Change in Out of Plane Forces on Edge of Plate

[Stezza]

Hi All,

Recently I modelled a plate with a small amount of curvature - i.e like an aircraft window. It has a uniform pressure applied to it. Around the edges grounded celas1 elements have been used with stiffness in the out of plane direction only, these elements simulate the rubber window seal. Rigid body motion is prevented at the center of the plate.

I ran a linear model with 2 different celas1 spring stiffnesses, I had thought beforehand that considering it was a linear analysis I would see the same constraint forces at the edge of the plate, but I don't, the corners especially have different constraint forces - the forces can even change sign if a very stiff spring is used.

Does anyone have a explanation of the reasons behind this behaviour?

Attached is a plot showing the difference in results.

Thanks

 

[gwolf2]

Your 'constraints' become part of the stiffness matrix because they are springs and not rigid constraints, the overall stiffness of the structure changes, therefore the reactions change.

 

[40818]

You would only see unchanging reaction forces if the plate was only carrying memrane tension. Your corner constraints can change sign because your preventing them from moving backwards. Imagine pressing down in the middle of a piece of paper (edges just simply supported), the corners move the opposite way to the force, just like you model shows.

 

[Stezza]

Hi Thanks you both for the posts.

40818 you helped me visulised what I think could be assumed to be happening. To paraphrase I think you could assume that the edge of the plate wants to take up a shape that looks like an arc when the pressure is lifting it off the grounded springs. When the springs are weak the arc shaped edge can lift completely off the ground so all reactions are pulling it back into the ground. When the springs are stiff the middle of the arc lifts off the ground but the ends are forced into contact with the ground - revering the sign of the constraint forces. I have drawn a quick sketch to illustrate.

gwolf2, while I am not expert in FEM theory, I think you can have situations where the stiffness matrix changes and the reactions do not change. For instance a beam with gounded springs at each end under a uniform running load, if the spring stiffness was doubled the reactions would still be the same but the stiffness matrix would have changed. It was thinking along this line that lead me to my initial incorrect thinking about the behaviour of the plate.

 

[johnhors]

Stezza, your simple example of a beam supported by springs at each end is a determinate structure, in which the reactions at the spring supports is independent of the spring stiffness used and can be easily calculated. Whilst your plate is an indeterminate structure and gwolf2's description is correct. With an indeterminate structure it is not possible to calculate reaction forces without including the relative stiffnesses of all parts of the structure.

 

[gwolf2]

Yes, I should have elaborated a little, my response was correct but too terse to aid comprehension. Johnhors' determinate/indeterminate structure point amplifies this well.

As another example closer to your problem: If your window was of constant curvature and the perimeter was circular then your spring stiffnesses would not have changed the reactions. The reason for this is because the structure is of equal stiffness around the 360 degrees perimiter. The edge deflections may change but the edge reactions must stay the same. Of course the symmetry introduces a relationship between all the edge reactions and the structure then becomes determinate.