Lecture 26: Identifying shell restraints | Finite Element Analysis of Plate and Shell Structures: Part 2 - Shells | EngineeringSkills.com

FINITE ELEMENT ANALYSIS OF SHELLS - NEW LECTURES

Section 1

Expanding the mechanics for flat shell elements

1. Welcome - the roadmap and pre-requisites

07:00 (Preview)

2. From plate to shell elements

12:10 (Preview)

3. The flat shell element displacement field

4. Shear, bending and membrane strains

5. The constitutive matrix, D

6. Moments, shears and membrane forces

7. Evaluating the constitutive matrices and shear correction

8. Building the strain-displacement matrix, B

9. A review of the element stiffness matrix

10. Quick recap of numerical integration and the Jacobian

Section 2

Building the shell element stiffness matrix

11. Section overview - Building the shell element stiffness matrix

01:19 (Preview)

12. The membrane stiffness

13. The bending and shear stiffness

14. The full element stiffness matrix

Section 3

Building the transformation matrix

15. Section overview - Building the transformation matrix

01:21 (Preview)

16. Defining the transformation matrix

17. Local element axes and direction cosines

18. Building edge-aligned local element axes

19. Building global axis-aligned local element axes

20. Calculating the transformation and global stiffness matrices

Section 4

Building a complete shell element solver

21. Section overview - Building a complete shell element solver

02:10 (Preview)

22. Procedurally generating a shell mesh

23. Visualising shell geometry

24. Calculating the transformation and rotation matrices

25. Visualising element reference frames

26. Identifying shell restraints

27. Building the structure stiffness matrix

28. Building the self-weight global force vector

29. Solving the system

30. The deflected shape and heatmap

31. Adding max displacement identifiers

32. Clean up and refactoring into the utils file

Section 5

Building an OpenSeesPy Shell for comparison

33. Section overview - Building an OpenSeesPy Shell for comparison

34. Setting up the structure

35. Building the OpenSeesPy model

36. Extracting reactions and max displacements

37. Plotting the structural response

38. Comparing our model with OpenSeesPy

Section 6

Drilling degrees of freedom and avoiding singularity

39. Section overview - drilling degrees of freedom and avoiding singularity

40. Drilling degrees of freedom and singularity

41. Drilling into stiffness matrix singularity

42. Decoding our structure's behaviour

43. A strategy for avoiding a singular stiffness matrix

44. Implementing drilling stabilisation

26. Identifying shell restraints

Building a complete shell element solver

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Summary

In this lecture, we will cover the following:

How to identify the indices of the nodes along the top and bottom straight edges of our shell,
How to define a restraint mask that controls which degrees of freedom are restrained at each node,
How to translate restrained node indices into a full list of restrained degrees of freedom for the structure.

In this lecture, we want to identify all of the restrained degrees of freedom for our shell. We will pin all the nodes along the two straight edges, meaning we restrain the three translational degrees of freedom at each of those nodes and leave the rotational degrees of freedom free.

With our restrained node indices identified, we build the list of restrained degrees of freedom. To do this, we define a restraint mask of [1, 1, 1, 0, 0, 0], which allows us to easily retain a list of the restrained degrees of freedom at each node and accumulate these for the whole structure.

Next up

In the next lecture, we will turn our attention to the structure stiffness matrix, building it up element by element and reducing it down using the restrained degrees of freedom we have just calculated.

Tags

restraintsboundary conditionsrestrained degrees of freedompinned supportsrestraint mask

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Finite Element Analysis of Plate and Shell Structures: Part 2 - Shells

Expanding from plate to shell elements - build a workflow that unlocks the behaviour of 3D shell structures

Finite Element Analysis of Plate and Shell Structures: Part 2 - Shells

After completing this course...

You will understand how we make the leap from Reissner-Mindlin plate elements to shell elements and what extra modelling fidelity that provides.
You will be comfortable using a combination of GMSH and the open-source 3D modelling software, Blender, to generate custom finite element meshes.
You will be able to use OpenSeesPy to model shell structures, as an alternative to your own custom finite element solver.
You will have a much greater understanding of what commercial finite element packages are doing, behind the UI, allowing you to authoritatively interrogate their results.

Next Lesson

27. Building the structure stiffness matrix