Section 1                Literature review 1.1                           IntroductionIn the following sections, the significance and possibilities of fabric formwork in the construction industry are outlined, and the concepts of ductility and moment redistribution in continuous concrete structures are introduced. An investigation has been carried out to identify the areas of this research that are in need of further development, and thus highlight the areas in which this dissertation will aim to expand on. An overview of the sectional analysis method that is used in the development of the computational model is also detailed below.1.2                           Fabric formwork for concrete 1.2.1                     HistoryThe initial use of fabric formwork can be traced back to 1899 by Gustav Lilienthal, who used fabric or paper between timber boards to cast a concrete slab. It was not until the work of Miguel Fisac that the true architectural potential of fabric formwork was revealed; Fisac, a Spanish architect, developed and patented a method for fabric forming prefabricated wall panels. Since then, there have been significant advancements in fabric forming techniques and the structures produced using such methods – notable works include the ‘zero waste formwork’ for fabric formed concrete walls by Japanese architect Kenzo Unno and the substantial research into fabric forming of a variety of structural elements by the Centre for Architectural Structures and Technology (CAST) at the University of Manitoba (Veenendaal et al., 2011; Orr, 2012).  Figure 1 – fabric formed T-beam cast using UHPFRC (left) (University of Bath, n.d.); fabric formed wall (right) (Orr, n.d.)1.2.2                     The rationale for fabric formworkThe agenda for climate change is an important consideration in the construction industry. Although concrete has a relatively low embodied carbon (Hammond and Jones, 2008), it is used in mass quantities in construction, resulting in a large carbon footprint. In 2016, approximately 12.8 million tonnes of CO2 were attributed to construction (ONS, 2018a). Though this quantity only composes of roughly 2% of the total carbon emissions in the UK (ONS, 2018b), statistics show that this figure has only illustrated a general increasing trend since the 1990s. Additionally, annual reports on the production of ready-mixed concrete by ERMCO (2003-2016) have shown that the UK has consistently produced approximately 25 million m3 of ready-mixed concrete every year in the past decade. There is a clear need for the UK construction industry to further their progression towards sustainable practices in order to meet the targets of the Climate Change Act (2008), which aims ‘to ensure that the net UK carbon account for the year 2050 is at least 80% lower than the 1990 baseline’.Fabric formwork for concrete structures is significantly advantageous in addressing these issues. Materials savings of up to 50% have be found in optimised fabric formed structures (Orr, 2012). In addition, replacing the requirement for fully rigid formwork with simple rectangular sheets of fabric that can be completely reused for multiple elements of different shapes (Orr et al., 2011) will promote further material and cost savings in the formwork, including reduced storage requirements, minimised labour times and intensities and improved ease of transportation of the formwork (Veenendaal et al., 2011).As established in research (Orr et al., 2011; Orr et al., 2013), there are also benefits with regards to enhanced durability performance of concrete structures formed using fabric compared to conventional rigid formwork. It was demonstrated that the use of permeable fabric allows water and air to escape, resulting in a greater reduction in the water:cement ratio towards the exterior surface of the form. Thus, the final concrete product produced from fabric forming has higher strength and lower surface porosity, and subsequently enhanced performance against chloride ingress, oxygenation and carbonation, leading to longer lasting structures. Moreover, it has been reported by Pallet (cited by Orr et al., 2011) that long term cost savings can be achieved with using permeable moulds in casting concrete forms, largely due to a lesser requirement for maintenance and repair.1.2.3                     The future of fabric formwork Though fabric formwork has been proven to have great potential for the future of construction as evidenced above, there is still plenty of research to be done before such practices can be confidently adopted commercially.Much of our understanding on flexible forms are primarily of flexural members and productions of such have been quite similar. There is a large scope for developing the potential for other forms, including shell structures, where it has been highlighted by Orr et al. (2011) that ‘it is in the design of shell structures that real material savings may be found’, as slabs in structural systems are typically where the largest quantities of concrete is used.It should also be noted that research on flexibly formed beams to date have only considered simply supported conditions and have only been optimised for ultimate state conditions (Tayfur, 2016). No investigation has been done into optimising such beams for serviceability apart from that done by Tayfur – the reduction in stiffness of a variable section fabric formed beam compared to its equivalent orthogonal beam of constant stiffness can raise some concerns with regards to the exceedance of deflection limits (Orr et al., 2011). Furthermore, Tayfur (2016) has identified that there has been no research into the ultimate or serviceability limit state performance of continuous fabric formed beams – our understanding in this area of fabric formwork is critical as continuity is so commonly found in many reinforced concrete structural systems.Although there have been many successful uses of fabric formwork in experimental and commercial applications, there is a need for standardised fabric forming methods to be developed due to the dependency of the final outcome on the boundary conditions, type of fabric used and any pre-stressing of the fabric, which can cause large variations. Establishment and optimisation of such methods, with rigorous testing to satisfy ultimate and serviceability limit state design and full consideration of risk, can lead to the widespread acceptance of fabric forming techniques to make it an industry standard (Hawkins et al., 2016).1.3                           Load-deflection behaviour of reinforced concrete beams 1.3.1                     The sectional analysis method The computational programme that has been written for this dissertation is based on the sectional analysis method. The method involves dividing a structure into multiple sections and analysing these local sections in response to the applied loads, with consideration for the non-linear material characteristics of reinforced concrete (Bentz, 2000), combining the behaviour of each section to provide a global response of the member.Tayfur (2016) has highlighted that unlike other methods, such as using the effective moment of inertia and empirical methods, the full load-deflection response of a flexural member can be observed up to failure. Sectional analysis provides a good compromise between lengthy hand calculations and using excessive computational power through non-linear finite element modelling. The method uses fundamental, well-established engineering concepts of axial loading, shearing and bending moments which can be utilised to analyse structures of varying geometry with good accuracy (Bentz, 2000). 1.3.1.1              AssumptionsBentz (2000), Kwak and Kim (2002) and Tayfur (2016) have outlined the following assumptions associated with the sectional analysis method:The perfect bond exists between concrete and steel – there is no bond slip between the two materials and thus there is full compatibility of strains in the section, and;The method is in agreement with the Euler-Bernoulli beam theory – plane sections remain plane in flexure and hence longitudinal strains are directly proportional to the distance from the neutral axis, and any transverse stresses are deemed negligible (Bauchau and Craig, 2009).1.3.1.2              Material models The non-linear material behaviour of reinforced concrete is considered in sectional analysis by the defined concrete and steel material models. As such commonly used materials in construction, the behaviours of steel and concrete have been thoroughly explored and is fairly predictable. However, in analysis and design, the tensile capacity of concrete is often ignored. However, concrete is capable of resisting small tensile stresses and it has been demonstrated that ignoring the tensile capabilities of concrete causes an underestimation of the stiffness of singly reinforced concrete beams that is considered as fairly substantial (Bazant and Oh, 1984), and thus modelling behaviour while neglecting the tensile capacity of concrete will lead to more conservative outcomes.A smeared crack model is used in the sectional analysis. Defined by Rots and Blaauwendraad (1989), the smeared crack model ‘imagines the cracked solid to be a continuum and permits a description in terms of stress-strain relations’. Individual microcracks are not modelled but rather considered through the reduction in concrete’s stiffness as it reaches its ultimate limit. However, because of this assumption, no information regarding crack patterns, spacings and widths can be deduced (Tayfur, 2016).1.3.2                     Analysis of deflectionsWhen a beam is subjected to uniform loading, the beam deflects, and the curvature of the deflected beam varies along its length. The relationship between the moment capacity of sections along the beam and the curvature can be related to the applied bending moments, and hence the full deflection profile of the beam can be found. An overview of the procedure as illustrated by Orr (2012) and Tayfur (2016) is summarised in the following sections.1.3.2.1              Moment-curvature relationshipConsider a simply supported fabric formed reinforced concrete beam. The beam is divided into multiple sections. Considering a single section along the beam, a value for the depth of the neutral axis is assumed. An incremental strain is applied to the compressive edge of the section, and using the defined concrete material model, the corresponding stress is found. Similarly, for the reinforcing steel, the strain at the reinforcement level is found using the assumed strain and neutral axis, and the stress is found using the steel material model.Once both stresses are known, compressive and tensile forces are found in the concrete and steel. The total force in the concrete is determined by considering small strips of the section. For each strip, the area of the strip and the stress at the depth of the strip is known from the defined material models – the forces for each of these strips across the entire concrete stress distribution are calculated and thus the total concrete force is equal to the sum of the forces of each strip. It should be noted that the strips considered must be small so that the stress distribution across the strip can be approximated as constant. As the area of reinforcement is small in comparison to the area of concrete, the stress distribution across the area of steel is assumed to be constant and a single value of stress at the centroid of the reinforcement can be taken to determine the tensile force.The resultant force of the tensile and compressive forces is evaluated by considering the horizontal force equilibrium in order to assess if the assumption has been satisfied. If the resultant force is not zero, the depth of the neutral axis as assumed initially is incorrect and a new depth is trialled. The previous steps are iterated with the new neutral axis depth and the same incremental strain until horizontal force equilibrium is achieved. With the correct neutral axis depth now known for the applied incremental strain, the moment capacity of the strain profile is determined by taking the summation of internal moments about the depth of reinforcement. For small deformations, the beam bends with a constant curvature along its length (Bauchau and Craig, 2009). With the assumption that the strain distribution along a section is linear, by geometry, the curvature of the beam can be determined as the angle of the strain distribution from the vertical. Thus, the curvature for the increment of strain being considered and the now known neutral axis depth is determined and recorded along with the calculated moment capacity of the section.This entire process is iterated for increasing increments of strain until the ultimate strain limits of concrete have been exceeded. For every iteration of strain, a different depth for the neutral axis and curvature and a different moment capacity for the strain profile is found, and therefore the complete moment-curvature relationship is obtained for a section. Each section along the beam will have its own unique moment-curvature relationship due to the variation in sectional geometry along the beam. 1.3.2.2              Double integration of curvatures Once the moment-curvature behaviour of all the sections along the beam has been found, this relationship can be linked to the applied loading on the beam to find the full load-deflection behaviour of the beam up to failure. The uniform load is applied incrementally and the curvature at each section is found for the applied moment using the established moment-curvature relationships. The distribution of curvature along the beam’s length is then integrated once to find the rotations along the beam, and then integrated a second time to arrive to the final deflection profile for the applied increment of load. The constants of integration can be found through the known boundary conditions of the behaviour of the beam – for the first constant, the rotation at midspan of the beam is known to be zero under uniform loading, and for the second constant, it is known that the ends of the beam at its supports do not deflect.This process is repeated for increasing increments of load until the required loading has been achieved or until the beam has failed. Failure is defined by the exceedance of the concrete strain limit – when any section along the beam has reached this limit, the concrete is deemed to have cracked. 1.4                           Ductility and moment redistribution 1.4.1                     Ductility Ductility is generally understood as the resistance the structure has against applied loading by plastic deformation (Tajaddini, 2015) or the ability of a structure to absorb and dissipate energy under loading (Mostofinejad, 1997). The flexural capacity of a reinforced concrete structure is one of the primary considerations in reinforced concrete design, commonly with less consideration for the flexural ductility of the structure. Nonetheless, flexural ductility is an equally significant aspect to consider, particularly with regards to the safety and ultimate capacity of the structure (Kwan et al., 2002), providing an allowance for the absorption and dissipation of energy prior to failure. Ductility of reinforced concrete structures is typically attributed to the yielding of steel reinforcement (Mostofinejad, 1997).As highlighted by Oehlers and Seracino (2004) and Tajaddini (2015), there are different types of ductility that can be identified. Material ductility is described by the stress-strain behaviour of a material; this is usually what is referred to as the general definition of the term ‘ductility’. Sectional ductility and beam ductility are closely related but can be distinguished through their definitions – beam ductility is known as a beam’s ability to absorb energy, which is observed through its non-linear deflection behaviour, whereas sectional ductility can be characterised by a section’s moment-curvature relationship. As discussed previously, it is known that the deflection profile of a given beam can be established from the moment-curvature relationship of numerous sections along the beam.1.4.2                     Redistribution of momentsPerhaps one of the most fundamental qualities of ductile behaviour in continuous reinforced concrete structures is the ability to redistribute internal stresses and moments to stiffer regions of the structure through the formation of plastic hinges (Mostofinejad, 1997). Consideration for the redistribution of bending moments in continuous structures can result in the more economical design of structures – smaller cross sections or a simplified reinforcement layout can be used due to the reduction in maximum design bending moments, as moments are transferred to areas that have yet reached their plastic limit (Mattock, 1959; Scott and Whittle, 2005; Tajaddini et al., 2016).Though seemingly trivial, quantifying the amount of moment redistribution in a structure has been a complex issue that has yet to be fully resolved and understood, as identified by Oehlers et al. (2010) and Tajaddini et al. (2016). Many national design standards across the world have established guidance to ensure that sufficient redistribution of moment has occurred through the yielding of steel by limiting the neutral axis depth (and thus avoiding over-reinforcement of the structure); however, the guidance seems to be founded on an empirical basis (Oehlers et al., 2010; Tajaddini, 2015). Section 2                ReferencesBauchau, O. A. and Craig, J. I., 2009. Structural Analysis. Dordrecht: Springer.Bazant, Z. P. and Oh, B. H., 1984. Deformation of progressively cracking reinforced concrete beams. Journal of the American Concrete Institute, 81(3), 268-278.Bentz, E., 2000. Sectional analysis of reinforced concrete members. Thesis (Ph.D.). University of Toronto.Climate Change Act 2008, c.1. UK: TSO.European Ready-Mixed Concrete Organisation (ERMCO), 2003-2016. Ready-mixed concrete industry statistics. Belgium: ERMCO.Hammond, G. P. and Jones, C. I., 2008. Embodied energy and carbon in construction materials. 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