Static stability: mechanisms and prestress modes

Mettupalayam V Sivaselvan, SUNY at Buffalo, Civil, Structural and Environmental Engineering

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Summary

Summary: The assignment is about detecting structural stability in a 3D truss by applying linear algebra concepts such as rank, column-space and null-space of a matrix.

Outcomes:

* Students should learn linear algebra concepts such as rank, column-space and nullspace of matrices, the four fundamental spaces of a matrix and orthonormal bases and Gram-Schmidt/QR decomposition.
* MATLAB is utilized for students to explore these concepts through computations. For example a vector or rigid-body displacements of a structure must be in the nullspace of its compatibility matrix; students can verify this with a simple computation.
* Higher-order thinking is invoked as students try to relate the above linear algebra concepts to the structural mechanics. For example, a vector in the nullspace of the compatibility matrix is a part of the structure that is not properly constrained from moving freely.
* Students present their work and findings in writing. They submit their work using the MATLAB Publish tool.

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Learning Goals

Upon completing this assignment, students will have further exercised abilities to
1. Use concepts of linear algebra to uncover features of linearized mechanics models. In particular

  1. Build the kinematic matrix, B, of a structure based on an understanding of its meaning
  2. Exemplify the linear algebra concepts (a) null space , (b) rank, (c) the four fundamental spaces through this matrix, and explain the significance of these concepts in mechanics
  3. Apply the concept of orthogonal projection to deduce unstable internal mechanisms in structures


2. Present formulation of mechanics models, computational strategies and numerical results in a logical and easy-to-follow manner
3. Demonstrate attributes of life-long learning including (a) confidence in understanding of fundamental principles, (b) willingness to make mistakes and learn from them, (c) perseverance in the face of frustration, (d) reason about mechanics/mathematics through interconnected principles and not isolated facts.

Context for Use

* This is assignment is part of a first year graduate-level class in civil (structural) engineering on mechanics and mathematics.
* The class is structured as a sequence of 2 week-long projects, and lectures and practice exercises in support of these projects. The assignment presented here is one such project.
* Students in this class have been exposed to MATLAB as undergraduates, but typically have not had sufficient practice and applications experience to be comfortable/proficient with MATLAB (and programming in general). So they are essentially learning MATLAB programming in parallel with the mathematics and mechanics concepts.
* Students in this class are expected to completed a sequence of mechanics classes (Statics and Mechanics of Solids) at the undergraduate level. When they encounter this assignment, they have also completed a previous assignment where they have exercised concepts of matrix structural analysis and its MATLAB implementation.
* This assignment would fit in a structural analysis or finite element analysis course in civil, mechanical or aerospace engineering, following students' mastery of basics matrix concepts.
* Before this assignments, students should have mastered matrix structural analysis concepts such as matrix assembly, and MATLAB concepts of matrix indexing with integer and logical arrays and writing and calling functions.

Description and Teaching Materials

This assignment has the following supporting material:

1. Linear Algebra [Part I] - vector spaces, matrix-vector multiplication, column-space of a matrix.
Video: https://youtu.be/9uG_kuU-JZU
There are also 3 MATLAB Grader exercises associated with this topic: Grader-Problem1.pdf (Acrobat (PDF) 1.3MB Nov8 21), Grader-Problem2.pdf (Acrobat (PDF) 1.3MB Nov8 21), Grader-Problem3.pdf (Acrobat (PDF) 1.5MB Nov8 21) (Grader-Problem3-mfiles.zip (Zip Archive 199bytes Nov8 21))

2. Linear Algebra [Part II] - Linear independence, Span, Basis, Dimension, Null space of a matrix, Rank of a matrix
Video: https://youtu.be/r5rowQtj51Q

3. The four fundamental spaces of a matrix
Video: https://youtu.be/jaYyp-5E1ig
There is 1 MATLAB Grader exercise associated with this topic: Grader-Problem4.pdf (Acrobat (PDF) 2.3MB Nov8 21) (Grader-Problem4-mfiles.zip (Zip Archive 197bytes Nov8 21))

4. The four fundamental spaces of the compatibility matrix
Video: https://youtu.be/QeW8JvDuXgU

5. Examples
Video: https://youtu.be/dpcKBO40hOM
There is also 1 MATLAB Grader exercise: Grader-Problem5.pdf (Acrobat (PDF) 2.2MB Nov8 21) (Grader-Problem5-mfiles.zip (Zip Archive 2kB Nov8 21))

6. Orthogonal (closest point) projection
Video: https://youtu.be/A8vL1Sf28Z4
1 MATLAB Grader exercise: Grader-Problem6.pdf (Acrobat (PDF) 1.5MB Nov8 21) (Grader-Problem6-mfiles.zip (Zip Archive 212bytes Nov8 21))

7. Rigid body modes
Video: https://youtu.be/eWNJzRaI_qU


Static stability (mechanisms and prestress modes) assignment (Acrobat (PDF) 185kB Sep9 21)
Rubric for the static stability assignement (Acrobat (PDF) 132kB Nov9 21)
MATLAB code for Static Stability Assignment (Zip Archive 4kB Sep9 21) 
Reference article (Acrobat (PDF) 878kB Sep9 21)

Students are expected to watch the videos to acquire the mathematics and mechanics concepts needed to complete the assignment. The MATLAB grader problems are intended for students to practice these concepts (they can be thought of as sideline exercises). With this preparation, students set out on the assignment. Class time is used to clarify concepts, to troubleshoot students' code (possibly by sharing their MATLAB screen via Zoom) and to explore any new approaches students may be considering. The entire process (watching the videos, doing the practice problems and completing the assignment) is to take two weeks.

The following may be used as an exercise for structured reflection: ReflectionExercise.pdf (Acrobat (PDF) 103kB Nov9 21)

Teaching Notes and Tips

The assignment is organized in the following way. Students review the supporting videos and practice using the MATLAB Grader exercises to prepare for the assignment. As they work on the assignment, class time to used to address conceptual questions, troubleshoot MATLAB code and reinforce concepts with additional discussion as needed.

The following is some context for the assignment.

1. Using linear algebra principles to make inferences about stability is a practical technique when working with structures that change form such as

  • Deployable structures (for example, the truss structure supporting the solar panels of the International Space Station)
  • Tensegrity structures and tensegrity robots

Photographs of these structures may be used as motivation.

2. The concept of the four fundamental spaces is quite central in applied mathematics and arises in different contexts, for example

  • Fredholm solvability condition for differential equations (a simple example of an undamped mass-spring with resonant forcing may be quickly discussed as illustration)
  • Concepts of controllability and observability in control systems theory

Assessment

Assessment is based on a rubric (attached above).

The assignment is assessed based on this rubric as satisfactory/unsatisfactory. If unsatisfactory, feedback is provided, and the student can resubmit in a week after incorporating the feedback. If the student has invested in significantly incorporating the feedback the assignment is assessed as satisfactory.

References and Resources

Two references are pertinent:
1. Calladine, C.R. and Pellegrino, S. (1991). "First-order infinitesimal mechanisms", International Journal of Solids and Structures, 27(4), 505-515.
The assignment is based on this article.
2. The textbook, Introduction to Linear Algebra by Gilbert Strang treats the relevant linear algebra concepts.