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The paper is a proposal of calculation of internal forces and dislocations in the reinforced concrete beams before and after cracking. For the ideally elastic bars transfer matrix proposed by Rakowski was applied. The effects associated with cracking were introduced by means of the Borcz's theory in the spectrally way. Numerical example was shown. The presented attitude also enables to calculate dynamic problems and those connected with the stability of the compressed and bending cracked beams and columns.

MODELLING THE PLASTIC HINGE IN THE STATICALLY INDETERMINABLE REINFORCED CONCRETE BAR ELEMENTS
(2006)

The paper presents the example numerical model to calculate the reinforced concrete bar structures. Usually applied methods of structure dimensioning do not include the case of plastic hinges occurrence under the limit load of construction. The model represented by A. Borcz is based on the differential equation of deflection line of the beam and it includes the effects of rearrangement of the internal forces and reological effects. The experimental parameters obtained in earlier investigations describe effects resulting from the rise of plastic hinges in the proposed equation.

Analysis of the reinforced concrete chimney geometry changes and their influence on the stresses in the chimney mantle was made. All the changes were introduced to a model chimney and compared. Relations between the stresses in the mantle of the chimney and the deformations determined by the change of the chimney's vertical axis geometry were investigated. The vertical axis of chimney was described by linear function (corresponding to the real rotation of the chimney together with the foundation), and by parabolic function (corresponding to the real dislocation of the chimney under the influence of the horizontal forces - wind). The positive stress pattern in the concrete as well as the negative stress pattern in the reinforcing steel have been presented. The two cases were compared. Analysis of the stress changes in the chimney mantle depending on the modification in the thickness of the mantle (the thickness of the chimney mantle was altered in the linear or the abrupt way) was carried out. The relation between the stresses and the chimney's diameter change from the bottom to the top of the chimney was investigated. All the analyses were conducted by means of a specially developed computer program created in Mathematica environment. The program makes it also possible to control calculations and to visualize the results of the calculations at every stage of the calculation process.

The paper contains a description of dynamic effects in the silo wall during the outflow of a stored material. The work allows for determining the danger of construction damage due to resonant vibrations and is of practical importance by determining the influence of cyclic pressures and vibro–creeping during prolonged use of a silo. The paper was devised as a result of tests on silo walls in semi-technical scale. The model is generally applicable and allows for identification of parameters in real- size silos as well.

FREE VIBRATION FREQUENCIES OF THE CRACKED REINFORCED CONCRETE BEAMS - METHODS OF CALCULATIONS
(2010)

The paper presents method of calculation of natural frequencies of the cracked reinforced concrete beams including discreet model of crack. The described method is based on the stiff finite elements method. It was modified in such a way as to take into account local discontinuities (ie. cracks). In addition, some theoretical studies as well as experimental tests of concrete mechanics based on discrete crack model were taken into consideration. The calculations were performed using the author’s own numerical algorithm. Moreover, other calculation methods of dynamic reinforced concrete beams presented in standards and guidelines are discussed. Calculations performed by using different methods are compared with the results obtained in experimental tests.

We present the way of calculation of displacement in the bent reinforced concrete bar elements where rearrangement of internal forces and plastic hinge occurred. The described solution is based on prof. Borcz’s mathematical model. It directly takes into consideration the effects connected with the occurrence of plastic hinge, such as for example a crack, by means of a differential equation of axis of the bent reinforced concrete beam. The EN Eurocode 2 makes it possible to consider the influence of plastic hinge on the values of the reinforced concrete structures. This influence can also be assumed using other analytical methods. However, the results obtained by the application of Eurocode 2 are higher from those received in testing. Just comparably big error level occurs when calculations are made by means of Borcz’s method, but in the latter case, the results depend on the assumptions made beforehand. This method makes it possible to apply the experimental results using parameters r1 i r0. When the experimental results are taken into account, one could observe the compatibility between the calculations and actual deflections of the structure.

The article presents analysis of stress distribution in the reinforced concrete support beam bracket which is a component of prefabricated reinforced concrete building. The building structure is spatial frame where dilatations were applied. The proper stiffness of its structure is provided by frames with stiff joints, monolithic lift shifts and staircases. The prefabricated slab floors are supported by beam shelves which are shaped as inverted letter ‘T’. Beams are supported by the column brackets. In order to lower the storey height and fulfill the architectural demands at the same time, the designer lowered the height of beam at the support zone. The analyzed case refers to the bracket zone where the slant crack. on the support beam bracket was observed. It could appear as a result of overcrossing of allowable tension stresses in reinforced concrete, in the bracket zone. It should be noted that the construction solution applied, i.e. concurrent support of the “undercut” beam on the column bracket causes local concentration of stresses in the undercut zone where the strongest transverse forces and tangent stresses occur concurrently. Some additional rectangular stresses being a result of placing the slab floors on the lower part of beam shelves sum up with those described above.

In this paper experimental studies and numerical analysis carried out on reinforced concrete beam are partially reported. They aimed to apply the rigid finite element method to calculations for reinforced concrete beams using discrete crack model. Hence rotational ductility resulting from crack occurrence had to be determined. A relationship for calculating it in static equilibrium was proposed. Laboratory experiments proved that dynamic ductility is considerably smaller. Therefore scaling of the empirical parameter was carried out. Consequently a formula for its value depending on reinforcement ratio was obtained.

MODEL DESCRIBING STATIC AND DYNAMIC DISPLACEMENTS OF SILOS WALL DURING THE FLOW OF LOOSE MATERIAL
(2012)

Correct evaluation of wall displacements is a key matter when designing silos. This issue is important from both the standpoint of design engineer (load-bearing capacity of structures) and end-consumer (durability of structures). Commonplace methods of silo design mainly focus on satisfying limit states of load-bearing capacity. Current standards fail to specify methods of dynamic displacements analysis. Measurements of stressacting on silo walls prove that the actual stress is sum of static and dynamic stresses. Janssen came up with differential equation describing state of static equilibrium in cross-section of a silo. By solving the equation static stress of granular solid on silo walls can be determined. Equations of motion were determined from equilibrium equations of feature objects. General solution, describing dynamic stresses was presented as parametric model. This paper presents particular integrals of differential equation, which enable analysing displacements and vibrations for different rigidities of silo walls, types of granular solid and its flow rate.

DISCRETE CRACK MODEL OF BORCZ FOR CALCULATING THE DEFLECTIONS OF BENDING REINFORCED CONCRETE BEAM
(2012)

In the design of the reinforced concrete beams loaded by the bending moment, it is assumed that the structure can be used at a level of load, that there are local discontinuities - cracks. Designing the element demands checking two limit states of construction, load capacity and usability. Limit states usability include also the deflection of the element. Deflections in the reinforced concrete beams with cracks are based on actual rigidity of the element. After cracking there is a local change in rigidity of the beam. The rigidity is variable in the element’s length and due to the heterogeneous structure of concrete, it is not possible to clearly describe those changes. Most standards of testing methods tend to simplify the calculations and take the average value of the beam’s rigidity on its entire length. The rigidity depends on the level of the maximal load of the beam. Experimental researches verify the value by inserting the coefficients into the formulas used in the theory of elasticity. The researches describe the changes in rigidity in the beam’s length more precisely. The authors take into consideration the change of rigidity, depending on the level of maximum load (continuum models), or localize the changes in rigidity in the area of the cracks (discrete models). This paper presents one of the discrete models. It is distinguished by the fact that the left side of the differential equation, that depends on the rigidity, is constant, and all effects associated with the scratches are taken as the external load and placed on the right side of the equation. This allows to generalize the description. The paper presents a particular integral of the differential equation, which allow analyzing the displacement and vibration for different rigidity of the silo’s walls, the flow rate and type of the flowing material.