Pitkaranta 200809

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We consider an old textbook example demonstrating the use of classical, simplidied mathematical models to analyze a shell roof. The roof consists of a spherical dome attached to a stiffening ring. The material of the structure is concrete, assumed homogeneous, isotropic and linearly elastic in the mathematical model. An equilibrium support at the foot of the ring is imposed to balance the dead load on the structure. The mathematical problem is then to find the shear force and moment acting at the junction of the dome and the ring.

The problem was recently announced as a challenge for the designers and users of finite element software. In the twenty contributions received so far the results vary by several orders of maginitude. This raises the question, whether one can rely at all on the much simpler classical models developed before the era of computers. In these models one ends up solving a linear system with only two unknowns, whereas in the finite element models the system sizes range from thousands to millions.

We show first how the shear force an moment at the junction are obtained by hand computation using various variants of classical models. For the best of these models we present an a posteriori error analysis with respect to full 2D linear elasticity. We use here the Hypercircle theorem of Prager and Synge. In comparison with the observed uncertainty of modern computer models, the a posteriori error bounds for the classical model turn out to be quite surprising.

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