Rigid-plastic Analysis Research Articles

Application and Validation of a Simplified Approach to Evaluate the Seismic Performances of Steel MR-Frames

The main aim of this work is to validate the application of a simplified performance-based method for assessing the seismic performance of steel buildings, focusing particularly on Moment Resisting Frames (MRFs) through nonlinear analyses. This simplified method defines the capacity curve of a structure through elastic and rigid-plastic analyses, calibrated by regression analyses conducted on 420 structures. To assess its accuracy, the method was compared with other analytical approaches, including incremental dynamic analyses (IDA) provided by existing codes. These analyses were performed on both real structures and simulated designs, considering recent and older codes. The comparison of capacity results derived from code-based approaches and IDA, aligned with the limit states outlined in current codes, showcased the high reliability of the proposed simplified assessment approach.

Three-dimensional plasticity-based topology optimization with smoothed finite element analysis

AbstractThis paper presents a novel plasticity-based formulation for three-dimensional (3D) topology optimization of continuum structures. The proposed formulation addresses the optimization problem by combining mixed rigid-plastic analysis with density-based topology optimization, resulting in a volume minimization approach. Unlike conventional stress-constrained topology optimization methods that rely on linear elastic structural analysis, our developed formulation focuses on enhancing the loading capacity of the designed structures based on the plastic limit theory, leading to more cost-effective designs. To improve computational efficiency, we employ the smoothed finite element technique in our proposed method, enabling the utilization of linear tetrahedral elements for 3D mesh refinement. Moreover, the final formulation of our developed method can be efficiently solved using the advanced primal–dual interior point method, eliminating the need for a separate nonlinear finite element structural analysis. Numerical examples are presented to demonstrate the effectiveness of the proposed approach in offering enhanced design possibilities for continuum structures.

An exact solution of active earth pressures based on a statically admissible stress field

Static lateral earth pressure problems are a central and long-standing topic in geotechnical engineering, but their exact solutions are lacking. This study is dedicated to providing an exact solution for active earth pressure exerted against rigid retaining walls with various wall geometries, backfill geometries, and wall roughness. A numerical model based on the rigid plastic analysis, namely a fully plastic solution, has been successfully developed to search out statically admissible stress fields in the backfills. In particular, Sokolovski’s theory coupled with the Mohr-Coulomb criterion has been adopted to enable the statically admissible stress fields to be developed. Robust numerical frameworks based on the fifth-order adaptive Runge-Kutta method, and a shooting technique has demonstrated efficacy in coping with the stress singularity that appeared in the analyzed domain, thus the admissible stress fields have been successfully figured out. The accuracy of the proposed solution is satisfactorily verified and validated by comparing the obtained results with exact values and those of experimental results available in the literature. The admissible stress fields explored in the backfills of rigid walls elucidate the load transfer mechanism of cohesionless materials in the backfills; the potential effects of various parameters e.g., wall geometries, backfill geometries, and wall roughness on the active earth pressure coefficients have been disclosed thereof. In essence, this study provided a conservative answer for the static active earth pressure problems whereby the exact values of active earth pressure coefficients are tabulated and provided for engineering practice.

Simplified Evaluation of Plastic Rotation Demand for Existing EBFs Equipped with Short Links

The seismic events that occurred in the last decades have highlighted the importance of a correct design of the structures in seismic areas and the seismic inadequacy of a large part of the built heritage. Modern codes are still lacking in terms of prescriptions for the evaluation of the seismic performance of existing buildings. The present work proposes a simplified method for the evaluation of the demand in terms of plastic rotation for short links of steel Eccentrically Braced Frames (EBFs). A relationship for the evaluation of the demand, that exploits elastic analysis and rigid-plastic analysis extended to the second-order effects, is proposed. The calibration of this relationship was carried out on 420 EBFs equipped with short links designed according to three different approaches. The 420 frames have been also used to analyze the behavior in the plastic range of EBF type structures equipped with short links. The study also provides an extensive analysis on the influence of plastic redistribution capacity on the demand in terms of plastic rotations of links, corresponding to the achievement of the maximum bearing capacity. The obtained relation can be exploited as an assessment tool by comparing the demand with the link capacity. Moreover, from a performance-based design point of view, the same can be used for predicting the required ultimate plastic rotation as a function of the plastic redistribution capacity of the structure.

On the composite behavior of a rebar truss ribbed slab with incorporated shuttering made of lipped channel section

A novel slab system composed by a cold-formed steel lipped channel section as permanent shuttering fastened to a rebar truss through plastic connectors was released in the market. However, the shear behavior between shuttering and concrete has not been fully investigated through experiments, reason why the system is still designed as a non-composite system. However, based on four-point bending tests, this research investigated the mechanical behavior of composite slabs in terms of both strength and stiffness. It was observed that the steel profile fully interacted with the concrete, increasing bending capacity by 2.61 times and ductility by five times when compared to prototypes without shuttering. Finally, the experimental results were compared to a proposed design procedure of the EN 1994-1-1:2004 based on rigid plastic analysis, which showed good agreement with theory, with average deviation of 6.5%.

Interfacial shear and flexural performances of steel–precast UHPC composite beams: Full-depth slabs with studs vs. demountable slabs with bolts

Two types of steel–precast ultra-high-performance concrete (UHPC) composite beams were investigated in this study, which can be vastly used in accelerated bridge construction (ABC). One of them contains full-depth slabs with stud shear connectors embedded in shear pockets, while the remaining one adopts demountable slabs with high-strength friction-grip bolts (HSFGBs). Four-point bending tests were conducted on six large-scale composite beams to investigate the interfacial shear and flexural performance of such innovative types of steel–precast UHPC composite structures. The failure modes, vertical deflection, horizontal interfacial slip and the stain distribution at mid-span cross-section were investigated. The experimental results indicated that the cracking patterns and failure modes of the composite beams were highly dependent on the slab concrete types and the degrees of shear connection. The composite beams adopting UHPC slabs achieved larger cracking resistance and higher ultimate strength than those containing normal-strength concrete slabs and favorable ductility. Similar trends were also presented by the composite beams with large degrees of shear connection. The composite beams with HSFGBs showed enhanced load capacities and larger vertical deflections and horizontal interface slips at peak load than those adopting stud shear connectors. Additionally, the composite beams that failed in shear connector fractures presented shear resistance–interface slip relationships comparable to their push-out counterparts. Finally, theoretical formulas based on rigid plastic analysis were proposed for predicting the flexural capacity of both types of composite beams.

Rigid-Plastic Analysis of Seismic Resistant T-Frame considering Moment-Shear Interaction

To select a seismic resistant system, in addition to strength and stiffness, ductility and energy dissipation are important to be considered. Structures have nonlinear behavior under the influence of moderate and strong earthquakes. One of the primary aims in designing seismic resistant structures is to prevent the formation of undesirable collapse mechanisms such as the collapse in only a few storeys of the structure that leads to low energy dissipation. In order to achieve a global collapse mechanism, modern seismic codes provide simple rules for design, which is called the hierarchy criteria. Although these simple criteria could prevent the formation of a soft storey mechanism, they could not lead to an optimal global collapse mechanism. In these mechanisms, the energy dissipation zones include all the yielding zones such as beams, while all other parts of the structure have remained in the elastic range. TRF (T-resisting frame) is an innovative lateral resistant system introduced for architectural reasons and to provide more energy dissipating capability. This system has several collapse mechanisms due to the moment, shear, or moment-shear behavior of its members. In this paper, within the framework of the theory of plastic mechanism control, the rigid-plastic analysis of the TRF system to achieve the desired collapse mechanism is used by considering the moment-shear interaction. According to these analyses, which are performed on a single storey frame, simple hierarchy criteria are developed to create the desired collapse mechanism. Also, these criteria prevent undesired collapse mechanisms in order to have more energy dissipation and more ductility. Finally, the validity of the proposed criteria has been verified by the pushover analysis.

Multi-criteria Estimation of Load-Bearing Capacity of Solids

The article discusses the problem of the load-bearing capacity of a deformable solid in the current configuration, which may be either reference (undeformed) or actual (deformed). An original variational approach is proposed, where, depending on different engineering considerations, the root-mean-square values (rms) of any stress components in various sub-domains are calculated and used to estimate the load-bearing capacity of the current configuration of a solid for given loads. For example, an estimate of the rms value of hydrostatic pressure is necessary for analyzing the load-bearing capacity of concrete. On the other hand, the rms value of all stress components in the vicinity of a concentrator (a neck or a groove) makes it possible to estimate the overall strength of a sample in full-scale rupture experiments. The proposed approach differs fundamentally from the classical rigid-plastic analysis of the load-bearing capacity of the reference configuration of a solid, within which a point criterion for the maximum intensity of tangential stresses or the local strain energy density is used. One of the drawbacks of rigid-plastic analysis is the uncertainty of the stress field in the rigid sub-domains of a solid. This effect is not present in the proposed method of assessment. The article provides informative and straightforward analytical and numerical examples showing the non-triviality of the proposed methodology.

Toward seismic design of tall steel moment resisting frames using the theory of plastic mechanism control

The kinematic approach and mechanism equilibrium curve in the rigid-plastic analysis framework are the basis of the theory of plastic mechanism control (TPMC). Applying this method for structural design will lead to a global collapse mechanism and prevent undesired collapse mechanisms. In this paper, to satisfy the TPMC conditions in tall moment-resisting frames (MRFs), linear programming (LP) is applied such that the TPMC conditions are defined as the optimization formulation. To investigate the effects of different structural constraints such as preventing the partial beam plastic hinge mechanisms (PBM), partial column plastic hinge mechanisms (PCM), and also to consider the strong-column weak-beam (SCWB) criterion in design MRFs by TPMC, a series of three (10, 15 and 20-story) MRFs is studied. Each frame is designed in the four different categories of the constraints: 1) PBM, 2) PBM + SCWB, 3) PBM + PCM, and 4) PBM + SCWB + PCM. The accuracy of the proposed design procedure is evaluated using pushover and nonlinear time history analyses. The results show that the designed structures reveal excellent seismic performance, and the global mechanisms are successfully achieved. The obtained nonlinear responses show that the seismic performances of the examined frames with different constraints are similar, and the PBM constraint is adequate for preventing undesired collapse mechanisms.

Rigid plastic analysis for the seismic performance evaluation of steel storage racks

The aim of the paper is the prediction of the seismic collapse mode of steel storage pallet racks under seismic loads. The attention paid by the researchers on the behaviour of the industrial steel storage pallets racks is increased over the years thanks to their high dead-to-live load ratio. In fact, these structures, generally made by cold-formed thin-walled profiles, present very low structural costs but can support large and expensive loads. The paper presents a prediction of the seismic collapse modes of multi-storey racks. The analysis of the possible collapse modes has been made by an approach based on the kinematic theorem of plastic collapse extended to the second order effects by means of the concept of collapse mechanism equilibrium curve. In this way, the dissipative behaviour of racks is determined with a simpler method than the pushover analysis. Parametric analyses have been performed on 24 racks, differing for the geometric layout and cross-section of the components, designed in according to the EN16618 and EN15512 requirements. The obtained results have highlighted that, in all the considered cases, the global collapse mechanism, that is the safest one, never develops, leading to a dangerous situation that must be avoided to preserve the structure during a seismic event. Although the studied racks follow all the codes prescriptions, the development of a dissipative collapse mechanism is not achieved. In addition, also the variability of load distribution has been considered, reflecting the different pallet positions assumed during the in-service life of the racks, to point out its influence on the collapse mechanism. The information carried out from the paper can be very useful for designers and manufacturers because it allows to better understand the racks behaviour in seismic load condition.

Dynamic modelling of elastic-plastic beams under impact

This study is aimed to investigate the dynamic response of a fully clamped beam subjected to impact loadings by a rigid mass. First, the analytical formulae are derived for predicting the quasi-static force-deformation relations of an elastic-plastic beam during loading and unloading. For a particular impact scenario in which the striker with a heavy mass impinges the mid-span of the beam at a low speed, it is assumed that the dynamic deformation mode remains identical to the quasi-static one. By employing energy method, the quasi-static characteristics are utilized to develop an elastic-plastic theoretical model of the beam under impact loadings so as to predict its dynamic behaviour during the loading, unloading and subsequent elastic vibration stages. Validation and comparison are conducted against numerical simulations using the commercial software package ABAQUS and it is shown that the proposed elastic-plastic theoretical model is capable of providing a good prediction of the overall dynamic behaviour of the beam under impact. In parallel, comparisons between the predictions of the current elastic-plastic model and those of the rigid-plastic analysis are carried out to highlight the elastic effect on the accuracy of modelling an elastic-plastic beam under impact.

Plastic design of moment resisting frames using mechanism control

In this paper, the plastic design with mechanism control (PDMC) for steel moment resisting frames (MRFs) is presented. The PDMC method in rigid plastic analysis, by considering the second-order effects, is guaranteed the achievement of global mechanism and prevented undesired mechanisms such as the soft story, which leads to decrease the ductility and energy dissipation capacity of the structures. The concept of PDMC is based on the plastic rotation of columns in developing a mechanism in which for a specified target displacement, the global mechanism has a minimum plastic rotation among all other undesired collapse mechanisms. In the presented method, the design conditions for story mechanisms are derived by using an energy equilibrium equation. For demonstrating the accuracy of the proposed design procedures, both pushover and nonlinear incremental dynamic analyses (IDAs) are applied to evaluate the structural performance. The results show that the designed structures comprise high seismic performance and the design objective that was to obtain a global mechanism is achieved.

Experimental, numerical, and theoretical studies of the response of short cylindrical stainless steel tubes under lateral air blast loading

This paper presents a series of experimental, numerical and theoretical analyses of cylindrical tubes subjected to lateral external air blast loading. The cylindrical tubes are laterally loaded in the mid span using cylindrical TNT with a charge mass of 20, 25, 30, and 40 g detonated at a stand-off distance of 50, 100, and 150 mm. The resultant impulses produce failures ranging from large inelastic deformation of the cylindrical tube to tearing of the structure. The results are compared with a modal solution and numerical simulations carried out in ABAQUS 6.12-1, where good correlations with the numerical simulations are observed. The experimentally obtained deformed cross-section can be divided into three parts: a flat top section, two arcs near the flat top and an undeformed circular arc at the bottom. The axial deformation section can also be divided into three parts: the first part is a large plastic deformation region at the center of the structure that divides two outward moving plastic hinges from the second part, which can be defined as a rigid section moving around the plastic hinge at the support, and the third part is the undeformed boundary region. The above deformation patterns were used to develop a theoretical model based on a rigid-plastic analysis. Since the experimental pressure of the cylindrical explosive was more localized than the theoretically defined pressure distribution, the modal solution together with the upper bound yield condition of the tube used in the analysis, overestimated the resistance of the tube.

Ultimate Strength of Hull Plates under Monotonic and Cyclic Uniaxial Compression

As the most fundamental structural components of the ship hull, the hull plates' ultimate strength not only affects their own load bearing capacity, but also contributes to the ultimate strength of the whole ship hull. Through the literature survey of the ultimate strength of hull plates, it can be found that most of the ultimate strength research studies focused on the ultimate strength behavior of hull plates under monotonic increasing loading. Considering the motion and deformation of hull girders among sea waves, the ultimate bearing capacity research of the hull structures under cyclic loading maybe more in line with reality. In this article, the local instability tests of plates under cyclic uniaxial loading carried out by Fukumoto and Kusama is reanalyzed. Based on the tests carried out by Fukumoto and Kusama, some fundamental behaviors about the ultimate strength and unloading and reloading paths of the plates under cyclic uniaxial compression are summarized. Then, a simple approach to construct the average stress-average strain relationship of the hull plates exposed to cyclic compression is proposed. Finally, tests on six square-column models are carried out to simulate the collapse behavior of the hull plates under cyclic uniaxial compression. The present study shows that the average stress-average strain relationship of hull plates under cyclic compression can be constructed by a simple approach conveniently and accurately. Meanwhile, the ultimate strength behavior of the hull plates under cyclic uniaxial compression can also be reflected well by the simple approach. 1. Introduction Ultimate strength is a critical and fundamental consideration in the design of ship and offshore structures. An appropriate level of the ultimate strength of a ship hull is perhaps its most important guarantee of structural safety. Accurate assessment of the ultimate strength and/or residual ultimate strength of a ship is pertinent not only for the initial design, but also for the operation, maintenance, repair, and even disaster management, rescue, and salvage assessment of the ship hulls in accident situations. The first attempt to calculate the ultimate hull girder strength was developed by Caldwell (1965), who applied "Rigid Plastic Mechanism Analysis" to evaluate the ultimate strength of the hull girder. From then on, the ultimate strength of ship structures (including plates, stiffened plates, and hull girders) was widely studied by numerous scholars. Certainly, a great deal of developments have made naval architects obtain a clearer understanding of the ultimate strength behaviors of ship structures. However, because of the complexity and importance of the problems related to the ultimate strength, they still need to be explored more deeply. As described in the report of 19th International Ship and Offshore Structures Congress (ISSC 2015), "with these developments ultimate strength assessment is now becoming a more important issue to ensure the safety of ship structures."

Ultimate Strength Assessment of Hull Girder under Cyclic Bending Based on Smith’s Method

Ultimate strength is a critical and fundamental consideration in the design of ship and offshore structures. Accurate assessment of the (residual) ultimate strength of a ship hull is important not only for the initial design, but also for the operation, maintenance, repair, and even for disaster management, rescue, and salvage assessment of the ship hulls. In fact, under ocean wave conditions, the hull girder is likely to withstand extreme cyclic bending. In this article, a simplified method based on Smith's method is proposed to assess the ultimate strength of the ship hulls under extreme cyclic bending. To obtain the nonlinear average stress-average strain relationship (σ-ε curve) of the basic structural element (consisting of a stiffener with its attaching plating) under cyclic in-plane compression and tension, systematic calculations are carried out by using a combination of the Beam-Column analysis and the nonlinear finite element method (NLFEM). By introducing the cyclic σ-ε curves of basic structural elements to a simplified calculation program, which is developed by the authors and based on Smith's method, the ultimate strength of hull girders exposed to extreme cyclic bending is subsequently determined. The ultimate strength of three box girder models and an actual ship are obtained under cyclic bending by using the authors' developed procedures including the simplified calculation program and are compared with the corresponding results from NLFEM calculations and also the box girder experiments. 1. Introduction Ultimate strength is a critical and fundamental consideration in the design of ship and offshore structures. An appropriate level of the ultimate strength of a ship hull is perhaps its most important guarantee of structural safety. Accurate assessment of the ultimate strength and/ or residual ultimate strength of a ship is pertinent not only for the initial design, but also for the operation, maintenance, repair, and even for disaster management, rescue, and salvage assessment of the ship hulls in accidental situations. Perhaps the first attempt to calculate ship hull girder ultimate strength was by Caldwell (1965), who applied a type of rigid plastic mechanism analysis to evaluate strength. Since then, naval architects have gradually realized various technology developments and advances of importance to the ultimate strength assessment of ship structures. Over the last two decades, the ultimate bearing capacity of ship structures (including plates, stiffened plates, and hull girders) in their many different modes of possible behavior has been widely studied, and a great deal of results developed, which have made possible an increasingly clearer understanding of the ultimate strength behavior of ship structures. However, because of the complexity and importance of the problem, and the need for continuing development is still felt, for example, as described in the 19th International Ship and Offshore Structures Congress (ISSC) report, which emphasizes that (even) "with these developments ultimate strength assessment is now becoming a more important issue to ensure the safety of ship structures."

Evaluation on prediction abilities of constitutive models considering FEA application

Constitutive model plays an important role in the numerical simulations of metal forming. However, the influence of the models on the calculation is vague. Based on the stress-strain data of Al 7050 and Ti-6Al-4V alloys generated by isothermal compressive tests, the Johnson-Cook (JC) and Arrhenius-type (A-type) hyperbolic sine models were fitted to obtain the constants. Flow stresses directly calculated by the equations were compared with the experiment results, and rigid-plastic finite element analyses (FEA) utilizing these models were employed to simulate the same compression processes. The results show that A-type model has higher accuracy in the direct prediction of flow stress, even outside of the fit domain. The simulation results using A-type model also have higher agreement with the experiment; however, the suitability is affected by the referential parameters employed in the regression process. In terms of the overall deformation and strain distributions, there are slight differences among the simulation results using these two models.

Positive bending capacity prediction of composite girders based on elastoplastic cross-sectional analysis

This paper investigates the positive bending moment capacity of composite plate girders with various geometric parameters as well as a wide range of concrete slab strengths and steel girder strengths. The elastoplastic cross-sectional analysis method is adopted to efficiently evaluate the positive bending capacity of the composite girder. The constitutive model of the steel girder is specified by a trilinear stress–strain relationship incorporating the strain hardening effect while the concrete slab based on the parabola-rectangle model. A large number of composite sections, which could represent the conventional cross-section of composite plate girders with medium spans, are designed along with various combinations of material strengths. For each composite section, its actual ultimate bending capacity and rigid plastic bending capacity are calculated and compared mutually. The analysis results indicate that the position of the plastic neutral axis has a significant effect on the ratio of the ultimate bending capacity to the rigid plastic bending capacity. The lower position of the plastic neutral axis locates, the more overestimate of the rigid plastic analysis method to predict the ultimate bending capacity. Besides, the yield strength of the steel girder also has an apparent effect on the ratio, of which the higher yield strength of the steel girder could lead to a lower ratio, while the concrete slab compressive strength has less influence on the ratio. The equation of predicting the ultimate bending capacity of the composite plate girder is proposed, in which a bending capacity reduction factor is introduced depending on the strength grade of the steel girder.

Dynamic response of stiffened plates under repeated impacts

This paper proposes some simple formulae based on the rigid-perfectly plastic method to examine the dynamic response of stiffened rectangular plates repeatedly impacted by a rigid knife-edged striker at any location. To validate the proposed theoretical method, experiments and numerical simulations are also conducted. The permanent deflection of the tested specimens and rebound velocity of the striker obtained from experiments show good agreement with the numerical simulations performed using the commercial software package ABAQUS. The influences of the material properties including the strain rate, strain hardening and material elasticity, on the accuracy of the theoretical predictions, are assessed by comparison with the numerical simulations. It shows that when accounting for these effects of the material properties in the rigid plastic analysis, the theoretical solutions could provide a good prediction of the structural dynamic behavior for repeated impacts.

On the influence of elasticity on the large deflections response of circular plates to uniform quasi-static pressure

This work presents an approximate theory to predict the large deflections and elastic spring-back of circular plates made of elastic, perfectly plastic material when subjected to uniform quasi-static pressure. A non-linear interaction between the membrane force and bending moment is assumed when the radial displacements are included in the analysis in order to obtain realistic strain distributions. It is shown that the stored elastic energy in the plate has a limit for the particular material model when sufficiently large deflection occurs and the plate becomes entirely plastic. Therefore, the spring-back of the plate during unloading decreases with the increase of the pressure level. It is also revealed that a plate thinning occurs at sufficiently large deflections, which has a non-negligible effect on the plate stiffness. The plate deflections predicted by the proposed elastic-plastic theoretical model in the loading phase are compared with those given by a rigid-plastic analysis reported in the literature so as to estimate the loading range where the material elasticity has an important effect.In parallel, numerical simulations are conducted using the commercial software package ABAQUS to verify the proposed theoretical model. It is shown that the theoretical predictions are in good agreement with the results from the numerical simulations in terms of the pressure-displacement dependence, accumulated elastic energy, elastic spring-back and permanent displacements.

Yield-line plasticity and tensile membrane action in lightly-reinforced rectangular concrete slabs

The paper provides the systematic derivation of a new analytical approach to tensile membrane action of lightly-reinforced thin concrete slabs at large deflections. The basic motivation for the work comes from the recent use of tensile membrane action as an enhancement, in the fire condition, of the capacity of the thin concrete slabs which are normally made composite with downstand steel beams, at temperatures which have substantially degraded the contribution of these steel beams. The method accepts as a premise that such slabs form a pattern of localized yield lines as an initial small-deflection failure mechanism, and that these yield lines retain their positions as subsequent deflection occurs. As the slab deflects, maintaining the correct kinematics of the articulation and displacement of the system of slab facets, interacting across the yield lines, is extremely important to the horizontal equilibrium of the slab. In this process it becomes necessary to re-think the basic assumption of traditional yield-line theory that any local cross-section of unit width along a yield line equilibrates the force of its concrete compression block with the yielded steel’s tension force, producing constant plastic moment capacities for the mesh in either direction along any yield line. In the approach set out in this paper only overall equilibrium of the system of facets needs to be maintained. As in normal rigid-plastic analysis, concrete acts only when compressed, and then at its compressive strength, and steel acts at its tensile yield strength whilst it remains intact. However, steel in either direction can fracture when the local crack-width causes its local strain to exceed its fracture ductility. When the rebar crossing the diagonal yield lines begins to fracture this generally indicates that the slab’s capacity is about to reduce with further deflection. The paper does not attempt to address how a rebar’s free length across a discrete crack is generated, or the limiting crack widths implied, but this is shown in a range of examples to be a major issue if tensile membrane action is to be used in practice to enhance the capacity of slabs, for example in hazard loading situations. It is important that principles be established in future to quantify this aspect of rebar ductility.