According to initiative, under the guidance of Vitalii Mitrofanov there have been multiple experimental and theoretical researches conducted in the following directions.
On this topic, the focus have been given to the following subjects:
It was worked out the experimental-theoretical method [15], allowing by the measured with the step 20 mm upper Ɛc,up(x) and lower Ɛc,low(x) fibres strains of reinforcement and its deflections Vs(x) to obtain its internal Ns(x), Ms(x), Qs(x) forces and transferring from concrete the external loads in the form of distributed compression q(x), tangential force t(x) and moment m(x), evoked by the non-uniform distribution of the bond shearing stress Ƭ between concrete and reinforcement on the latter section contour. It was obtained the essential significance of longitudinal reinforcement cross bend near the DIC tensile end, when the reinforcement had not curtailment on its length [16]. Here with the interaction forces q, t, m between reinforcement and concrete cover were considerable and testified about their joint work in composition of so called «reinforcement chord». The shear force Qs depended on mainly rigidity and length of reinforcement chord and in the beams failure state the one may have the considerable relative value Qs/Q=0,2-0,4, although the distributed moment m(x)<0 decreased the Qs=dMs/dx+m [15, 17]. When the part of longitudinal reinforcement was curtailed with reliable anchorage on the DIC tensile end, the influence of reinforcement cross bend was decreased especially after beginning of steel yielding [15].
A series of experiments have proved existence of three possible kinematic failure schemes and the corresponding relations between ultimate forces Qc and Nc have been obtained, which were in addition verified on the beams with artificial DIC, excepting the interlock of roughness on the DIC sides [13-15, 17].
It was introduced the notion «roughness character of concrete rupture surface» as the relationship between mutual shear of adjacent crack surfaces Δch and crack opening width Wcr under mutual crack surfaces sliding without deformation of roughness juts Δch=φ(Wcr) [15]. Such relation allows to ascertain the interlock existence fact of adjacent crack surfaces when their actual shear Δact>Δchφ(Wcr). It was suggested the experimental setup and method of roughness character determination and relationship for estimation of interlock forces in the DIC. This has proved existence of the DIC sides interlock with the longitudinal reinforcement without curtailment. The interlock was absent, if the longitudinal reinforcement part was curtailed with reliable anchorage on the DIC tensile end, when the reinforcement reached the yielding state and take place Δact< <Δch =φ(Wcr) consequent on the mutual rotation of the RCE parts divided by the DIC [15, 18].
The concrete failure zone above the end of normal (cross) crack is often subjected to action of considerable shear force Q. Depending on direction Q relative to the concrete failure zone, force Q can either press the noted zone to the RCE block or tear off the same zone from the mentioned above block. In the first case, according to the Plasticity Theory solution [13, 15], concrete strength is enhanced and in the second case strength is decreased, inducing the corresponding essential strength change of all normal section. The experiments [19] confirmed the noted influence of shear force Q on the normal section strength, what is taken into account by the considered lower the OSTRCE (see point 1.6).
We have tested the beams and columns with longitudinal reinforcement without curtailment and with the latter, with straight and bent-up bars, in the form of different composite bars: bundles of two non-staggered bars without gap and the staggered ones, bundles of two not staggered bars with gap and welded keys between the bars. In the staggered bars bundles the end of the curtailed bar was anchored by means of weld to the continuing bar. We used various longitudinal reinforcement when we have been studying the latter under the its bending near the DIC tensile end (point 1.1). The longitudinal and transverse reinforcement percentage was changed on the broad interval. It is necessary to draw attention to the obtained important experimental fact [15]: the RCE failure on the DIC with the shear force action may be the plastic character under certain conditions: 1) the longitudinal staggered reinforcement is curtailed near middle point of RCE part with decreasing moment M up to 40-60% of the area in the section of maximum M; 2) the end of the curtailed bar is anchored by the reliable way to the continuing bar (by the welding, Lenton couplings, Ancon MBT coupler system et al.); 3) the transverse reinforcement intensity is normal, i.e. the one reaches the yielding state in the RCE failure stage. The noted failure means the formation in the RCE DIC the plastic hinge as a consequence the simultaneous yielding both longitudinal and transverse reinforcement. The mentioned above experimental fact contradicts to the known usual notion about the brittle failure on the DIC under the shear force action. Nevertheless, it is necessary to take into account that traditional notion has arisen as a result of the numerous tests involving beams with longitudinal reinforcement without curtailment, when the beams failure on the DIC is substantially brittle. The situation changes abruptly when the proper curtailment of longitudinal reinforcement was combined with the due anchorage of the curtailed bars and with the normal transverse reinforcement.
The mentioned above research has allowed us to work out the OSTRCE [4, 5, 20]. The first thesis of OSTRCE is the RCE classification depending on quantity of longitudinal and transvers reinforcement and corresponding behavior under the loading and failure types. This classification is based on generalization of the RCE tests results, accumulated by many researchers. The classification allows natural branching of the design types and prompts to the opportunity of using more economical reinforcing classes. The OSTRCE has passed extensive and multi-sided examination and has shown considerable advantages such as: significant concordance with experiments, essential economic effect, taking into account some new factors, the general initial fundamentals excepting the necessity of additional empiric relationships, solution method unity for different strength problems, for example, on the inclined and on the cross (normal) sections (cracks). Additionally, we have developed the algorithms for problems solution optimization for the strength control and for the selection of required longitudinal and transverse reinforcement. Two design methods are recommended and available for application by designers and students: «manual» intended for mastering of the OSTRCE notions and the «computing» method, which applies simple optimization software for resolution of practical problems.
The insufficiently precise Deformation Strength Criterion (DSC), which is currently used in the codes of many countries, does not ensure the continuous transition from the pure bending to the pure axial compression. This essential disadvantage is being resolved by the ESC which was grounded [12] and is characteristic for pseudoplastic materials such as concrete and rocks, which displays in failure stage the descending branch of the compression diagram σc-Ɛc. Therefore we suggest instead of using the DSC to apply the ESC in strength design of RCE in which the concrete is exposed to the axial compression. As a result the known «deformation model» of the ultimate state of the RCE normal (cross) sections was improved and the one acquired the significant positive features. In the designs on the ESC basis the ultimate deformation Ɛcu of the compressed concrete fibre turns out to be a sought-for value, depending on, except concrete strength, the character of RCE stress-strain state (pure bending, eccentric compression, axial compression), section shape, type of reinforcement tension diagram σs-Ɛs, quantity of tensile and compressed reinforcement, prestressing intensity and another factors. For designs that utilizes the ECS, we have developed algorithms for optimization problem solution, which are intended for the strength control and selection of necessary reinforcement with use of simple software accessible for designers and students [21].
We have conducted systematic experiments on the determination of the critical stress intensity factor KIC for the usual concrete (fcm=10…80 MPa) and the concrete with the light porous aggregate claydite (fcm=5…50 MPa). As part of the research we have tested the beam specimens 100x200x800(1000) mm under 4-point bend and the tensile slab specimens 30x300x600 mm with edge notch. The beam specimens have been tested utilizing the steel tie for the prevention of the sudden notch propagation and to allow for the possibility of the repeated measurement of KIC value. The obtained by the beam and by the slab specimens the KIC values turned out to be close. We have suggested the empirical relationships for KIC of usual and claydite concretes depending on the cubic strength [15, 22 - 24]. The obtained relationship has found itself in perfect agreement with the test data derived from testing substantial number of specimens performed by other researchers.
For those designs physical relationships σij-Ɛij of the concrete is necessary. However the ones are highly complicated and depend on many conditions, and the SSS character, in particular. Thus, when mean (hydrostatic) stress (pressure) σ is changed from tensile to gradually increasing compressive one, the change of the behavior character under loading and failure type [10, 11] takes place. At the tensile σ the brittle behavior with failure on the rupture crack takes places. At the compressive σ of mean values the pseudoplastic behavior of concrete has been observed. The pseudoplastic behavior is distinguished by the exposure of the dilatation consequent on the micro-cracks system formation, development of the deformation anisotropy evoked by the certain micro-cracks orientation [25] and the descending branch of the physical relations σij-Ɛij. Here with the macroscopic destruction surface (which is to be more precise is a fairly thin layer of destroyed material [26]), frequently observed in the tests and that looks like as the shear plane, on the microscopic level consist of the alternating micro-areas of rupture and shear. Under high compressive σ the dilatation is impossible because the micro-cracking is suppressed, the descending branch of σij-Ɛij curves is absent and on the micro-level the plastic shears is developed, leading to the actual plastic strains.
Thus, determining the ultimate load under tensile σ action the Fracture Mechanics is most suitable, which in corresponding cases ought to take into account the development of the irreversible strains zone ahead of crack end. Under high compressive σ the Theory of Plasticity is applicable. The pseudoplasticity case is a most complicated instance, however it corresponds to the usual conditions in the RCE and RCS where the concrete is mainly intended for resistance to compression. One of the complex aspects of concrete physical pseudoplastic relations, which must be taken into account, is the change of the descending branch steepness which is decreased along with the compressive σ increase. Here with the ultimate load is determined by the Extreme Strength Criterion (ESC) which indissolubly relates to manifestation of the descending branch σij-Ɛij curves. Thus the adequate solution of the concrete strength problems in pseudoplastic state is possible on the basis of physical relations with descending branch and the ESC use which together reflect the specificity of the pseudoplastic failure.
Nowadays the physical relation of concrete under axial compression (compression diagram σc-Ɛc) has been most studied and approximated. Therefore the adequate strength problems solution is possible for the RCE where the concrete is undergone the axial compression. These problems concern the strength designs of RCE normal (cross) sections under forces M and N joint action. This problem has been resolved using the compression diagram according to fib as well as by suggested ESC [12]. The design obtained as a result has some important merits (see point 2).
Nevertheless under multi-axial SSS the complicated physical relations of concrete in pseudoplastic state are being in the development stage despite of large number of published recommendations. As a result, the strength designs on the basis of the ESC use turned out to be difficultly accessible. That is why there is necessity to address simplified models, among of which the model of rigid-plastic body combines the considerable simplicity with satisfactory exactness at certain conditions.
The applicability of rigid-plastic body model often meets objections induced, on one hand, by its connection with Prandtle’s diagram with unlimited plastic yielding and, on another hand, by the usual notion about concrete as brittle material with little ultimate strains. Nevertheless there are many Theory Plasticity solutions for the CE and RCE, which demonstrate good agreement with the tests results: concrete wedges [13-15], different cases of CE shear and crushing [9, 27], beam shear, shear in joints, punching shear [28], continuous beams and frames, slabs, bunkers, siloses, shells, the ultimate load of which is found by the known Ultimate Balance Method (UBM). Therefore there must be an explanation of the possible conditions of the Theory Plasticity applicability to the CE, RCE and RCS.
In each element under the non-uniform (heterogeneous) SSS there is the part with the highest strains level, which enters into the process of irreversible deformation earlier than the rest. This dangerous part in case of pseudoplastic behavior controls the development of the ultimate (failure) state of the element as a whole. This happens because the ultimate strain Ɛcu is reached in the dangerous part in the failure moment. It is obvious that for applicability of the plasticity theory the dangerous element part has to be capable of carrying plastic strains of such limited value when the formation of the complete failure zone (CFZ) can take place.
The element’s CFZ is the ultimate state zone or the failure surface, which completely crosses and divides the element in pieces that move one relative to other as rigid bodies at the expense of plastic (irreversible) strains localized in the CFZ. Using other words, the plastic strains resource of the element’s dangerous part has to ensure development of the ultimate plastic kinematic mechanism of the element. It is important to emphasize that the duration of time existence of the element CFZ does not influence the Plasticity Theory applicability, i.e. it does not matter what will happen with element material after formation of the CFZ: «either it will yield unlimitedly or it will fly to bits» [29]. Thus it is sufficient even to have the instantaneous existence of the element CFZ in order the element ultimate load can be determined on the basis of the Plasticity Theory. That is why the outwardly brittle (instantaneous) element failure does not mean that the Plasticity Theory cannot be applied to that element.
As a character of the resource of element plastic strains we assume the length of the «Conditional Yielding Plateau (CYP)» [10, 11], which is the plateau on the maximum level either curve «load parameter F – characteristic strain Ɛch» or sometimes curves σij-Ɛij. The CYP has to be close enough to the noted curves with divergence, for example, of less than 5%. The CYP has the strain Ɛbeg in its beginning, the strain Ɛend on the end and length Ɛcyp=Ɛend – Ɛbeg. When the CYP length is considerable, the plastic strains resource may be sufficient for the development of the CFZ and the plastic kinematic element mechanism. In those cases the Plasticity Theory gives satisfactory results. On the contrary, at the short CYP the descending branch of F – Ɛch and σij-Ɛij curves essentially influences and Plasticity Theory overstates the ultimate load.
. The presence of tensile part in the pseudoplastic CFZ reduces the CYP and Plasticity Theory applicability. At the compressive CFZ only the most favorable conditions take place for Plasticity Theory applicability, however the influence of the descending branch cannot be excluded. When the CFZ includes compressive and tensile parts, there can be variations. If the strains level in the compressive part exceeds the strain level in the tensile part, the application of Plasticity Theory is possible. If the strains level is higher in the tensile part, the brittle failure takes place and the Fracture Mechanics must be used.
For the Plasticity Theory problems solution it is deemed [30] that the most productive method is the variation method grounded on the corresponding variation principles of mechanics. Here with the considerable simplicity of the solution is reached if discontinuous velocities field are used. In this case plastic kinematic mechanism of the element consists of rigid parts divided by the CFZ, i.e. by the failure surfaces, which is often observed in the CE, RCE and RCS. Such approach is widely used in the known Theory of ultimate balance. We have applied this method to massive two- and three-dimensional elements, which are subject to various cases of shear, crushing, punching shear and others [9, 13, 27]. A satisfactory closeness of theoretical strength to the test ones have been observed herein [14, 15, 27].
. In spite of Plasticity Theory applicability at certain conditions, the required research direction is the development of the concrete pseudoplasticity theory. On another hand, it is also important to emphasize today’s necessity in implementing into practice the designs grounded on the Plasticity Theory. These designs are not only more precise than existing ones, but will also enhance scientific expertise level of designers and students and will set them forth towards perception of more advanced level of designs, connected with the concrete pseudoplasticity theory implementation in the future.
The Concrete Pseudoplasticity Theory development is not only important research direction. Although the concrete is mainly intended for the compression resistance, the tension in the RCS is of essential importance, where the Fracture Mechanics plays the determining role. Nevertheless, the applicability of Fracture Mechanics to concrete is in the development stage despite known positive results. Here taking into account of the irreversible (plastic) strains zone in front of the crack deadlock end is essential (as well as for metals). Although the role of the noted zone is smaller in concrete than in plastic metals, however there is no reason to neglect that either.
Thus Pseudoplasticity Theory along with Plasticity Theory, as its first approximation, and Fracture Mechanics of concrete are the corner-stones for the improvement of designs system of the CE, RCE, CS and RCS. Without these development directions it is impossible to reach a more advanced design system in comparison with the existing ones.