Buckling Theory used in Spreadsheet:
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:
- The compressed concrete zone creeps;
- The concrete and the reinforcement behave elasto-plastically;
- 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.
About The Spreadsheet:
This spreadsheet can be used in either English or SI units.
I have highlighted the cells that require input in blue. They are:
- Diameter of Dome
- Height of Dome
- Added Dead Load
- Live Load
- Rebar Size
- Number of Layers of Rebar
- Rebar Spacing
- Concrete Thickness
- Strength of Concrete
- The ratio of the eccentricity of the compressive force belonging to the initial imperfection to the initial imperfection amplitude, c.
Note: c will always be 0.67 for domes.
- The variable, a. This variable represents the influence of the accuracy of the erection method.
Use a = 1 for reinforced concrete shells with rigid formwork, while for sliding shuttering we can take a = 6.
The safety factor, k, is highlighted in yellow at the end of the spreadsheet.
Note: The critical loads of several erected large reinforced concrete domes were determined in Kollar and Dulacska’s book The data showed that most structures have a safety factor greater than two. Two domes exhibited a safety factor somewhat inferior to two, and one showed a safety factor inferior to one. This latter structure, in fact, collapsed. On the whole, they assert, a safety factor between 2.5 and 3.5 seems to be realistic for shells with decreasing post-buckling load bearing capacity.
Link to the Spreadsheet:
Click on the link below and you can download the spreadsheet. I wrote the spreadsheet in Open Office but the linked version is saved in Excel format. It can be opened in either program.
Note: This is intellectual property of Nanette South Clark, freely shared. Please give proper credit when referring to this spreadsheet.
The accuracy of the results obtained using this spreadsheet is in no way guaranteed.
Please verify all calculations. If you find an error or have a suggestion for improvement, please E-mail me at [email protected]. Thank you!