Failure modes
Insufficient thickness
A minimum thickness of '4.2% of the radius' (Jacques Heyman, 1995) is required to ensure the thrust line remains inside the masonry. This prevents the distortion of forces and maintains equilibrium. 'The required minimum thickness falls sharply if the dome does not embrace the full 180 degrees'. Jacques Heyman 'The Stone skeleton'(1995).
Buckling.
Continuing with the analysis presented by A.Muttoni, F.Lurati, M.F.Ruiz 'Concrete shells-towards efficient structures'(2012). Buckling needs to be considered in areas of minimum curvature and largest compression (normally at the crown of a shell) taking into consideration initial imperfections, creep strains of concrete and the non-linear behaviour of concrete.
Too greater slope
A.Muttoni, F.Lurati, M.F.Ruiz 'Concrete shells-towards efficient structures' , (2012) specifies that an angle of <20 degrees was not viable in design. A slope of this inclination would induce high stresses resulting in buckling. It may also prevent the even curing of concrete.
The inner forces
Overall the membrane in plane tensile forces are low and can be supported by minimal reinforcement, however post tensioning may be required at the equator to support the presence of large tensile forces. Additionally, Insufficient reinforcement at the dome base junctions could lead to the formation of cracks as a result of shrinkage.
Edge forces
The edge forces (bending and shear) appear at the boundaries of a shell, they vary depending on the support conditions. Throughout my thesis I will be anaylsing the relationship between support conditions and egde stresses. In order to control these edge conditions the thickness of the shell locally can be increased. For cost and aesthetic reasons this solution may not be valid therefore an alternative solution can be used using suitable amounts of reinforcement. The amount of reinforcement is related to shape and corresponds to the serviceability and ultimate limit states of the chosen shape. Analysis of Bending moments and shear forces are used to determine appropriate amounts.
Cracking of domes
Although cracking presents signs of failure not all cracks will lead to collapse, for example those in St Peters, Rome. If yielding occurs due to compressive forces in the dome then further analysis is required to determine the stability of the structure. Figure 6 take from Jacques Heyman, (1995) identifies the exaggerated meridian cracking patterns; the dome is intact from the crown down to an arbitrary value (related to the thickness) then cracks into 'slices'. These slices act as flying buttresses. In 1748 an account by Poleni reported the cracks in the dome of St Peters, Rome and performed a comprehensive review on the stability of the dome. In order to maintain a domes stability 'the line of thrust should lie everywhere within the masonry' (Poleni, 1748). He establishes that the cracks in St Peters are not critical but proposes the addition of encircling ties to contain the horizontal thrust occurring at the supports.
A minimum thickness of '4.2% of the radius' (Jacques Heyman, 1995) is required to ensure the thrust line remains inside the masonry. This prevents the distortion of forces and maintains equilibrium. 'The required minimum thickness falls sharply if the dome does not embrace the full 180 degrees'. Jacques Heyman 'The Stone skeleton'(1995).
Buckling.
Continuing with the analysis presented by A.Muttoni, F.Lurati, M.F.Ruiz 'Concrete shells-towards efficient structures'(2012). Buckling needs to be considered in areas of minimum curvature and largest compression (normally at the crown of a shell) taking into consideration initial imperfections, creep strains of concrete and the non-linear behaviour of concrete.
Too greater slope
A.Muttoni, F.Lurati, M.F.Ruiz 'Concrete shells-towards efficient structures' , (2012) specifies that an angle of <20 degrees was not viable in design. A slope of this inclination would induce high stresses resulting in buckling. It may also prevent the even curing of concrete.
The inner forces
Overall the membrane in plane tensile forces are low and can be supported by minimal reinforcement, however post tensioning may be required at the equator to support the presence of large tensile forces. Additionally, Insufficient reinforcement at the dome base junctions could lead to the formation of cracks as a result of shrinkage.
Edge forces
The edge forces (bending and shear) appear at the boundaries of a shell, they vary depending on the support conditions. Throughout my thesis I will be anaylsing the relationship between support conditions and egde stresses. In order to control these edge conditions the thickness of the shell locally can be increased. For cost and aesthetic reasons this solution may not be valid therefore an alternative solution can be used using suitable amounts of reinforcement. The amount of reinforcement is related to shape and corresponds to the serviceability and ultimate limit states of the chosen shape. Analysis of Bending moments and shear forces are used to determine appropriate amounts.
Cracking of domes
Although cracking presents signs of failure not all cracks will lead to collapse, for example those in St Peters, Rome. If yielding occurs due to compressive forces in the dome then further analysis is required to determine the stability of the structure. Figure 6 take from Jacques Heyman, (1995) identifies the exaggerated meridian cracking patterns; the dome is intact from the crown down to an arbitrary value (related to the thickness) then cracks into 'slices'. These slices act as flying buttresses. In 1748 an account by Poleni reported the cracks in the dome of St Peters, Rome and performed a comprehensive review on the stability of the dome. In order to maintain a domes stability 'the line of thrust should lie everywhere within the masonry' (Poleni, 1748). He establishes that the cracks in St Peters are not critical but proposes the addition of encircling ties to contain the horizontal thrust occurring at the supports.