From:

http://www.monolithic.org/stories/buckling-analysis-of-reinforced-concrete-domes-excel-spreadsheet

Nannette South Clark writes:

"All calculations in this spreadsheet are according to theory in “Buckling of Shells for Engineers” by L. Kollar and E. Dulacska.

"The first factor Kollar and Dulacska considers is that the materials of shells are elastic at most only up to a certain limit; after this they become plastic (“physical nonlinearity”). Due to the intricacy of shell-buckling problems, only a few attempts have been made to assess theoretically the effects of plastic behaviour. Hence, they use a simple approximate method that corrects the results of elastic stability theory by taking the effects of plastic behaviour of the material into account.

"The creep of the materials also substantially reduces the critical load intensity of shells. Due to difficulties similar to those connected with plasticity, they take this into account only approximately.

"Cracks occurring in concrete diminish the stiffness of reinforced concrete shells considerably, as compared to the uncracked section, so they reduce the critical load intensity as well. Kollar and Dulacska show how this unstiffening effect of cracks (together with the stiffening effect of the reinforcement) can be taken into account.

"The experimental determination of the critical load is also treated briefly.

"Finally, all circumstances (post-buckling behaviour of the shell, etc.) determining a suitable magnitude for the safety factor are examined.

"The deformation properties of the material of reinforced concrete shells which are necessary for buckling analysis cannot be defined as simply as for other materials, because the deformation depends on the cracks, the reinforcement, and the creep of the concrete as well, so that it becomes a nonlinear function of the load.

"On the whole, reinforced concrete differs from elastic homogeneous material in the following ways:

1. The compressed concrete zone creeps;

2. The concrete and the reinforcement behave elasto-plastically;

3. The tension zone of the concrete cracks, the stiffness of the cross section drops, and the position, quantity and quality of the reinforcement plays an important role."

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