# Stiffness Method and Finite Elements

Due to the ease of implementation, computational efficiency and the proven track record, the Stiffness Method is most commonly utilised in computer applications and is the method used implemented in the developed application. Over the years Matrix Structural Analysis has morphed with the Finite Element Method to become the de-facto standard for computer Structural Analysis tools (Felippa, 2000), with very few tools developed utilising different methods.

The general procedure for solving structural problems using the Direct Stiffness Method is as follows:

1) Establish the number of unknown joint displacements to define the scale of the problem.

2) A number of restraints are introduced to create a kinematically determinate structure (primary structure), and Member forces for this structure are then found forming a particular solution for each of the elements.

3) The structure stiffness matrix K is formed by assembling the element stiffness matrices/coefficients and the load vector P is formed by finding the equivalent joint forces.

4)The complimentary solution is a set of forces required to remove the restraints and re-establish equilibrium. Finding these forces is normally a two-step process. Initially, joint displacements are found by solving the equation d = K-1P, where P is a load vector, K is a structure stiffness matrix, and d is a displacement vector. Joint displacements are considered the primary solution to the stiffness method problem. Based on the joint displacements, it is possible to calculate the member forces due to those displacements using the equation P = Kd.

5) Finally a total solution is found: Total Solution = Particular Solution + Complementary Solution.

Structural analysis and the stiffness method is a well-established subject with numerous textbooks published on the topic. It is studied as a part of all undergraduate and post-graduate civil and structural engineer courses. The recent research on the topic is mostly concerned on introducing new types of elements, non-linear analysis, dynamic analysis and optimisation of the calculation method, but these are mostly beyond the scope of this project.

The Finite Element Method (Zienkiewicz & Taylor, 2004) is possibly the most comprehensive textbook on the subject and contains a wealth of information not available in other publications, however it is very broad. Although an older book, and not republished recently, (Coates, et al., 1987) provides an excellent introduction to structural analysis and also gives guidance on computer applications in structural analysis and was used in developing this application extensively. This is omitted from many modern textbooks, as most new engineers elect to use off-the shelf analysis packages. The book still remains a recommended textbook by many lecturers of structural design.

The essential sources of information are the Lecture Notes for the Structural Mechanics and Finite Elements module (Sagaseta, 2012). For this particular project it provides a useful introduction to the stiffness method as well as detailing many mathematical concepts (coordinate transformations, shape functions, etc) required to successfully tackle a project of this type. Furthermore, many lecturers have published their lecture notes on the internet (Gavin, 2009), (Chandra & Namilae, n.d.), (Felippa, 2004) and many others, and these provide different viewpoints on the similar subject and expand on the basic syllabus in different ways.

Although the core procedure is common for both hand-calculations and for computer programme, there are a number of differences along the way. These all reflect the differences in the ability of an engineer to selectively apply simplifications and of computers to perform a large number of calculations quickly and accurately.

Firstly, in hand calculations all axial strains in the members are usually disregarded. This is a valid assumption, as they contribute very little to the actual displacements. This assumption is used to reduce the number of displacements to be considered in hand calculations. However, in computer software, all displacements are taken into account to simplify the calculations procedure and the extra computational requirements are normally not an issue with modern computers. These differences are illustrated below. Only a brief introduction to the stiffness method is presented here. The details of the programmatic implementation, as well as a more detailed description of the method and the techniques employed is detailed in other chapters.

### Structural Analysis – Example Hand Calculations

In order to demonstrate the utility and validity of the application’s structural analysis algorithm a check was carried out against the simple example hand calculations. The example was part of a 2nd year structural analysis coursework at Kingston University.

The simplified excel-based calculations is reproduced below. This represents how stiffness method is applied in hand calculations. These are based on the first principles of Structural Analysis and the Stiffness Method.      