# Technical Notes

## Introduction

Technical Notes provide information on a technical issue of interest to RMA and, potentially, to a wider audience.

### The Navier Solution for the SSSS Plate

### Uniformly Loaded Rectangular Plate on Three Corner Supports: Part 1 - Error in Deflection

### Large Thermal Strains in FEA

The practising engineer will probably be familiar with the idea of thermal expansion or contraction of structural members or mechanical components. If this deformation is constrained then stresses will be induced in the member or component. In finite element (FE) analysis the idea of thermal expansion/contraction can be used as a rather straightforward device to model a number of phenomena. The interference fit between rotating components, for example, is a good example where the standard small thermal strain approximation is generally appropriate. There are other cases where the use of the small thermal strain approximation can lead to significant errors. For example, the extension/contraction of a hydraulic ram can be modelled as an axial element with temperature change used to control the ram length. In this case it is necessary to adopt the proper thermal strain equation in order to obtain accurate results in an FE model which includes large displacements and rotations. This technical note reminds readers of the difference between the small and large thermal strain theories and presents examples where small strain theory is appropriate and where large strain theory is required to obtain the correct solution.

### The Inertial Ellipsoid

The practising engineer might find it useful, particularly when dealing with skeletal structural models, beams, plates, point masses etc, to be able to visualise the inertial properties of his/her structure. For a three-dimensional structure the ellipsoid provides an opportunity to satisfy this desire. This technical note illustrates how this may be done.

### Revision Request for ASME B&PV Code

The low precision thermal strain data offered in the ASME Boiler & Pressure Vessel code of practice is such that it can create a rather significant uncertainty in the results of an engineering analysis. In the example chosen the uncertainty is plus or minus 25%!

### Accuracy of Thermal Expansion Properties in ASME B&PV Code.

Thermal strains listed in the ASME Boiler & Pressure Vessel Code are given in units of mm/m and to 1 decimal place accuracy. For small temperature rises, e.g., 20oC to 50oC, the thermal strain can be as low as 0.3mm/m. This implies an accuracy of +/- 16.6'% which seems rather large given that thermal strains can, presumably, be measured to much greater accuracy.

### ASME/ANSYS: Potential Issue with Thermal Expansion Calculations

This technical note identifies a potential issue with the way in which ASME converts thermal strains to mean coefficients of linear thermal expansion.

### ANSYS: Potential Issue with Element Table Operation

This technical note checks the integration scheme used in the ANSYS Element Table functions and finds that the method is not exact. The approximation is small but it does point to the potential for other approximate numerical schemes creeping into commercial software that might cause potential problems for the practising engineer.

### Dimensional Analysis & Numerical Experiments for a Rotating Disc

This technical note has been prepared to demonstrate how dimensional analysis can lead to a more rational and simplified way of understanding and describing the mechanics of a problem in structural analysis. The case of a rotating, solid, parallel-sided disc is considered. Finite element analysis is used to explore the relationship between the two non-dimensional groups for this problem.

### On the Uniqueness of Limit Analysis Solutions

Whereas elastic solutions are unique, limit analysis solutions are only unique in the sense of the collapse load. This means that there can be many different solutions (yield line patterns or moment fields) each having the same collapse load. This document shows some examples of multiple solutions in terms of yield line patterns and moment fields.

### Stress Linearisation for Practising Engineers

The process of stress linearisation was originally developed to assist practising engineers working in the design and analysis of pressure retaining equipment (pressure vessels, pipes, pumps, etc.) and, using general finite element models, to predict the stresses in these structures. In a mechanics of materials approach, structural forms such as pressure vessels are considered as shells and the codified assessment procedures, such as ASME, require the stresses to be cast in the form of stress resultants found in a shell member, i.e., membrane, bending shear resultants etc. When a pressure vessel, or similar, is analysed using continuum finite elements, then these stress resultants are not part of the standard output. These stress resultants may, however, be recovered by operating on the finite element stress field by the process of stress linearisation. The stress resultants may then further be operated on to obtain stress measures suitable for comparison with allowable limits prescribed in the codes of practice.