# Aeronautical Engineering (MEng)

## Structures 2

### Module aims

This module builds on Mechanics and Structures 1, and provides a basis for Structures 3. It introduces matrix methods in overall structural analysis (stiffness and flexibility methods), structural dynamics (equations of motion for multi degree-of-freedom problems using force equilibrium, Hamilton’s principle and Lagrange’s equation; free vibrations; periodic and transient loads; modal analysis; damping; continuum) and buckling (stability analysis; buckling of struts; critical buckling load; failure envelopes).

### Learning outcomes

On successfully completing this module, you should be able to:
1. formulate and solve static structural problems using both the stiffness and flexibility matrix methods;
2. obtain the equations of motion for a dynamics multi degree-of-freedom problem using force equilibrium, Hamilton’s principle and Lagrange’s equation;
3. solve dynamics multi degree-of-freedom problems (i) under free vibrations to obtain the natural frequencies and associated mode shapes; (ii) under periodic loads, both in the physical and modal space and (iii) under transient loads using the convolution integral and modal analysis;
4. formulate the continuum equations of motion for a beam, and solve them under free vibrations to obtain the natural frequencies and associated mode shapes;
5. formulate the static continuum equations of equilibrium for a beam under compression, and solve them to obtain the buckling loads and associated buckling shapes.
6. explain the failure mechanisms in compressive members and the interaction between different failure modes.
7. use Finite Element software packages to analyse problems involving pin-jointed frameworks, vibrations and buckling.

### Module syllabus

This course covers matrix methods in Overall Structural Analysis, Structural Dynamics and Buckling. The contents of the three areas covered in this module are:

1) Overall Structural Analysis
- Stiffness method: introduction to multi degree-of-freedom problems; review of unit displacement method; forming the stiffness matrix directly for a problem; forming the stiffness matrix for a problem by assembling the stiffness matrix for a generic rod element.
- Flexibility method: review of unit load method; forming the flexibility matrix for a statically-determined problem; forming the flexibility matrix for a problem with a generic number of redundancies.

2) Structural Dynamics
- Equations of motion. Introduction to dynamics for problems with n degree-of-freedom; obtaining equations of motion via force equilibrium; derivation of Hamilton’s principle and Lagrange’s equations; obtaining equations of motion via Hamilton’s principle and via Lagrange’s equations.
- Free vibrations. Formulation of the free vibrations problem as an eigenvalue problem. Solution of the undamped problem in terms of natural frequencies and mode shapes.
- Periodic load. Solution of the single degree-of-freedom problem and corresponding Nyquist plots; solution of multi degree-of-freedom problem in both physical and modal space.
- Transient load. Derivation of the convolution (Duhamel’s) integral. Use of the convolution integral and modal analysis to solve transient multi degree-of-freedom problems.
- Continuum beams. Derivation of the continuum equation of motion for a beam. Typical boundary conditions. Analytical solution for free vibrations, leading to natural frequencies and corresponding mode shapes.

3) Buckling
- Flexural Buckling of Struts: With various boundary conditions, discontinuities, supports, imperfection, inelastic material behaviour.
- Failure modes of a strut under compression: introducing the concept of a failure envelope defined by elastic buckling and plastic collapse.

### Teaching methods

The module will be delivered primarily through large-class lectures introducing the key concepts and methods, supported by a variety of delivery methods combining the traditional and the technological. The content is presented via a combination of slides, whiteboard and visualiser.

Learning will be reinforced via two laboratories. Finally, Finite Element software packages will also be used to demonstrate the numerical solutions to various problems representative of those encountered by practising engineers, and how they relate to respective analytical and experimental solutions.

### Assessments

This module presents opportunities for both formative and summative assessment.
You will be formatively assessed through progress tests and tutorial sessions.
You will have additional opportunities to self-assess your learning via tutorial problem sheets. You will be summatively assessed by a written examination at the end of the module as well as through practical laboratory assessments and a written laboratory report.

 Assessment type Assessment description Weighting Pass mark Examination Written examination 80% 40% Practical Laboratory assessment 12% 40% Coursework Laboratory report 8% 40%

You will receive feedback both during the laboratory sessions and following the coursework submission.
You will receive feedback on examinations in the form of an examination feedback report on the performance of the entire cohort.
You will receive feedback on your performance whilst undertaking tutorial exercises, during which you will also receive instruction on the correct solution to tutorial problems.
Further individual feedback will be available to you on request via this module’s online feedback forum, through staff office hours and discussions with tutors.