# Detailed module information

Module information on this degree can be found below, separated by year of study.

The module information below applies for the current academic year. The academic year runs from August to July; the 'current year' switches over at the end of July.

Students select optional courses subject to rules specified in the Mechanical Engineering Student Handbook,  for example at most three Design and Business courses. Please note that numbers are limited on some optional courses and selection criteria will apply.

### Module aims

To introduce the student to fundamental elasticity and plasticity theory, problems and solutions. Stress function formulation and methods will be presented for plane stress, plane strain and torsional loading and the solution of a range of problems with be developed. The analysis of torsion will also be developed to treat thin walled sections of arbitrary but uniform, cross section. The Convolution Integral will be developed and used to analyse time dependent effects. A review of plasticity will be made and the Levy-Mises equations will be derived and used for problem solving.

ECTS units:    5

### Learning outcomes

On successfully completing this module, students will be able to:

Explain the method of solution using stress functions for two dimensional problems and torsion, involving bi-harmonic and Laplace equations

Explain the time dependent analysis using the Convolution integral

Explain the rationale behind the Levy-Mises equations

Propose stress functions and assess their suitability to solve 2D problems, by appropriate consideration of the bi-harmonic condition, boundary conditions, geometric and loading symmetries, and load decomposition;

Obtain exact solutions to stress analysis problems such as the bending of rectangular beams; loading of thick walled cylinders, split and curved rings; loaded wedges, holes in plates, discs between opposing loadsl and, approximately, the torsion of thin-walled non-circular section

Apply the Convolution Integral to the analysis of creep and relaxation phenomena in visco-elastic materials.

Solve residual stress problems and plasticity deformation problems using the Levy-Mises equations

### Module syllabus

Stress function methods in two dimensions: equilibrium, compatibility and constitutive equations; combination for linear elastic materials in Cartesian co-ordinates to produce biharmonic equation using Airy stress function; solution by polynomials, application to beam problems. Transformation to cylindrical polar co-ordinates; general solution; application to selected problems from tick walled cylinders; split and curved rings, wedges, elastic half-space, cracks, discs between opposing loads; superposition for arbitrary loading.

Torsion of non-circular sections: torsion of solid bars; warping function and Prandtl stress function; solutions for elliptical and triangular bars. Torsion of thin walled sections — membrane analogy, relationship between shear stress, applied torque, geometry and angle of twist.

Convolution Integral: why an alternative to previous stress analysis is needed. Development and application to simple problems. Study of more advanced problems.

Plastic Deformation: review of yielding and plasticity of bending of non-rectangular beams; development of the Prandtl-Reuss and Levy-Mises equations; solving problems for examples from the preceding situations.

### Pre-requisites

Pre-requisites: ME1-hSAN; ME2-hSAN

### Teaching methods

• Duration: Autumn and Spring terms (22 weeks)
• Lectures: 1 x 1h/week
• Tutorials: 1 x 1h/week maximum
 Hours Lectures 22 Group teaching Lab/ practical Other scheduled Independent study 81 Placement Total hours 125 ECTS ratio 25

### Assessments

 Written examinations: Date (approx.) Max. mark Pass mark Advanced Stress Analysis (3h) A Data and Formulae book and Stress Analysis Information Sheets are provided. This is a CLOSED BOOK Examination April/ May 200 n/a