The performance of conventionally designed reinforced concrete (RC) structures during the 2011 Christchurch earthquake has demonstrated that there is greater uncertainty in the seismic performance of RC components than previously understood. RC frame and wall structures in the Christchurch central business district were observed to form undesirable cracks patterns in the plastic hinge region while yield penetration either side of cracks, and into development zones, were less than theoretical predictions. The implications of this unexpected behaviour: (i) significantly less available ductility; (ii) less hysteretic energy dissipation; and (iii) the localization of peak reinforcement strains, results in considerable doubt for the residual capacity of RC structures. The significance of these consequences has prompted a review of potential sources of uncertainty in seismic experimentation with the intention to improve the current confidence level for newly designed conventional RC structures. This paper attempts to revisit the principles of RC mechanics, in particular, to consider the influence of loading history, concrete tensile strength, and reinforcement ratio on the performance of ‘real’ RC structures compared to experimental test specimens.
One of the most controversial issues highlighted by the 2010-2011 Christchurch earthquake series and more recently the 2016 Kaikoura earthquake, has been the evident difficulty and lack of knowledge and guidelines for: a) evaluation of the residual capacity damaged buildings to sustain future aftershocks; b) selection and implementation of a series of reliable repairing techniques to bring back the structure to a condition substantially the same as prior to the earthquake; and c) predicting the cost (or cost-effectiveness) of such repair intervention, when compared to fully replacement costs while accounting for potential aftershocks in the near future. As a result of such complexity and uncertainty (i.e., risk), in combination with the possibility (unique in New Zealand when compared to most of the seismic-prone countries) to rely on financial support from the insurance companies, many modern buildings, in a number exceeding typical expectations from past experiences at an international level, have ended up being demolished. This has resulted in additional time and indirect losses prior to the full reconstruction, as well as in an increase in uncertainty on the actual relocation of the investment. This research project provides the main end-users and stakeholders (practitioner engineers, owners, local and government authorities, insurers, and regulatory agencies) with comprehensive evidence-based information to assess the residual capacity of damage reinforced concrete buildings, and to evaluate the feasibility of repairing techniques, in order to support their delicate decision-making process of repair vs. demolition or replacement. Literature review on effectiveness of epoxy injection repairs, as well as experimental tests on full-scale beam-column joints shows that repaired specimens have a reduced initial stiffness compared with the undamaged specimen, with no apparent strength reduction, sometimes exhibiting higher displacement ductility capacities. Although the bond between the steel and concrete is only partially restored, it still allows the repaired specimen to dissipate at least the same amount of hysteretic energy. Experimental tests on buildings subjected to earthquake loading demonstrate that even for severe damage levels, the ability of the epoxy injection to restore the initial stiffness of the structure is significant. Literature review on damage assessment and repair guidelines suggests that there is consensus within the international community that concrete elements with cracks less than 0.2 mm wide only require cosmetic repairs; epoxy injection repairs of cracks less and 2.0 mm wide and concrete patching of spalled cover concrete (i.e., minor to moderate damage) is an appropiate repair strategy; and for severe damaged components (e.g., cracks greater than 2.0 mm wide, crushing of the concrete core, buckling of the longitudinal reinforcement) local replacement of steel and/or concrete in addition to epoxy crack injection is more appropriate. In terms of expected cracking patterns, non-linear finite element investigations on well-designed reinforced concrete beam-to-column joints, have shown that lower number of cracks but with wider openings are expected to occur for larger compressive concrete strength, f’c, and lower reinforcement content, ρs. It was also observed that the tensile concrete strength, ft, strongly affects the expected cracking pattern in the beam-column joints, the latter being more uniformly distributed for lower ft values. Strain rate effects do not seem to play an important role on the cracking pattern. However, small variations in the cracking pattern were observed for low reinforcement content as it approaches to the minimum required as per NZS 3101:2006. Simple equations are proposed in this research project to relate the maximum and residual crack widths with the steel strain at peak displacement, with or without axial load. A literature review on fracture of reinforcing steel due to low-cycle fatigue, including recent research using steel manufactured per New Zealand standards is also presented. Experimental results describing the influence of the cyclic effect on the ultimate strain capacity of the steel are also discussed, and preliminary equations to account for that effect are proposed. A literature review on the current practice to assess the seismic residual capacity of structures is also presented. The various factors affecting the residual fatigue life at a component level (i.e., plastic hinge) of well-designed reinforced concrete frames are discussed, and equations to quantify each of them are proposed, as well as a methodology to incorporate them into a full displacement-based procedure for pre-earthquake and post-earthquake seismic assessment.
Gravelly soils’ liquefaction has been documented since early 19th century with however the focus being sand-silts mixture – coarse documentation of this topic, that gravels do in fact liquefy was only acknowledged as an observation. With time, we have been impacted by earthquakes, EQ causing more damage to our property: life and environment-natural and built. In this realm of EQ related-damage the ground or soils in general act as buffer between the epicentre and the structures at a concerned site. Further, in this area, upon the eventual acknowledgement of liquefaction of soils as a problem, massive efforts were undertaken to understand its mechanics, what causes and thereby how to mitigate its ill-effects. Down that lane, gravelly soils’ liquefaction was another milestone covered in early 20th century, and thus regarded as a problematic subject. This being a fairly recent acknowledgement, efforts have initiated in this direction (or area of particle size under consideration-gravels>2mm), with this research outputs intended to complement that research for the betterment of our understanding of what’s happening and how shall we best address it, given the circumstances: socio (life) - environment (structures) - economic (cost or cost-“effectiveness’) and of course political (our “willingness” to want to address the problem). Case histories from at least 29 earthquakes worldwide have indicated that liquefaction can occur in gravelly soils (both in natural deposits and manmade reclamations) inducing large ground deformation and causing severe damage to civil infrastructures. However, the evaluation of the liquefaction resistance of gravelly soils remains to be a major challenge in geotechnical earthquake engineering. To date, laboratory tests aimed at evaluating the liquefaction resistance of gravelly soils are still very limited, as compared to the large body of investigations carried out on assessing the liquefaction resistance of sandy soils. While there is a general agreement that the liquefaction resistance of gravelly soils can be as low as that of clean sands, previous studies suggested that the liquefaction behaviour of gravelly soils is significantly affected by two key factors, namely relative density (Dr) and gravel content (Gc). While it is clear that the liquefaction resistance of gravels increases with the increasing Dr, there are inconclusive and/or contradictory results regarding the effect of Gc on the liquefaction resistance of gravelly soils. Aimed at addressing this important topic, an investigation is being currently carried out by researchers at the University of Canterbury, UC. As a first step, a series of undrained cyclic triaxial tests were conducted on selected sand-gravel mixtures (SGMs), and inter-grain state framework concepts such as the equivalent and skeleton void ratios were used to describe the joint effects of Gc and Dr on the liquefaction resistance of SGMs. Following such experimental effort, this study is aimed at providing new and useful insights, by developing a critical state-based method combined with the inter-grain state framework to uniquely describe the liquefaction resistance of gravelly soils. To do so, a series of monotonic drained triaxial tests will be carried out on selected SGMs. The outcomes of this study, combined with those obtained to date by UC researchers, will greatly contribute to the expansion of a worldwide assessment database, and also towards the development of a reliable liquefaction triggering procedure for characterising the liquefaction potential of gravelly soils, which is of paramount importance not only for the New Zealand context, but worldwide. This will make it possible for practising engineers to identify liquefiable gravelly soils in advance and make sound recommendations to minimise the impact of such hazards on land, and civil infrastructures.
There is a relationship between inelastic deformation and energy dissipation in structures that are subjected to earthquake ground motions. Thus, if seismic energy dissipation can be achieved by means of a separate non-load bearing supplementary damping system, the load bearing structure can remain elastic with continuing serviceability following the design level earthquake. This research was carried out to investigate the advantages of using added damping in structures. The control system consists of passive friction dampers called ring spring dampers installed in the ground floor of the structure using a tendon to transmit the forces to the other parts of the structure. The ring springs dampers are friction devices consisting of inner and outer ring elements assembled to form a spring stack. External load applied to the spring produces sliding action across mating ring interfaces. The damping forces generated by the dampers and transferred in the supplemental system to the structure by the tendon and horizontal links oppose the internal loads. A four storey-two bay steel frame structure was used in the study. Experimental and analytical studies to investigate the effectiveness of a supplemental control system are presented. The model was subjected to a series of earthquake simulations on the shaking table in the Structural Laboratory of the Civil Engineering Department, at the University of Canterbury. The earthquake simulation tests have been performed on the structure both with and without the supplemental control system. The earthquake simulations were a series of gradually increasing intensity replications of two commonly used earthquake records. This thesis includes detailed description of the structural model, the supplemental control system, the ring springs dampers and the data obtained during the testing. Analyses were then carried out on a twelve storey framed structure to investigate the possible tendon arrangements and the size and type of dampers required to control the response of a real building. Guidelines for determining the appropriateness of including a supplemental damping system have been investigated. The main features of the supplemental control system adopted in this research are: • It is a passive control system with extreme reliability and having no dependence on external power sources to effect the control action. These power sources may not be available during a major earthquake. • Ring springs are steel friction devices capable of absorbing large amounts of input energy. No liquid leakage can occur and minimal maintenance is required for the ring spring dampers. • With a damper-tendon system, the distribution of the dampers throughout the structure is not so critical. Only one or two dampers are used to produce the damping forces needed, and forces are then transferred to the rest of the building by the tendon system. • It is a relatively inexpensive control system with a long useful life.
Geosynthetic reinforced soil (GRS) walls involve the use of geosynthetic reinforcement (polymer material) within the retained backfill, forming a reinforced soil block where transmission of overturning and sliding forces on the wall to the backfill occurs. Key advantages of GRS systems include the reduced need for large foundations, cost reduction (up to 50%), lower environmental costs, faster construction and significantly improved seismic performance as observed in previous earthquakes. Design methods in New Zealand have not been well established and as a result, GRS structures do not have a uniform level of seismic and static resistance; hence involve different risks of failure. Further research is required to better understand the seismic behaviour of GRS structures to advance design practices. The experimental study of this research involved a series of twelve 1-g shake table tests on reduced-scale (1:5) GRS wall models using the University of Canterbury shake-table. The seismic excitation of the models was unidirectional sinusoidal input motion with a predominant frequency of 5Hz and 10s duration. Seismic excitation of the model commenced at an acceleration amplitude level of 0.1g and was incrementally increased by 0.1g in subsequent excitation levels up to failure (excessive displacement of the wall panel). The wall models were 900mm high with a full-height rigid facing panel and five layers of Microgird reinforcement (reinforcement spacing of 150mm). The wall panel toe was founded on a rigid foundation and was free to slide. The backfill deposit was constructed from dry Albany sand to a backfill relative density, Dr = 85% or 50% through model vibration. The influence of GRS wall parameters such as reinforcement length and layout, backfill density and application of a 3kPa surcharge on the backfill surface was investigated in the testing sequence. Through extensive instrumentation of the wall models, the wall facing displacements, backfill accelerations, earth pressures and reinforcement loads were recorded at the varying levels of model excitation. Additionally, backfill deformation was also measured through high-speed imaging and Geotechnical Particle Image Velocimetry (GeoPIV) analysis. The GeoPIV analysis enabled the identification of the evolution of shear strains and volumetric strains within the backfill at low strain levels before failure of the wall thus allowing interpretations to be made regarding the strain development and shear band progression within the retained backfill. Rotation about the wall toe was the predominant failure mechanism in all excitation level with sliding only significant in the last two excitation levels, resulting in a bi-linear displacement acceleration curve. An increase in acceleration amplification with increasing excitation was observed with amplification factors of up to 1.5 recorded. Maximum seismic and static horizontal earth pressures were recorded at failure and were recorded at the wall toe. The highest reinforcement load was recorded at the lowest (deepest in the backfill) reinforcement layer with a decrease in peak load observed at failure, possibly due to pullout failure of the reinforcement layer. Conversely, peak reinforcement load was recorded at failure for the top reinforcement layer. The staggered reinforcement models exhibited greater wall stability than the uniform reinforcement models of L/H=0.75. However, similar critical accelerations were determined for the two wall models due to the coarseness of excitation level increments of 0.1g. The extended top reinforcements were found to restrict the rotational component of displacement and prevented the development of a preliminary shear band at the middle reinforcement layer, contributing positively to wall stability. Lower acceleration amplification factors were determined for the longer uniform reinforcement length models due to reduced model deformation. A greater distribution of reinforcement load towards the top two extended reinforcement layers was also observed in the staggered wall models. An increase in model backfill density was observed to result in greater wall stability than an increase in uniform reinforcement length. Greater acceleration amplification was observed in looser backfill models due to their lower model stiffness. Due to greater confinement of the reinforcement layers, greater reinforcement loads were developed in higher density wall models with less wall movement required to engage the reinforcement layers and mobilise their resistance. The application of surcharge on the backfill was observed to initially increase the wall stability due to greater normal stresses within the backfill but at greater excitation levels, the surcharge contribution to wall destabilising inertial forces outweighs its contribution to wall stability. As a result, no clear influence of surcharge on the critical acceleration of the wall models was observed. Lower acceleration amplification factors were observed for the surcharged models as the surcharge acts as a damper during excitation. The application of the surcharge also increases the magnitude of reinforcement load developed due to greater confinement and increased wall destabilising forces. The rotation of the wall panel resulted in the progressive development of shears surface with depth that extended from the backfill surface to the ends of the reinforcement (edge of the reinforced soil block). The resultant failure plane would have extended from the backfill surface to the lowest reinforcement layer before developing at the toe of the wall, forming a two-wedge failure mechanism. This is confirmed by development of failure planes at the lowest reinforcement layer (deepest with the backfill) and at the wall toe observed at the critical acceleration level. Key observations of the effect of different wall parameters from the GeoPIV results are found to be in good agreement with conclusions developed from the other forms of instrumentation. Further research is required to achieve the goal of developing seismic guidelines for GRS walls in geotechnical structures in New Zealand. This includes developing and testing wall models with a different facing type (segmental or wrap-around facing), load cell instrumentation of all reinforcement layers, dynamic loading on the wall panel and the use of local soils as the backfill material. Lastly, the limitations of the experimental procedure and wall models should be understood.
The effects of soil-foundation-structure interaction (SFSI) have been a topic of discussion amongst the structural and geotechnical community for many decades. The complexity of the mechanisms, as well as the need for inter-disciplinary knowledge of geotechnical and structural dynamics has plagued the advancement and the consequent inclusion of SFSI effects in design. A rigorous performance-based design methodology should not just consider the performance of the superstructure but the supporting foundation system as well. Case studies throughout history (eg. Kobe 1995, Kocaeli 1999 and Christchurch earthquakes 2010-2011) have demonstrated that a poor performance at the foundation level can result in a full demolition of the structure and, in general terms, that the extent of damage to, and repairability of, the building system as a whole, is given by the combination of the damage to the soil, foundation and superstructure. The lack of consideration of the modifying factors of SFSI and an absence of intuitive performance levels for controlling foundation and soil behaviour under seismic loads has resulted in inadequate designs for buildings sited on soft soil. For engineers to be satisfied that their designs meet the given performance levels they must first, understand how SFSI affects the overall system performance and secondly have tools available to adequately account for it in their design/assessment. This dissertation presents an integrated performance-based design procedure for buildingfoundation systems that considers all of the major mechanisms of SFSI. A new soil-foundation macro-element model was implemented into a nonlinear finite element software and validated against several experimental tests. The numerical model was used to provide insights in to the mechanisms of SFSI and statistical analysis on the results yielded simple expressions that allow the behaviour to be quantified. Particular attention was paid to the effects of shear force on the foundation response and the quantification of the rocking mode of response. The residual deformations of the superstructure and distribution of forces up the structure were also investigated. All of the major SFSI mechanisms are discussed in detail and targeted numerical studies are used to explain and demonstrate concepts. The design procedure was validated through the design and assessment of a series of concrete buildings that were designed to account for the effects of SFSI.
This thesis studies the behaviour of diaphragms in multi-storey timber buildings by providing methods for the estimation of the diaphragm force demand, developing an Equivalent Truss Method for the analysis of timber diaphragms, and experimentally investigating the effects of displacement incompatibilities between the diaphragm and the lateral load resisting system and developing methods for their mitigation. The need to better understand the behaviour of diaphragms in timber buildings was highlighted by the recent 2010-2011 Canterbury Earthquake series, where a number of diaphragms in traditional concrete buildings performed poorly, compromising the lateral load resistance of the structure. Although shortcomings in the estimation of force demand, and in the analysis and design of concrete floor diaphragms have already been partially addressed by other researchers, the behaviour of diaphragms in modern multi-storey timber buildings in general, and in low damage Pres-Lam buildings (consisting of post-tensioned timber members) in particular is still unknown. The recent demand of mid-rise commercial timber buildings of ten storeys and beyond has further highlighted the lack of appropriate methods to analyse timber diaphragms with irregular floor geometries and large spans made of both light timber framing and massive timber panels. Due to the lower stiffness of timber lateral load resisting systems, compared with traditional construction materials, and the addition of in-plane flexible diaphragms, the effect of higher modes on the global dynamic behaviour of a structure becomes more critical. The results from a parametric non-linear time-history analysis on a series of timber frame and wall structures showed increased storey shear and moment demands even for four storey structures when compared to simplistic equivalent static analysis. This effect could successfully be predicted with methods available in literature. The presence of diaphragm flexibility increased diaphragm inter-storey drifts and the peak diaphragm demand in stiff wall structures, but had less influence on the storey shears and moments. Diaphragm force demands proved to be significantly higher than the forces derived from equivalent static analysis, leading to potentially unsafe designs. It is suggested to design all diaphragms for the same peak demand; a simplified approach to estimate these diaphragm forces is proposed for both frame and wall structures. Modern architecture often requires complex floor geometries with long spans leading to stress concentrations, high force demands and potentially large deformations in the diaphragms. There is a lack of guidance and regulation regarding the analysis and design of timber diaphragms and a practical alternative to the simplistic equivalent deep beam analysis or costly finite element modelling is required. An Equivalent Truss Method for the analysis of both light timber framed and massive timber diaphragms is proposed, based on analytical formulations and verified against finite element models. With this method the panel unit shear forces (shear flow) and therefore the fastener demand, chord forces and reaction forces can be evaluated. Because the panel stiffness and fastener stiffness are accounted for, diaphragm deflection, torsional effects and transfer forces can also be assessed. The proposed analysis method is intuitive and can be used with basic analysis software. If required, it can easily be adapted for the use with diaphragms working in the non-linear range. Damage to floor diaphragms resulting from displacement incompatibilities due to frame elongation or out-of plane deformation of walls can compromise the transfer of inertial forces to the lateral load resisting system as well as the stability of other structural elements. Two post-tensioned timber frame structures under quasi-static cyclic and dynamic load, respectively, were tested with different diaphragm panel layouts and connections investigating their ability to accommodate frame elongations. Additionally, a post-tensioned timber wall was loaded under horizontal cyclic loads through two pairs of collector beams. Several different connection details between the wall and the beams were tested, and no damage to the collector beams or connections was observed in any of the tests. To evaluate the increased strength and stiffness due to the wall-beam interaction an analytical procedure is presented. Finally, a timber staircase core was tested under bi-directional loading. Different connection details were used to study the effect of displacement incompatibilities between the orthogonal collector beams. These experiments showed that floor damage due to displacement incompatibilities can be prevented, even with high levels of lateral drift, by the flexibility of well-designed connections and the flexibility of the timber elements. It can be concluded that the flexibility of timber members and the flexibility of their connections play a major role in the behaviour of timber buildings in general and of diaphragms specifically under seismic loads. The increased flexibility enhances higher mode effects and alters the diaphragm force demand. Simple methods are provided to account for this effect on the storey shear, moment and drift demands as well as the diaphragm force demands. The analysis of light timber framing and massive timber diaphragms can be successfully analysed with an Equivalent Truss Method, which is calibrated by accounting for the panel shear and fastener stiffnesses. Finally, displacement incompatibilities in frame and wall structures can be accommodated by the flexibilities of the diaphragm panels and relative connections. A design recommendations chapter summarizes all findings and allows a designer to estimate diaphragm forces, to analyse the force path in timber diaphragms and to detail the connections to allow for displacement incompatibilities in multi-storey timber buildings.
Timber has experienced renewed interests as a sustainable building material in recent times. Although traditionally it has been the prime choice for residential construction in New Zealand and some other parts of the world, its use can be increased significantly in the future through a wider range of applications, particularly when adopting engineered wood material, Research has been started on the development of innovative solutions for multi-storey non-residential timber buildings in recent years and this study is part of that initiative. Application of timber in commercial and office spaces posed some challenges with requirements of large column-free spaces. The current construction practice with timber is not properly suited for structures with the aforementioned required characteristics and new type of structures has to be developed for this type of applications. Any new structural system has to have adequate capacity for carry the gravity and lateral loads due to occupancy and the environmental effects. Along with wind loading, one of the major sources of lateral loads is earthquakes. New Zealand, being located in a seismically active region, has significant risk of earthquake hazard specially in the central region of the country and any structure has be designed for the seismic loading appropriate for the locality. There have been some significant developments in precast concrete in terms of solutions for earthquake resistant structures in the last decade. The “Hybrid” concept combining post-tensioning and energy dissipating elements with structural members has been introduced in the late 1990s by the precast concrete industry to achieve moment-resistant connections based on dry jointed ductile connections. Recent research at the University of Canterbury has shown that the concept can be adopted for timber for similar applications. Hybrid timber frames using post-tensioned beams and dissipaters have the potential to allow longer spans and smaller cross sections than other forms of solid timber frames. Buildings with post-tensioned frames and walls can have larger column-free spaces which is a particular advantage for non-residential applications. While other researchers are focusing on whole structural systems, this research concentrated on the analysis and design of individual members and connections between members or between member and foundation. This thesis extends existing knowledge on the seismic behaviour and response of post-tensioned single walls, columns under uni-direction loads and small scale beam-column joint connections into the response and design of post-tensioned coupled walls, columns under bi-directional loading and full-scale beam-column joints, as well as to generate further insight into practical applications of the design concept for subassemblies. Extensive experimental investigation of walls, column and beam-column joints provided valuable confirmation of the satisfactory performance of these systems. In general, they all exhibited almost complete re-centering capacity and significant energy dissipation, without resulting into structural damage. The different configurations tested also demonstrated the flexibility in design and possibilities for applications in practical structures. Based on the experimental results, numerical models were developed and refined from previous literature in precast concrete jointed ductile connections to predict the behaviour of post-tensioned timber subassemblies. The calibrated models also suggest the values of relevant parameters for applications in further analysis and design. Section analyses involving those parameters are performed to develop procedures to calculate moment capacities of the subassemblies. The typical features and geometric configurations the different types of subassemblies are similar with the only major difference in the connection interfaces. With adoption of appropriate values representing the corresponding connection interface and incorporation of the details of geometry and configurations, moment capacities of all the subassemblies can be calculated with the same scheme. That is found to be true for both post-tensioned-only and hybrid specimens and also applied for both uni-directional and bi-directional loading. The common section analysis and moment capacity calculation procedure is applied in the general design approach for subassemblies.
Recent experiences from the Darfield and Canterbury, New Zealand earthquakes have shown that the soft soil condition of saturated liquefiable sand has a profound effect on seismic response of buildings, bridges and other lifeline infrastructure. For detailed evaluation of seismic response three dimensional integrated analysis comprising structure, foundation and soil is required; such an integrated analysis is referred to as Soil Foundation Structure Interaction (SFSI) in literatures. SFSI is a three-dimensional problem because of three primary reasons: first, foundation systems are three-dimensional in form and geometry; second, ground motions are three-dimensional, producing complex multiaxial stresses in soils, foundations and structure; and third, soils in particular are sensitive to complex stress because of heterogeneity of soils leading to a highly anisotropic constitutive behaviour. In literatures the majority of seismic response analyses are limited to plane strain configuration because of lack of adequate constitutive models both for soils and structures, and computational limitation. Such two-dimensional analyses do not represent a complete view of the problem for the three reasons noted above. In this context, the present research aims to develop a three-dimensional mathematical formulation of an existing plane-strain elasto-plastic constitutive model of sand developed by Cubrinovski and Ishihara (1998b). This model has been specially formulated to simulate liquefaction behaviour of sand under ground motion induced earthquake loading, and has been well-validated and widely implemented in verifcation of shake table and centrifuge tests, as well as conventional ground response analysis and evaluation of case histories. The approach adopted herein is based entirely on the mathematical theory of plasticity and utilises some unique features of the bounding surface plasticity formalised by Dafalias (1986). The principal constitutive parameters, equations, assumptions and empiricism of the existing plane-strain model are adopted in their exact form in the three-dimensional version. Therefore, the original two-dimensional model can be considered as a true subset of the three-dimensional form; the original model can be retrieved when the tensorial quantities of the three dimensional version are reduced to that of the plane-strain configuration. Anisotropic Drucker-Prager type failure surface has been adopted for the three-dimensional version to accommodate triaxial stress path. Accordingly, a new mixed hardening rule based on Mroz’s approach of homogeneous surfaces (Mroz, 1967) has been introduced for the virgin loading surface. The three-dimensional version is validated against experimental data for cyclic torsional and triaxial stress paths.
