Crossbars of frame buildings when changing the sign of forces

. A large number of reinforced concrete frame buildings are located in seismically dangerous regions. In case of seismic impacts plastic strains and local destructions occur in elements of such buildings, which influence sufficiently on their bearing capacity and stress-strain state when changing the sign of internal forces. This fact is not taken into account in current Russian norms on earthquake-resistant construction. The numerical research of reinforced concrete beams behavior under action of alternating load with regard to elastic-plastic properties of concrete and reinforcement has been conducted. The evaluation of influence of plastic strain maximum value in tensed reinforcement on the relative bearing capacity of the beam when changing the force sign and on the limit value of coefficient of plasticity corresponding to the destruction of compressed zone of concrete has been made. The comparison of results of numerical simulation with the results of experimental tests of beams with the same parameters has been made, which has demonstrated good convergence of numerical calculations with experimental data both in terms of bearing capacity and deformations for plasticity coefficients less than 2.5. During the experiment it has been found that when changing the sign of internal forces the through cracks form in bending reinforced concrete elements throughout the entire section height, which close afterwards if the coefficient of plasticity for reinforcement deformations does not exceed 2.5. For greater values of coefficient of plasticity the through cracks closing doesn’t take place until the destruction of the specimens due to the rupture of reinforcement bars. According to the results of the study it was concluded, that the decrease by 40% of relative bearing capacity of beams compared with reference samples as the increase of coefficient of plasticity in the first semi cycle of loading takes place. Also with the increase of the coefficient of plasticity the three times decrease of the limit values of coefficients of plasticity, corresponding to destruction of compressed concrete, when changing the sign of forces is observed. The numerical calculations of beams models for 10 and 50 alternating load cycles have been conducted with cycle asymmetry coefficient equal to -1. The conclusion has been made that under the action of alternating load


Introduction
Significant area of Russian Federation is located in seismically dangerous regions.Buildings and structures, located in these regions, can be affected to alternate effects of high intensity.During the strong earthquakes significant plastic deformations develop in the bearing elements of frame buildings, and this fact is taken into account yet at the design stage [8].And when choosing design solutions of frame buildings the preference is given to those, in which plastic deformations occur in crossbars.
The accelerograms analysis shows that the number of cycles of loading during the earthquake with a 95% probability does not exceed 50 cycles, and according to the research results of behavior of buildings during the real earthquakes the number of oscillations cycles of elastic plastic systems does not exceed 20 [1].As the results of frames calculation with the regard to physical non-linearity of materials show the maximum displacements were observed during the first two semi cycles of loading, and then they decreased due to the sharp decrease of rigidity of system in plastic stage [2,3,5].
The results of calculation of two-span three-storey reinforced concrete beam with the span size of 4.8 m and storey height of 3.4 m made of concrete B25 and reinforcement А500С for the effect of harmonic vibrations of the base with regard to physical nonlinearity of materials are represented in fig. 1.The cross section size of columns is 400400 mm, the cross section size of crossbars is also 400400 mm.The reinforcement is symmetric  =0.01.As it can be seen from the graph there is a decrease of oscillations amplitude after the maximum displacement is reached.In current Russian norms the two levels of design impacts conception is adopted: for a project earthquake and for a control earthquake.The calculation for a design earthquake is carried out mainly by the spectral method.In this case the effect of plastic strains is considered only when determining forces by introduction of reducing factor К1 [7,9].The calculation for a control earthquake is conducted in the time domain considering the non-linear dependence between the external effect and the level of deformations on a basis of direct dynamic methods of calculation [10].
In fact the values of plastic deformations under alternate effects of high intensity of seismic type influence both bearing capacity and the stress-strain state of bending elements when changing the sign of forces [6].In spite of the fact that a significant amount of research is devoted to the study of constructions behavior under seismic impacts and to seismic design of structures as well as to the behavior of constructions under action of alternate loads, the effect of plastic deformations on behavior of constructions under seismic loads is insufficiently studied.
Consequently the study of behavior of beams in framed buildings when changing the sign of internal forces and the estimation of influence of plastic strains on the bearing capacity and on the stress-strain state is interesting.A number of experiments have shown that the limiting states of reinforced concrete elements during quasi-static tests can be considered as the lower limit of the bearing capacity of structures under alternating effects.[1].

Methods
In order to study the behavior of reinforced concrete bending elements under low cycle alternate effect and to estimate the influence of plastic deformations on the bearing capacity and on the limit value of plastic strains of reinforcement, corresponding to destruction, the calculation of reinforced concrete beams models in the computing complex SIMULIA Abaqus, using the finite element method, has been made.All beams had the same geometric dimensions, reinforcement scheme and materials parameters.Dimensions of the crosssection of the beams were 200200 mm.The load application diagram and reinforcement scheme of the beams are represented in fig. 2.
Reinforcement bars A500C with a diameter 10 mm were accepted as a symmetric longitudinal reinforcement.In order to supply necessary anchoring transverse plates 1501504 mm were welded at the ends of these bars.Rods with diameter 6 mm, installed in the support zones with a pitch of 70 mm, were used as shear reinforcement.
Concrete and reinforcement bars were modelled by three-dimensional finite elements.Loading was modelled as a concentrated force transmitted to the beams through two load distribution plates 50×200 mm.For modelling of concrete behavior the model «concrete damaged plasticity» has been used, and the behavior of reinforcement has been modelled by the bilinear diagram with small increase in plastic stage.

Fig 2. (а) -Load application diagram; (b) -Reinforcement scheme
The calculation results have been verified by comparison with experimental data, obtained by tests under the action of two concentrated forces of 15 beams with the same characteristics as computer models had [4].
Physical and mechanical characteristics of concrete were determined before the beginning of experiment by tests of six concrete cubes 100100100 mm and of three concrete prisms 100100400 mm.According to the results of the tests the prism strength of concrete was 21.3 MPa.In order to determine physical and mechanical properties of reinforcement experimental tests of six rods 400 mm long were carried out.Average yield strength of reinforcement was R s = 657 MPa.
The beams both during numerical calculations and in the experiment were divided in 7 series.The first series, considered as reference, was testes for the effect of monotonically increasing load up to destruction.During the test the load was applied in steps of 0.1 from the destructive load with the delay between the steps of 10 minutes.The stress-strain state of beams during loading and the destruction mechanism were studied.
Strain gages were glued to measure longitudinal strains of reinforcement and of compressed zone concrete.Their testimony was duplicated by testimony of digital indicators with a base of 150 mm and a division price of 0.001 mm.The arrangement of strain gages on reinforcement and concrete and of digital indicators is represented in fig. 3. Fig. 3 The digital indicators arrangement scheme and strain gages arrangement scheme on concrete and reinforcement of reference beams.
The beams of series II-VII were tested by two semi cycles of loading (from the forward and from the reverse side).In the first semi cycle the beams were loaded in steps of 0.1 from the maximum load up to the achievement of the set value of the coefficient of plasticity by reinforcement deformations  pl,1 , equal to maximum deformations of stretched reinforcement to yield deformation ratio.The values of this coefficient varied from 1.21 to 5.51 (table 1).Deformations of concrete of compressed zone and of reinforcement were determined using strain gages, located as shown in fig. 4. The same as in the reference beams their testimony was duplicated by testimony of digital clock-type indicators.Then the unloading was carried out, after which the values of residual deformations were determined.
Fig. 4 The digital indicators arrangement scheme and strain gages arrangement scheme on concrete and reinforcement of beams of series II-VII After that the beam was turned over, and strain gages were glued along the banks of cracks, formed in the first semi cycle of loading.Then the inverted beam was loaded according to the same scheme up to the destruction.The stress-strain state of be beams after changing the sign of the internal forces was studied.It was assumed, that the closure of the banks of cracks and inclusion in the work of compressed concrete took place, when the residual strains after the first semi cycle of loading were compensated.
Besides the value of destructive load after changing the sign of internal forces and the limit value of coefficient of plasticity by reinforcement deformations  pl,2 , equal to maximum deformations of reinforcement at the moment of destruction to yield deformation ratio, were determined.
The parameters of beams, for which the numerical calculations in Abaqus were conducted, and the scheme of loads application corresponded to the conditions of experiment.The comparison of results of calculation with results of tests has been made.
After that the numerical calculations of the same beams for 5, 10 and 50 cycles of loading with a cycle asymmetry coefficient -1 have been conducted.The comparison of obtained results with the results of the calculation for one cycle and with the results of experiment has been made.In order to determine the relative bearing capacity comparing with bearing capacity of beams loaded with only one cycle of alternate load, the models were brought to destruction on the 10 th and 50 th cycle of loading.

Results
According to the results of experimental study it has been found that in all beams when changing the sign of forces the formation of through crack for the entire height of the section took place.When the values of the plasticity coefficient in the first half-cycle of loading were less than 2.14 the closure of cracks and destruction of beams due to crushing of compressed zone concrete was observed.
For greater values of plasticity coefficient there was no closing of residual cracks up to destruction.After the formation of a through crack the reverse sign load in the second semi cycle of loading was perceived only by compressed and stretched reinforcement due to shutting down from work of concrete layers.The yield in compressed reinforcement began at a load of about 52% of the breaking load in the reference beams.After that the beams continued to work up to the destruction due to the rupture of stretched reinforcement.The load, corresponding to destruction, exceeded by 12 % the breaking load in reference beams because of the release of reinforcement to the branch of hardening.
The comparison of results of numerical calculation in Abaqus of reference beams and of beams of the rest series in the first semi cycle of loading with experiment data has shown their satisfactory convergence.Maximum deviation by deflections for all beams didn't exceed 12 %.
When changing the sign of internal forces the results of numerical calculations has shown satisfactory convergence with experimental data at the value of the coefficient of plasticity by reinforcement deformations in the first semi cycle of loading not exceeding 2.5.
At the coefficient of plasticity in the first semi cycle of loading greater than 2.5 the relative bearing capacity and the limit value of the coefficient of plasticity in the second semi cycle was significantly different form the experimental data.This is most likely caused by the features of the design model of concrete and by the difference in destruction mechanism.According to numerical calculations results at large values of coefficient of plasticity the destruction took place due to brittle destruction of compressed concrete before the yield of stretched reinforcement, whereas in the experiment the destruction took place due to the rupture of stretched reinforcement after the formation through non-closing crack.
The dependence of the bearing capacity of beams of II-VII series to the reference beams bearing capacity ratio after changing the sign of internal forces on maximum value of coefficient of plasticity for reinforcement deformations in the first semi cycle is represented in fig. 5.The values of relative load, corresponding to yield of compressed reinforcement in specimens of series VI-VII, which can be considered as limit values, are marked by crosses on the graph.Fig. 5 The dependence of the breaking load in the second semi cycle to reference sample breaking load ratio on coefficient of plasticity in the first semi cycle.
The dependence of plasticity coefficient, corresponding to crushing of compressed zone concrete when changing the sign of internal forces,  pl,2 on maximum plasticity coefficient in the first semi cycle  pl,1 is represented in fig.6.As one can see from the graph the numerical calculations has shown satisfactory convergence with experiment at  pl,1  2.5.The deviation doesn't exceed 12.5%.At the coefficient of plasticity in the first semi cycle of loading greater than 2.5 the deviation of the calculated values from the experimental ones increases significantly in connection with different mechanisms of destruction in numerical calculations and in experiment.
Points, obtained by calculations of the same specimens for 10 and 50 cycles with the cycle asymmetry coefficient equal to -1 in all calculations, are also plotted on the graph.The calculations were conducted for the maximum values of plasticity coefficient in the first semi cycle of loading, not exceeding 1.62, because just in this range the results of numerical calculation for one cycle of loading are the closest to the results of experiment.Fig. 6 The dependence of the limit value of plasticity coefficient (calculated and experimental), corresponding to destruction of compressed zone of concrete in the second semi cycle, on plasticity coefficient in the first semi cycle.

Discussions
As one can see from the graph in fig.5, as plastic deformations in the first semi cycle increase the considerable (to 40%) reduction of bearing capacity in the second semi cycle, associated with a decrease in the cross-section height due to an unclosed normal crack in the compressed zone.
If the coefficient of plasticity in the first semi cycle of loading is less than 2.5, the results of numerical calculations when changing the sign of internal forces are close to experimental data, and the deviation in relative breaking load doesn't exceed 14%.At greater values of plasticity coefficient significant deviation is observed, connected with different destruction mechanisms.
As one can see from the graph in fig.6, with an increase of plastic strains in the first semi cycle of loading the limit values of plasticity coefficient, corresponding to the destruction of compressed zone concrete when changing the sign of internal forces, decrease intensively.This is evidenced by both the results of numerical calculations and experimental data.
The results of beams calculation for 10 and 50 cycles of alternate loading has shown satisfactory convergence with results of calculation of the same beams for one cycle of alternate loading.
The numerical study has demonstrated, that at coefficient of plasticity  pl,1  1.62 the reinforcement plastic strains increments almost stabilized by the third cycle of loading and increase insignificantly by the 50 th cycle.For example at coefficient of plasticity in the first semi cycle  pl,1 = 1.21 when changing the sign of internal forces the residual strains in the first cycle were equal to 1,56•10 -3 .By the 10-th cycle of loading the residual strains were 1,88•10 -3 and by the 50-th cycle they were 2,01•10 -3 .
At coefficient of plasticity in the first semi cycle of loading  pl,1  2.5 the destruction of beams wasn't observed up to the 50-th cycle of the alternate loading.At greater values of coefficient of plasticity according to the results of calculation the brittle destruction of compressed zone concrete took place in the first cycle of loading after changing the sign of internal forces before the beginning of rebar yield.This did not correspond to the results of the experiment.
As the result of numerical calculations have shown, at small values of plastic strains with an increase in the number of load cycles the insignificant reduce of relative bearing capacity of beams takes place, which on the 50-th cycle of loading doesn't exceed 10%.

Conclusions
1.The results of numerical calculations in the Abaqus software package of models, identical to experimental samples, provide the satisfactory convergence of determining parameters with experimental data under monotonous loading, both in terms of bearing capacity and deformations.2. The limit values of coefficient of plasticity by reinforcement deformations, corresponding to destruction in the second semi cycle of loading, and the relative bearing capacity when changing the sign of internal forces depending on the coefficient of plasticity at the first semi cycle of loading, have shown satisfactory convergence with experimental points at  pl1  2.5.3. Increasing the coefficient of plasticity by reinforcement deformations in the first semi cycle of loading up to values of 2.5 the 40% reduction of relative bearing capacity of reinforced concrete specimens when changing the sign of internal forces and three times decrease of coefficient of plasticity, corresponding to the compressed zone concrete destruction, takes place.4.Under the action of alternating loads with a number of loading cycles not exceeding 50 and the maximum values of the coefficient of plasticity not exceeding 1.62 there is a slight decrease in the relative bearing capacity of the samples and in the limit values of plasticity coefficient, not exceeding 10 %, with increase of number of cycles.

Fig 1 .
Fig 1. Dependence of horizontal displacement of the upper frame unit on time.

Table 1 .
Series and the maximum value of coefficient of plasticity of the reference specimens and of specimens of the rest series in the first semi cycle of loading