmiany sztywności i odkształcalności węzłów na redystrybucję sił wewnętrznych w wielokondygnacyjnej konstrukcji ramowej, Budownictwo i Inżynieria Środowiska 2(1) (2011) 65-69 [7] W Starosolski, Konstrukcje żelbetowe według Eurokodu i norm związanych, Wydawnictwo Naukowe PWN, Warszawa, 2012 [8] M Tichý, J Rákosník, Obliczenia ramowych konstrukcji żelbetowych z uwzględnieniem odkształceń plastycznych, Arkady, Warszawa, 1971 [9] ISO 6784:1982 Concrete Determination of static modulus of elasticity in compression [10] PN-B 03264:2002 Konstrukcje betonowe, żelbetowe i sprężone Obliczenia statyczne i projektowanie [11] PN-EN 1992-1-1 Eurocode 2: Design of concrete structures - Part 1-1: Genaral rules and rules for buildings ... that the direction of redistribution of bending moments in a beam, early loaded, was opposite to the Eurocode standard specifications Distribution of bending moments in the beam, statically indeterminate,... load at the designed level was settled 4.2 Redistribution of bending moments The redistribution of bending moments is presented as a relation of the supporting MB and the span M1 moment on the chart... the impact of changes in stiffness and deformability of joints on the redistribution of internal forces in a multi-frame structure In his example, he obtained a change of bending moments at the