Bond of reinforcement in concrete under high loading rates

The bond between concrete and reinforcing steel is fundamental to the load bearing capacity of reinforced concrete structures. Several experimental studies indicate strength or rather resistance enhancements coming with increasingly dynamic loading. The phenomenon is known as strain or loading rate effect and its causes are still not fully clarified. The work presented herein provides a numerical view of the bond of reinforcement in concrete and investigates its loading rate dependent behaviour. Finite element analyses focusing on structural and inertia effects are carried out. Modelling is conducted at the rib scale, where bond is predominately controlled by mechanical interaction. In the first step, the model is developed and calibrated. Its quality, credibility, and limitations are assessed by a series of numerical case studies and the results are compared with available experimental data. Numerical parametric studies follow. The loading rate dependence of bond is featured, loading rate dependent characteristics are identified, and conclusions on causes of the phenomenon drawn. It is shown that structural effects are strongly involved and the same holds for hydrostatic pressure stress states and inertia effects. The thesis concludes in reviewing currently available methods for incorporating the results into large-scale simulations and highlighting further investigations and developments that are necessary in order to design dynamic loading-resistant structures in the future

Relationships between algebra, differential equations and logic in England 1800-1860.

This thesis surveys the links between mathematics and algebraic logic in England in the first half of the 19th century. In particular, we show the impact that De Morgan's work on the calculus of functions in 1836 had on the shaping of his logic of relations in 1860. Similarly we study Boole’s background in D-operational methods and its impact on his calculus of logic in 1847. The starting point of the thesis is Lagrange’s algebraic calculus and Laplace’s analytical methods prominent in late 18th century French mathematics. Revival in mathematical research in early 19th century England was mainly effected through the diffusion of Lagrange’s calculus of operations as further developed by Arbogast, Servois and others in the 1800’s and of Laplace’s theory of attractions. .Lagrange’s algebraic calculus and Laplace’s methods in analysis – particularly on functional equations – were considerably developed by Herschel and Babbage during the period 1812-1820. Further research on the foundations of the calculus of operations and functions was provided by Murphy, De Morgan and Gregory in the late 1830’s. .Symbolic methods in analysis were further extended by Boole in 1844. Boole was followed by several analysts distinguished in their obsession in further vindicating these methods through applications on two differential equations which originally appeared in Laplace’s planetary physics. We record the main issues of De Morgan’s logic and their mathematical background. Special reference is given to his logic of relations and its connection with his foundational study of the calculus of functions. On similar lines we study Boole’s algebraic cast of logic drawing consequently a comparison between his two major works on logic. Moreover we emphasise his epistemological views and his evaluation of symbolical methods within logic and analysis

Bond of reinforcement in concrete under high loading rates

The bond between concrete and reinforcing steel is fundamental to the load bearing capacity of reinforced concrete structures. Several experimental studies indicate strength or rather resistance enhancements coming with increasingly dynamic loading. The phenomenon is known as strain or loading rate effect and its causes are still not fully clarified. The work presented herein provides a numerical view of the bond of reinforcement in concrete and investigates its loading rate dependent behaviour. Finite element analyses focusing on structural and inertia effects are carried out. Modelling is conducted at the rib scale, where bond is predominately controlled by mechanical interaction. In the first step, the model is developed and calibrated. Its quality, credibility, and limitations are assessed by a series of numerical case studies and the results are compared with available experimental data. Numerical parametric studies follow. The loading rate dependence of bond is featured, loading rate dependent characteristics are identified, and conclusions on causes of the phenomenon drawn. It is shown that structural effects are strongly involved and the same holds for hydrostatic pressure stress states and inertia effects. The thesis concludes in reviewing currently available methods for incorporating the results into large-scale simulations and highlighting further investigations and developments that are necessary in order to design dynamic loading-resistant structures in the future

Strain rate effects for spallation of concrete

Appropriate triaxial constitutive laws are the key for a realistic simulation of high speed dynamics of concrete. The strain rate effect is still an open issue within this context. In particular the question whether it is a material property – which can be covered by rate dependent stress strain relations – or mainly an effect of inertia is still under discussion. Experimental and theoretical investigations of spallation of concrete specimen in a Hopkinson Bar setup may bring some evidence into this question. For this purpose the paper describes the VERD model, a newly developed constitutive law for concrete based on a damage approach with included strain rate effects [1]. In contrast to other approaches the dynamic strength increase is not directly coupled to strain rate values but related to physical mechanisms like the retarded movement of water in capillary systems and delayed microcracking. The constitutive law is fully triaxial and implemented into explicit finite element codes for the investigation of a wide range of concrete structures exposed to impact and explosions. The current setup models spallation experiments with concrete specimen [2]. The results of such experiments are mainly related to the dynamic tensile strength and the crack energy of concrete which may be derived from, e.g., the velocity of spalled concrete fragments. The experimental results are compared to the VERD model and two further constitutive laws implemented in LS-Dyna. The results indicate that both viscosity and retarded damage are required for a realistic description of the material behaviour of concrete exposed to high strain effects [3]

Strain rate effects for spallation of concrete

Appropriate triaxial constitutive laws are the key for a realistic simulation of high speed dynamics of concrete. The strain rate effect is still an open issue within this context. In particular the question whether it is a material property – which can be covered by rate dependent stress strain relations – or mainly an effect of inertia is still under discussion. Experimental and theoretical investigations of spallation of concrete specimen in a Hopkinson Bar setup may bring some evidence into this question. For this purpose the paper describes the VERD model, a newly developed constitutive law for concrete based on a damage approach with included strain rate effects [1]. In contrast to other approaches the dynamic strength increase is not directly coupled to strain rate values but related to physical mechanisms like the retarded movement of water in capillary systems and delayed microcracking. The constitutive law is fully triaxial and implemented into explicit finite element codes for the investigation of a wide range of concrete structures exposed to impact and explosions. The current setup models spallation experiments with concrete specimen [2]. The results of such experiments are mainly related to the dynamic tensile strength and the crack energy of concrete which may be derived from, e.g., the velocity of spalled concrete fragments. The experimental results are compared to the VERD model and two further constitutive laws implemented in LS-Dyna. The results indicate that both viscosity and retarded damage are required for a realistic description of the material behaviour of concrete exposed to high strain effects [3]

Relationships between algebra, differential equations and logic in England, 1800-1860

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