Numerical modelling of braided fibres for reinforced concrete

Fire has been always a major concern for designers of steel and concrete structures. Designing fire-resistant structural elements is not an easy task due to several limitations such as the lack of fire-resistant construction materials. Concrete reinforcement cover and external insulation are the most commonly adopted systems to protect concrete and steel from overheating, while spalling of concrete is minimised by using HPFRC instead of standard concrete. Although these methodologies work very well for low rise concrete structures, this is not the case for high-rise and inaccessible buildings where fire loading is much longer. Fire can permanently damage structures that cost a lot of money. This is unsafe and can lead to loss of life. In this research, the author proposes a new type of main reinforcement for concrete structures which can provide better fire-resistance than steel or FRP re-bars. This consists of continuous braided fibre rope, generally made from fire-resistant materials such as carbon or glass fibre. These fibres have excellent tensile strengths, sometimes in excess of ten times greater than steel. In addition to fire-resistance, these ropes can produce lighter and corrosive resistant structures. Avoiding the use of expensive resin binders, fibres are easily bound together using braiding techniques, ensuring that tensile stress is evenly distributed throughout the reinforcement. In order to consider braided ropes as a form of reinforcement it is first necessary to establish the mechanical performance at room temperature and investigate the pull-out resistance for both unribbed and ribbed ropes. Ribbing of ropes was achieved by braiding the rope over a series of glass beads. Adhesion between the rope and concrete was drastically improved due to ribbing, and further improved by pre-stressing ropes and reducing the slacked fibres. Two types of material have been considered for the ropes: carbon and aramid. An implicit finite element approach is proposed to model braided fibres using Total Lagrangian formulation, based on the theory of small strains and large rotations. Modelling tows and strands as elastic transversely isotropic materials was a good assumption when stiff and brittle fibres such as carbon and glass fibres are considered. The rope-to-concrete and strand-to-strand bond interaction/adhesion was numerically simulated using newly proposed hierarchical higher order interface elements. Elastic and linear damage cohesive models were used effectively to simulate non-penetrative 'free' sliding interaction between strands, and the adhesion between ropes and concrete respectively. Numerical simulation showed similar de-bonding features when compared with experimental pull-out results of braided ribbed rope reinforced concrete.

Connected Hopf algebras of finite Gelfand-Kirillov dimension

Following the seminal work of Zhuang, connected Hopf algebras of finite GK-dimension over algebraically closed fields of characteristic zero have been the subject of several recent papers. This thesis is concerned with continuing this line of research and promoting connected Hopf algebras as a natural, intricate and interesting class of algebras. We begin by discussing the theory of connected Hopf algebras which are either commutative or cocommutative, and then proceed to review the modern theory of arbitrary connected Hopf algebras of finite GK-dimension initiated by Zhuang. We next focus on the (left) coideal subalgebras of connected Hopf algebras of finite GK-dimension. They are shown to be deformations of commutative polynomial algebras. A number of homological properties follow immediately from this fact. Further properties are described, examples are considered and invariants are constructed. A connected Hopf algebra is said to be "primitively thick" if the difference between its GK-dimension and the vector-space dimension of its primitive space is precisely one . Building on the results of Wang, Zhang and Zhuang,, we describe a method of constructing such a Hopf algebra, and as a result obtain a host of new examples of such objects. Moreover, we prove that such a Hopf algebra can never be isomorphic to the enveloping algebra of a semisimple Lie algebra, nor can a semisimple Lie algebra appear as its primitive space. It has been asked in the literature whether connected Hopf algebras of finite GK-dimension are always isomorphic as algebras to enveloping algebras of Lie algebras. We provide a negative answer to this question by constructing a counterexample of GK-dimension 5. Substantial progress was made in determining the order of the antipode of a finite dimensional pointed Hopf algebra by Taft and Wilson in the 1970s. Our final main result is to show that the proof of their result can be generalised to give an analogous result for arbitrary pointed Hopf algebras.