Unloading Branches Research Articles

Critical behavior of thick rubber bearings under compression and shear

Thick rubber bearings (TRBs) are used for seismic isolation as well as for subway-induced vertical vibration mitigation. Under earthquake ground shaking, TRBs typically undergo large horizontal displacements that lead to a reduction of their vertical bearing capacity, which should be carefully evaluated during design. This study investigates the performance of TRBs under compression and compression-shear loading. Eight full-scale TRB specimens were designed and tested. The test results show that increasing the axial load increases the compressive stiffness of TRBs. During compression-shear loading, TRBs under design level axial loads failed due to rubber rupture at shear strain γ > 350 %, while under higher axial loads, they failed because of instability at γ ＜ 300 %. In addition, in cyclic shear deformation under axial loads larger than twice the design level axial load, the TRBs exhibited negative stiffness at the unloading branch of the F-u loop. The effective damping ratio significantly increased with increased axial load. Then, the test data were used to validate existing mechanical models for the prediction of the compression-shear behavior of elastomeric bearings. Existing analytical models were modified by accounting for the compressive stiffness change due to increased axial loads, leading to more accurate prediction. Finally, models for the critical load of TRBs under zero and lateral displacement were examined and improved.

A microstructural insight into the compression behaviour of scaly clays

Scaly clays are intensely fissured clays with lens-shaped elements of millimetre size and they show a complex compression behaviour that poses challenges to the design and construction of geostructures (excavations, retaining diaphragms and tunnels). Scaly clays show a normal compression line (NCL) where plastic deformation accumulates, as typically observed in non-scaly clays. Yet the response observed upon unloading and subsequent reloading is very peculiar, (a) the unloading–reloading cycle is typically a closed loop with relatively large hysteresis; (b) the compressibility recorded at high overconsolidation ratio of the unloading or reloading branches is close to the NCL compressibility. This paper presents a microstructural study on an Italian scaly clay where scanning electron microscope observations are integrated with mercury intrusion porosimetry analyses and X-ray computed tomography images. The mechanism associated with the closing of inter-scale porosity and the generation of new intra-scale porosity was identified as the process responsible for the plastic deformation. Experimental observation of reconstituted clay showed a ‘quasi-reversible’ behaviour upon loading and unloading and a pore size distribution characterised only by interparticle porosity. The observation that unloading and reloading curves are parallel in natural and reconstituted clays, led to the postulation that the interparticle porosity is controlling the elastic response.

Numerical modelling of fatigue crack closure and its implication on crack front curvature using ΔCTODp

One of the widely used approaches to characterize fatigue crack propagation is the use of the effective stress intensity factor range ΔKeff, which relies on determination of crack closure value Kcl. The used models of crack closure are most frequently-two-dimensional, however, for real cracks, the 3D effects should be taken into account. The paper presents 3D finite element analyses of the influences of crack front shape, inserted cycles (loading cycles between node releases) and crack closure on the crack driving force in terms of ΔKeff and ΔCTODp (plastic part of crack tip opening displacement range). Numerically obtained crack closure depended on the simulation strategy. In the case of inserted cycles, crack closure disappeared in the internal part of the specimen and remained only near the free edges. The use of ΔCTODp had the advantage of a well-defined parameter in situations where ΔKeff was problematic, namely at the corner points, which did not allow finding of equalized crack driving force along the whole crack front using ΔKeff. Equalized crack driving force in terms of ΔCTODp was found for crack front curvature with the edge angle 15.4° in simulation with crack closure, which was in good agreement with the experimentally measured value of 16°. Loops produced by loading and unloading branches of the force vs CTOD diagrams helped to describe the crack closure process and magnitude. Actual values of CTOD did not agree with the classical idea of 2D solutions under plane strain and plane stress. CTOD was larger in the internal part of the specimen than at the free edges, even in simulations with no crack face contact.

Monotonic and cyclic constitutive behaviour of rubberised one-part alkali-activated concrete

This study presents an experimental assessment into the monotonic and cyclic compressive stress-strain rate-dependent response of rubberised one-part alkali-activated concrete. Three different volumetric crumb rubber replacement ratios of total natural aggregates (0%, 30% and 60%), and three different strain rates accounting for quasi-static, moderate seismic and severe seismic conditions are considered. The results indicate a reduction in the elastic modulus, compressive strength, and crushing energy in proportion to the rubber content regardless of the strain rate or loading condition, monotonic or cyclic. A reduction in the axial crushing strain is also obtained with the increase in rubber content within the ranges considered. The increase in strain rate leads to a proportional enhancement in elastic modulus, compressive strength, and axial crushing strain. Under cyclic conditions, the unloading and reloading branches of the stress-strain response fall within the monotonic curves. The cumulative energy dissipation from each first cyclic loop reduces with the increase in rubber content, whilst an increase in the loading rate results in a proportional increase in the cumulative energy dissipation. The unloading modulus is shown to be sensitive to the unloading strain, rubber content and strain rate, while the plastic residual strain is mainly influenced by the unloading strain. Analytical expressions to predict the reduction in elastic modulus, compressive strength, axial crushing strain, unloading modulus and plastic residual strain, with varying rubber contents and for the different strain rates considered, are proposed. Constitutive models representing the monotonic stress-strain response of rubberised alkali-activated concrete materials, as well as the unloading and reloading branches of the cyclic stress-strain response, are also given. Finally, formulations for the strain rate-dependent dynamic increase factors for the elastic modulus, compressive strength, and axial crushing strain are also provided.

SCIENTIFIC AND TECHNICAL SUBSTANTIATION OF SOIL BASES STRENGTHENING WITH REINFORCED SOIL-CEMENT PILES

Purpose. The aim of the work is to substantiate the technological parameters of effective use of reinforced soil-cement piles obtained by jet mixing technology. Such studies are needed to determine the possibility of increasing the bearing capacity of pile foundations in the body of earthen structures of buildings and roads. Methodology. Determination of bearing capacity of reinforced soil-cement piles was carried out by reasonable choice of structure, soil base, material, depth of piles in accordance with engineering and geological conditions, structural scheme of the structure and method of their arrangement. Findings. Significant differences in soil engineering and geological elements in plan and depth necessitated an in-depth study of soil characteristics, so it was decided to determine their strength and deformability based on static probing of reinforced piles. Analysis of the results of static studies showed that the soil-cement pile, reinforced with a reinforcing frame, has a higher bearing capacity in soil and material, and the unloading branch confirms the fact that in specific conditions the soil and piles work in elastic mode. Originality. It consists of the obtained dependences of the change of vertical loads on the reinforced soil-cement pile, created on the basis of brown mixing technology. Graphs of the dependence of the values of deformations occurring during the action on the reinforced soil-cement pile under static loading and unloading are obtained. Practical value. In practical conditions, the solution for the manufacture and testing of static load of the experimental reinforced soil-cement pile was implemented at the construction site. The manufacturing technology is designed so that the spatial frames of the soil-cement element with a length of 5…7 meters are immersed under their own weight, and the frames of 12 and more meters are lowered with a vibrator.

On Stress–Strain Function of Geomaterials Subjected to Blast-Loads

An appropriate stress–strain relationship of geomaterials subjected to blast loading is essential for the design of underground protective structures. Previous experimental and theoretical research efforts indicate that the constitutive behavior of geomaterials under blast loading depends upon strain rate, stress level, and interaction among the three phases (solid, liquid, and gases). In current state-of-the-art, various advanced constitutive models are available to model the stress–strain behavior of geomaterials under blast-loads. However, considering the cost of computation associated with such models, a functional form is discussed to model the loading and unloading branches of stress–strain curve of geomaterials subjected to blast load based on the three parameters: weight factor, initial modulus ratio, and strain recovery ratio. It is observed that the new functional form reasonably captures the mean trend of the experimentally obtained or simulated stress–strain data. This paper further investigates the applicability of this functional form and provides a catalog of the model parameters for direct use by practicing engineers. The dependence of function parameters on strain rate, lateral confinement, degree of saturation, initial compaction, and locking initiation stress is investigated, and some simple rules are proposed for reasonable estimation of the three parameters. The proposed functional form would be quite useful in practical design problems specially where cost of computation associated with advanced constitutive models is too high. In addition to this, the proposed simple parametrization of complex nonlinear stress–strain behavior also provides an opportunity to investigate the effect of uncertainties in soil parameters in a computationally efficient manner.

Cyclic behaviour and hysteretic model of austenitic stainless steel bolted T-stubs

The aim of this paper is to investigate the hysteretic behaviour of stainless steel bolted T-stub connections. A total of 22 bolted T-stubs in austenitic grade EN 1.4301 were tested under monotonic loading and cyclic loading under both variable and constant amplitudes. The tested T-stubs were carefully designed with two different bolt configurations and were fabricated with two types of stainless steel bolts – A4–70 and A4–80 thus allowing different failure modes to be obtained. Three additional bolted T-stubs in carbon steel grade Q345B were used as benchmark specimens to facilitate a direct comparison of the performance of carbon and stainless steel T-stubs under cyclic loads. Monotonic loading tests were conducted to acquire basic parameters of the cyclic loading protocols employed in the cyclic loading tests. The hysteretic responses of the tested T-stubs under two variable amplitude loading histories and three proportional constant amplitude loading histories were recorded and were further analysed in terms of failure characteristic, hysteresis response, skeleton curve, energy dissipation and strength degradation. Based on the evolution of the hysteresis curves, standard hysteresis loops were acquired and were used to develop a novel hysteretic model able to simulate the initial loading branch, the unloading branch and reloading branch of the hysteretic response. It has been demonstrated that the developed hysteretic models provide a close agreement with the obtained experimental results and can thus be used to estimate the expected energy dissipation of bolted stainless steel connections subjected to cyclic loads.

Residual deformations in obliquely reinforced fibrous composites: experiments on cyclic stretching

Experimental studies of test samples from a obliquely reinforced composite based on carbon tape on cyclic stretching were carried out. It is established that the loading and unloading branches do not coincide, forming a hysteresis loop, and the axial deformation at each subsequent cycle at the same stress level in the cycle gains an increment that decreases in the initial cycles and tends to a constant value with a further increase in the number of cycles. This process is called the process of adaptability of the composite material. On the basis of tests on multi-cycle stretching of a composite with an aging after unloading, a viscoelastic component was distinguished from accumulated strain at each loading cycle. It is shown that at the initial cycles, the total strain accumulated in the sample by the end of the unloading process can be represented as a sum consisting of residual strain formed due to structural changes in the composite and creep deformation that disappears in the sample after unloading and holding during a long time.

Influence of Fiber Distribution and Orientation in the Fracture Behavior of Polyolefin Fiber-Reinforced Concrete.

Polyolefin fiber-reinforced concrete (PFRC) has become an attractive alternative to steel for the reinforcement of concrete elements, mainly due to its chemical stability and the residual strengths that can be reached with lower weights. The use of polyolefin fibers can meet the requirements of standards, although the main constitutive relations are based on experience with steel fibers. Therefore, the structural contributions of the fibers should be assessed by inverse analysis. In this study, the fiber dosage was fixed at 6 kg/m3, and both self-compacting concrete and conventional concrete were used to compare the influence of the positioning of the fibers. An idealized homogeneous distribution of the fibers with such fibers crossing from side to side of the specimen was added to self-compacting concrete. The experimental results of three-point bending tests on notched specimens were reproduced by using the cohesive crack approach. Hence, constitutive relations were found. The significance of this research relies on the verification of the formulations found to build constitutive relations. Moreover, with these results, it is possible to establish a higher threshold for the performance of PFRC and the difficulties of limiting the first unloading branch typical of fracture tests of PFRC.

In‐plane–out‐of‐plane non‐linear model of masonry infills in the seismic analysis of r.c.‐framed buildings

SummaryThe out‐of‐plane (OOP) behaviour of masonry infills (MIs), inserted in reinforced concrete (r.c.)–framed buildings, is recognized as one of the most important failure modes of this nonstructural element during an earthquake, which may be a consequence of simultaneous or prior in‐plane (IP) damage. A five‐element macro‐model, with four diagonal OOP non‐linear beams and one horizontal IP non‐linear truss, with an equivalent mass of the infill panel divided between two central nodes, takes into account the IP and OOP failure modes occurring in the event of seismic loading. Pivot hysteretic models predict the non‐linear IP and OOP force‐displacement laws of the infill panel, based on geometrical rules defining loading and unloading branches. Firstly, a calibration of the proposed IP‐OOP interaction model of MIs is carried out considering full‐scale experimental results of traditional masonry typologies. Each specimen is initially subjected to in‐plane quasi‐static cyclic loading, until a maximum drift is reached, and then one‐sided OOP cycles are imposed pushing in the horizontal direction and back to zero force. Then a numerical investigation considers masonry infills of an existing six‐storey r.c.‐framed building designed in compliance with a former Italian seismic code. To evaluate the interaction, the results of simultaneous IP and OOP cyclic tests on MIs at the top, intermediate, and lowest levels of the test structure are presented, assuming different displacement histories: (1) OOP loading faster than IP, at the sixth storey; (2) equal IP and OOP loading, at the third storey; (3) IP loading faster than OOP, at the first storey. Finally, attention is focused on the contribution of masonry infills to the IP and OOP energy dissipation of r.c.‐framed structures.

Nonlinear modelling of the in-plane-out-of-plane interaction in the seismic analysis of masonry infills in r.c. framed buildings

A five-element macro-model, with four diagonal out-of-plane (OP) nonlinear beams and one horizontal in-plane (IP) nonlinear truss, takes into account the OP and IP failure modes occurring, in the event of seismic loading, for masonry infills (MIs) inserted in reinforced concrete (r.c.) framed buildings. Pivot hysteretic models predict the nonlinear IP and OP force-displacement laws of the infill panel, based on geometrical rules defining loading and unloading branches. Firstly, a calibration of the proposed IP-OP interaction model of MIs is carried out considering full-scale experimental results of traditional masonry typologies. To evaluate the interaction, the numerical results of simultaneous IP and OP cyclic tests on MIs at the top, intermediate and lowest levels of an existing six-storey r.c. framed building are presented, assuming different displacement histories: i) OP loading faster than IP, at the sixth storey; ii) equal IP and OP loading, at the third storey; iii) IP loading faster than OP, at the first storey.

Prediction of the Bond–Slip Law Between Externally Bonded Concrete Substrates and CFRP Plates Under Fatigue Loading

This study proposes a realistic model for the bond stress–slip relation between concrete and carbon fiber-reinforced polymer plates under fatigue loading. Double-lap shear experimental tests were implemented to explore the bond stress–slip behavior under fatigue loading. The bond stress–slip relation under fatigue loading was derived as a function of the loading/unloading linear slip, the stiffness of the unloading branch in slip and the number of fatigue cycles. In this model, the stiffness of the unloading branch in slip was derived from the simplest linear fitting of the experimental data based on the bond stress–slip relationship. Numerical analyses using the proposed model compared well overall to the experimental fatigue results.

Hysteretic Model for Flexure-shear Critical Reinforced Concrete Columns

ABSTRACTA model for predicting the cyclic lateral load-deformation response of flexure-shear critical reinforced concrete (RC) columns subjected to combined axial load and cyclic shear is proposed. Strength deterioration in the primary curve due to the effect of shear after yielding is considered by a modification coefficient. Rules for unloading and reloading branches of the hysteretic curve are obtained from regression analysis of test results. Unloading stiffness is fitted as a function of displacement ductility and secant stiffness of the point with maximum displacement in the primary curve. Pinching is simulated by changing the slope of reloading branch. Path-based cyclic strength deterioration is incorporated in the proposed model. In the expression of cyclic strength deterioration, the effects of aspect ratio and axial-load ratio are considered. Comparison between the predicted cyclic response and experimental results indicates that the proposed model can predict the observed hysteretic response of flexure-shear critical RC columns well.

Contact stiffness depth-sensing indentation: Understanding of material properties of thin films attached to substrates

The classic version of the depth-sensing indentation techniques assumes the estimation of the elastic contact modulus of a material sample by measuring the slope (the contact stiffness) of the initial part of the unloading branch of the force-displacement curve. This approach assumes that the curve at loading reflects both elastic and plastic deformations of the material, while the unloading is taking place elastically. Therefore, neglecting the plastic deformations, one can assume that the structure of the material is the same at both branches and the assumptions of the Hertz-type contact theory are valid for the unloading branch. However, the contact problem for an elastic film attached to a substrate depends on the properties of the substrate. Hence, the film contact modulus is usually estimated by measuring the slopes of the initial unloading force-displacement curves obtained for different maximal values of indentation depth, and fitting the experimental points by various empirical analytical dimensionless functions of the ratio between the contact radius, a, and the layer thickness, t. Here, analytical analysis of contact problems for coated materials is performed. Both re-scaling and asymptotic techniques are employed. Asymptotic analysis of the contact at the small-scale indentation range (the ratioa/t is small) shows that the formula of the contact stiffness derived for an elastic half-space, has to be multiplied by the so-called indentation scaling factor that is a function of a/t. Thus, the asymptotic approach allows us to take into account analytically the substrate effect. The analytical fitting function obtained agrees with both some known semi-empirical functional forms and the published experimental results on depth-sensing nanoindentation of thin metallic layers, while the function is in a disagreement with results obtained for inhomogeneous films of brittle materials such as coals. It is argued that the disagreement is caused by structural transformations (crushing) of the coals during loading.

Rigid projectile penetration into a concrete medium: A new model

A new simplified model to analyze the penetration of a rigid projectile into a concrete thick medium is presented. The concrete medium is modeled by a set of discs, responding in the radial direction under plain strain conditions. A convenient mathematical formulation is derived based on some simplifying assumptions. A major new feature of the present model is that it hybridizes two different analytical and numerical approaches that have been developed by the authors. In the present model, the Riemann problem exact solution is applied to the interaction of the projectile nose tip with the target. It is assumed that when the nose tip meets a new undisturbed disc, loading occurs during its interaction with the concrete, and therefore the unloading branch of the equation of state is not required in the Riemann problem. The other discs that maintain contact with the projectile nose and had been loaded in earlier time steps undergo unloading, and a special approach has been developed to compute their contact stresses. This entirely new formulation has not been proposed earlier. The present model enables calculations of the projectile motion time history (i.e. deceleration, velocity, and depth). Comparison of the present model results with experimental data shows very good agreement. Contrary to many other simplified models, the present approach does not require any empirical constants or any preliminary assumptions concerning the contact pressures acting on the penetrating projectile nose, as other models do. Computer time of less than 1 min is needed for a complete analysis.

A procedure for determining long-term creep rates of soft clays by triaxial testing

The paper presents a procedure for determining a significantly larger range of creep rates (isotaches) of soft clays than found from standard incremental loading oedometer tests. The test is performed in a triaxial apparatus by first loading up to the virgin compression line at a relevant effective stress level and then determine the creep rates at different stress states along an unloading branch keeping the horizontal to vertical effective stress ratio constant. The test procedure is tested on a soft Norwegian plastic marine clay. Challenges due to the non-linear behaviour of natural soft clays where the standard creep and compression indexes are not constant values are demonstrated, and recommendations for how to account for this effect are given. The obtained results from the test are also compared with results from a parallel long-term creep phase in a standard oedometer apparatus taken from the same block sample.

Superelastic Behavior in NiTi Shape Memory Alloy Wires and Ribbons

Shape memory alloy ribbons in austenitic state were studied in a tensile testing machine in order to assess their superelastic behavior. They were compared with conventional materials and hair wire. The shape memoy alloy ribbon shows a particular behavior, with an ultimate tensile stress of about 1450 MPa reached at 9.5 % strain. The superelastic plateau was recorded around 590 MPa on loading and around 350 MPa on the unloading branch. Following multiple loading and unloading cycles, the superelastic behavior was not affected, nor was affected the ultimate tensile strength, that remained in the same range as for the one tested before cycling. The advantages of the superelastic ribbon compared to the ones of conventional materials are discussed. The superelastic properties of the ribbon are in the range of single crystals on what concerns the recoverable strains. This is attributed to the particular fine microstructure of the NiTi ribbon.

The Mach stem phenomenon for shaped obstacles buried in soil

The paper investigates the explosion characteristics of a below ground event of an explosive charge in proximity to a rigid cylindrical obstacle. The two-dimensional study simulates a line explosive and a parallel long cylindrical structure. The investigation shows that the unloading branch has a negligible effect on the peak pressure envelope whereas shear behavior and explosives burning have a considerable effect and should not be disregarded. The effect of the soil's equation of state and especially the full locking parameter on the pressure distribution on an obstacle has been studied. At a short standoff distance where a steep pressure growth beyond the full compaction point is developed, the pressure distributions envelope shows three maxima values that are located at some distance away from the axis of symmetry. It is different than the common single peak along the axis, in the case of a distant explosion. This effect is more pronounced for a medium having sharper pressure growth in the EOS beyond the full compaction point and for smaller charge standoff distances. The pressure distributions analysis indicates that the appearance of second (absolute-primary) and third (secondary) peaks are caused by the Mach stem effect appearing in a soil medium with full locking. The secondary peak pressure envelope maximum corresponds to the secondary Mach stem phenomenon that does not appear in the case of a planar wall, where the incident angle depends on the wave front curvature only.

Cyclic Stress–Strain Relationships of FRP Confined Concrete Members

External confinement of columns of existing reinforced concrete buildings using fiber-reinforced polymer (FRP) composites can significantly increase the ductility and compressive strength of structural members against seismic actions. Therefore, knowledge on the behavior of FRP confined concrete/reinforced concrete members under cyclic stresses is vitally important for realistic prediction of seismic performance of retrofitted structures. In this study, available experimental data on the cyclic stress–strain behavior of FRP confined concrete/reinforced concrete members are evaluated in a comparative manner, and accuracy of several stress–strain models proposed for FRP confined concrete is discussed in terms of prediction of the skeleton curves, as well as the unloading–reloading branches of the cyclic stress–strain relationships of the tested samples. Furthermore, a new model is proposed for the prediction of unloading and reloading branches based on the cyclic tests carried out specifically for this purpose. The predictions of the proposed model and two other available models proposed for cyclic loading are compared with experimental data, and a satisfactory agreement is observed between the predictions and experimental data.

A new continuous–discontinuous damage model: Cohesive cracks via an accurate energy-transfer process

A new continuous–discontinuous strategy to describe failure of quasi-brittle materials is presented. For the early stages of the failure process, a gradient-enhanced model based on smoothed displacements is employed. As soon as the damage parameter exceeds a critical value Dcrit<1, a cohesive crack is introduced. A new criterion to estimate the energy not yet dissipated by the bulk when switching models—from continuous to continuous–discontinuous—is proposed. Then, this energy is transferred to the cohesive crack thus ensuring that the continuous and the continuous–discontinuous strategies are energetically equivalent. Compared to other existing techniques, this new strategy accounts for the different unloading branches of damage models and thus, a more accurate estimation of the energy that has to be transferred is obtained. The performance of this technique is illustrated with one- and two-dimensional examples.