Note: Cross references are to sections in the book
Structural engineers' are normally introduced to structural analysis as part of a 3 or 4 year degree course in civil engineering. This document discusses how the curriculum for modern structural analysis in such a course may be structured.
Basic objectives
Graduates from the course should have:
Making room for modelling
Items 2 and 3 on this list are not normally covered in the curriculum and It is unlikely that extra time will be made available for them. Something has to go. It is common to spend quite a lot of time on solution processes; it is in this area that accommodation can be made. Matrix analysis is an essential topic for anyone who will be involved in software development but software development is a specialist skill which need not be addressed in an undergraduate civil engineering course.
Matrix analysis is useful for developing a basic understanding of the process but the way that it should be taught for such an objective is different to that for software developers. For the latter situation one would cover the direct stiffness method and other computing techniques. For developing basic understanding one would adopt a more general approach. For example, the transformation of the element stiffnesses to the structural stiffness matrix can be written as:
P = CT k C D (a)
Where P is the system load vector, k is the diagonally partitioned matrix of element stiffnesses, C is the connection matrix and D is the vector of system displacements. CT is the equilibrium condition relating the system loading to the element end actions and C is the corresponding compatibility condition. CT k C is of course the structural stiffness matrix. Expression (a) shows how equilibrium, compatibility and element force deformation relationships are combined. It is not used directly in software but is useful for explaining the fundamentals.
Manipulation of matrices is for computers. Explicit matrices are not really of any interest or value to software users. For demonstration of matrix manipulation, the order of the matrices should kept low - preferably 2 and not more than 3.
The minimal approach
The simplest strategy is to replace some of the time spent working on solution processes on modelling. When I taught at Strathclyde I latterly took the 3rd and 4th years for structural analysis. In the 3rd year we worked on a plane frame problem based on a simple machinery support system. In the examination the students were required to validate the model and verify the results for a model of frame similar to the one on which they had already done a class exercise. This appeared to be making it too easy for but the standard deviation for the marks in the final examination was quite high. The poorer students could get a pass if they did a reasonable amount of study and the better students were able to achieve high marks. I believe that the examination was successful in identifying those with better intellectual capacity. I used a similar strategy in the 4th year with a 3D model of a building as the example structure and extending the model to cover dynamic behaviour.
I later asked graduates who had taken these classes what they had found useful. One woman had become a stress analyst in aircraft design. She said that she found the idea of checking models most important. Another student explained how he was given a analysis of a long span roof to check because the results showed that the system was unsafe. He said to himself “What question would MacLeod ask? He would ask: Are you sure that the supports have been properly defined?” He looked at the survey drawing and found that the supports had indeed been wrongly specified; a re-analysis showed that the structure was OK.
The fundamental approach
If I was given a clean sheet to develop a curriculum for structural analysis in a civil engineering degree course, I would do something along the following lines:
I would start with structural mechanics with emphasis on the principle of equilibrium. This is the most important topic in mechanics and it should be an aim that graduates from the course should be able to apply equilibrium with confidence. Later in the first year the principles of modelling would be introduced and from then on mechanics and modelling would be covered in parallel or together. By ‘together’ I mean that where practical, topics in mechanics would be used as a natural consequence of the modelling process.
Groups of students would work together to model and to investigate a structure using checking models (e.g as in the examples in Chapter 12 and in the equivalent beam examples in the supplementary information for Chapter 5), sensitivity analysis (as in the examples in Chapter 12) and producing modelling reviews (as defined in Chapter 3). A main aim would be to encourage them to get ‘inside’ the structures, to develop deep understanding. Each group would work on a different structure and at the end of the exercise they would make presentations to share their discoveries.
Each examination might be based on a single practical structure with questions related to mechanics and to modelling (The tradition in examinations for structural analysis is to use mostly impractical structures to test ability in the use of processes that are seldom used.)
Solution process would be addressed but with less emphasis than is conventional.
It is not possible within a civil engineering undergraduate course for the students to spend enough time to develop understanding of the behaviour of a range of structural forms. The aim should be to equip them with strategies which would help them to develop such understanding when needed in practice.
The natural ability threshold problem
Young people in education are brought up on a diet of determinate problem solving. When presented with a problem they say “Show me the method for solving this and I will learn how to do it obtain the answer” From this starting point the find it unsettling when there is no unique answer to a problem. It is not just a matter of implementing an algorithm. Messy complexity must intervene. The members of the undergraduate classes to whom I taught modern structural analysis, demonstrated a spectrum of ability to cope with this situation. At one end of the spectrum some really enjoyed the intellectual challenge of working with uncertainty. At the other end students found it very difficult to handle the non-determinate problems. They would rather not have to bother with them.
So if you adopt a modern approach to the teaching of structural analysis there may be a problem that a proportion of the class will be negative about this way of treating the subject. When I taught modelling at Strathclyde I was feeling my way with it. Students would ask me where they could read more about modelling and I had to admit that there was very little in the way of source material. I remember in the early stages being asked by a student why one can accept 10% difference between a checking model and an element model as a good result but that an equilibrium check which was only good to 5 significant figures was not adequate. At the time I did not have a good answer. It was later that I was able to explain the difference between uncertainty (about the relationship between the checking and element models) and error (in relation to the accuracy of the equation solver) See Section 3.1.4.
I recommend that in using the modern approach in teaching that feedback responses from the students are considered with a view to making changes to help those who find that they are having difficulty working with it. Explaining the difference between determinate and non-determinate problems at the outset should be helpful.
I MacLeod 15.08.05