Mithila is a lecturer in Structural Engineering at the University of Southampton, UK. He obtained his PhD from Cambridge University in 2008, and prior to the current employment he was a Post-Doctoral Researcher at University of Oxford. Mithila is a Chartered Structural Engineer and a Fellow of the Higher Education Academy, UK. His current research interests are ‘modelling residual stress in glass’, ‘joints in glass structures’, ‘chemically-toughened glass’ and ‘glass-GFRP composites. At Southampton University, Mithils leads a team of researchers looking at various issues related to the structural use of glass in buildings and structures. He has been awarded and managed a total research income of over £500k, including funding from European Commission, Innovate UK, A*STAR (Singapore) and professional institutions such as ICE and IStructE. Mithila has authored or co-authored over 60 publications, and delivered over 25 conference papers in over 10 countries.

**Mithila Achintha **

Bogdan Balan School of Engineering University of Southampton University Road Southampton SO17 1BJ

A comprehensive modelling technique for incorporating the effects of residual stresses (RS) in commercially available float glass (annealed glass and thermally-strengthened glass) is presented. RS are developed in glass due to the differential cooling experienced by glass during manufacturing (i.e. outer regions of glass cool and solidify first while the inner regions remain hot). Glass manufactures expect compressive surface RS of magnitude of ~80-150 MPa in thermally-strengthened glass. Surface compressive prestress enhances resistance against the development and propagation of surface cracks in thermally-strengthened glass. Although no RS are expected in annealed glass, research showed some RS in annealed glass, albeit of small magnitudes (usually <10 MPa).

The effects of RS are critical for the structural performance and failure behaviour of glass structures. But, there is no accurate method to incorporate RS in stress analysis. Accurate stress analysis is required for design since glass is brittle and failures trigger at locations where high tensile stresses present. Therefore, the existing design practice of the ignorance of RS in annealed glass, and limiting the maximum design surface tensile stress determined from an analysis without incorporating RS to the surface RS specified by the manufactures of heat-strengthened glass usually result in structurally inefficient and cost expensive structures.

Analysis of RS in glass by explicitly modelling differential cooling and the complex multi-physics process of glass manufacturing is virtually impossible, since the exact details of the thermal parameters and the viscous response of glass at different temperatures are unknowable. These parameters are difficult to be determined as they depend on complex phenomena, such as convection, radiation, thermal vibration and microstructural rearrangements. Rather than attempting to model the unknowable multiphysics phenomena of the generation of RS, we propose to model the RS distribution as the static elastic response of glass if a misfit strain (i.e. eigenstrains) representing the effects of all the mechanisms that contribute to the generation of RS in glass.

Knowledge of a representative eigenstrains distribution in a given glass is a prerequisite in the proposed RS modelling technique. However, the representative eigenstrains distribution cannot be known at the beginning of an analysis and the determination of the eigenstarins is not trivial. We propose to determine a representative eigenstrains distribution by matching (in a least squares sense) the resultant RS distribution for an sensibly chosen initial assumed eigenstrains distribution with experimentally measured RS values at a finite number of locations. Finite element (FE)-based analyses are proposed for the stress analysis given the complex nature of the problem. In the present study, a scattered-light-polariscope (SCALP) was used to experimentally obtain RS values at a few selected locations in a given glass specimen. The recent advances in the SCALP techniques enable accurate measurements of RS in glass, in particular close to the surface regions. After an accurate estimate of the eigenstrain distribution has established, the full RS distribution in the glass specimen can be determined from a FE analysis by incorporating the estimated eigenstrains as a misfit strain distribution. The solution formulated this way satisfies equilibrium, compatibility, boundary conditions of the glass specimen, and the resultant RS distribution is entirely self-consistent. The step-by-step procedure of the proposed RS modelling technique is presented in Figure 1.

Figure 2a shows the RS values measured at seven discrete points (up to 2 mm deep from the surface) of a 10 mm thick annealed glass specimen. The stress values were measured using a SCLAP with an accuracy of ±2MPa (as reported by the manufacture). Using the eigenstrains analysis described above, an estimate of the actual eigenstrains in the glass specimen was determined (Figure 2b). The estimated eigenstrains distribution was then incorporated in a 3D FE model to determine the full RS distribution. Figure 2c shows the full RS distribution in the middle region of the mid-plane (xz plane) of the glass specimen. Figure 2a also shows the comparison between the RS depth profile predicted from the eigenstrain analysis and the experimentally measured values. As can be seen from Figure 2a, the stresses predicted by the proposed model compared well with the measured values. Using the eigenstrains technique the RS distribution in thermally-strengthened glass were also determined. For example, Figure 3d shows the comparison between the predicted RS depth profiles in 10 mm thick annealed and fully-tempered glass.

The eigenstrains technique of RS modelling was extended to analyse stress states in practical applications of glass structures, For example, Figure 3 shows the predicted RS distribution around a central hole (e.g. a hole drilled in glass for a bolted connection) in a fully-tempered glass piece. A 3D FE model was used in the analysis, and only a quarter of the specimen is modelled due to symmetry. The results showed a clear interaction between the geometry and the RS distribution. For example, RS distribution is not uniform over the internal surface of the hole, and this suggests the actual spatial distribution of RS must be taken into account in design in order to ensure structurally efficient, safe structures.

The finding of this paper shows RS distributions in annealed and thermally-strengthened glass can be modelled using the knowledge of eigenstrain depth profile that may be determined from an inverse eigenstrain analysis using RS values measured at a few location of a given glass specimen. The paper also shows that the eigenstrain analysis can be implemented in FE models for accurate stress analysis of glass structures.