Boundary conditions

The classification of Som and Deb (2014) [som2014] has been used for the boundary conditions.

The simply-supported boundary conditions are \(u_v = test\):

\[\begin{split}\begin{array}{l c r} Name & Displ. Vector & Elastic Constants \\ \hline SS1 & u=v=w=0 & K^u=K^v=K^w=\infty \\ & & K^{\phi_x}=K^{\phi_\theta}=0 \\ \hline SS2 & v=w=0 & K^v=K^w=\infty \\ & & K^u=K^{\phi_x}=K^{\phi_\theta}=0 \\ \hline SS3 & u=w=0 & K^u=K^w=\infty \\ & & K^v=K^{\phi_x}=K^{\phi_\theta}=0 \\ \hline SS4 & w=0 & K^w=\infty \\ & & K^u=K^v=K^{\phi_x}=K^{\phi_\theta}=0 \end{array}\end{split}\]

and the clamped are:

\[\begin{split}\begin{array}{l c r} Name & Displ. Vector & Elastic Constants \\ \hline CC1 & u=v=w=w_{,x}=0 & K^u=K^v=K^w=K^{\phi_x}=\infty \\ & & K^{\phi_\theta}=0 \\ \hline CC2 & v=w=w_{,x}=0 & K^v=K^w=K^{\phi_x}=\infty \\ & & K^u=K^{\phi_\theta}=0 \\ \hline CC3 & u=w=w_{,x}=0 & K^u=K^w=K^{\phi_x}=\infty \\ & & K^v=K^{\phi_\theta}=0 \\ \hline CC4 & w=w_{,x}=0 & K^w=K^{\phi_x}=\infty \\ & & K^u=K^v=K^{\phi_\theta}=0 \end{array}\end{split}\]

Using the default boundary conditions

The analyst may set the boundary conditions specifying the parameter bc in the ConeCyl object, using the same names specified in the tables above:

>>> cc = ConeCyl()
>>> cc.model = 'fsdt_donnell_bc1'
>>> cc.bc = 'ss1'

Setting a different boundary condition for the bottom and top edges is possible using a hyphen - or an underscore _ to separate them, obtaining 'bcBot-bcTop' or bcBot_bcTop:

>>> cc.bc = 'ss1-cc1'
>>> cc.bc = 'ss1-ss2'

Note

When using boundary conditions from different types in the same model the analyst must select the most flexible model to use, for example when using cc.bc = 'ss1-ss2', the analyst must use cc.model = 'fsdt_donnell_bc2', a similar one or a more flexible model.

The model selected should be compatible with the boundary conditions that one whishes to simulate. For example, for the SS4 or CC4 boundary conditions, it is recommended to use the clpt_donnell_bc4 or the fsdt_donnell_bc4 models, and so forth.

The more flexible models can be used to simulate the more rigid boundary conditions, since elastic constraints are ajusted in order to provide the right set of boundary conditions, as shown in the table above. The table below shows the models that can be used for each boundary condition:

\[\begin{split}\begin{array}{l c r} Name & Model \\ \hline SS1 / CC1 & bc2, bc3, bc4 \\ \hline SS2 / CC2 & bc2, bc4 \\ \hline SS3 / CC3 & bc3, bc4 \\ \hline SS4 / CC4 & bc4 \end{array}\end{split}\]

Note that the models bc4 can be used for all the cases listed above. It is expected that more terms are required in the approximation when a model from another group is used.

Using arbitrary boundary conditions

When no value is given to the parameter bc the model will run by default with the SS1 boundary conditions. The analyst must change the elastic stiffnesses by changing the following parameters:

>>> cc = ConeCyl()
>>> cc.kuBot
>>> cc.kuTop
>>> cc.kuBot
>>> cc.kvTop
>>> cc.kvBot
>>> cc.kphixTop
>>> cc.kphixTop

In order to achieve the desired results.

The other stiffnesses kwBot, kwTop, kphitBot and kphitTop will affect only models clpt_donnell_bcn and fsdt_donnell_bcn (or the counterparts that use the Sander’s equations).