Basic MathCad Analysis Examples
Introduction
The bulk of the examples presented in the book employ MathCad for two main reasons. First the MathCad system explicitly presents the mathematical formulation so that a user can see the mathematical basis of the subject under discussion. By using MathCad it is possible to present the finite element solution process with complete clarity: this is not a 'black box' approach. The second reason is that the input data, that sets up the problem, can be easily changed allowing alternative problems to be run to those presented in the book. This is exploited in the sensitivities page where the reader is encouraged to play with the methods for generating sensitivities to control the impact of errors or uncertainties on the results obtained from a finite element analysis.
This part of the web page presents a set of MathCad programs. The first analyses the behaviour of a simple 2-dimensional pin-jointed structure when subjected to static loads applied at the framework nodes and separately its free-vibration behaviour. The second is concerned with the analysis of 1-dimension beams that can be modelled by simple beam elements. The reader can adapt the models to allow an analysis of any other simple 2-dimensional pin-jointed framework or 1-dimensional beam problems to be undertaken.
Before starting to use the models presented below there is a word of warning. The matrix and vector nomenclature employed by MathCad is slightly unusual. Conventionally a matrix or vector would be numbered as shown in the first matrix/vector combination shown below whilst the MathCad way of numbering matrices and vectors is shown in the second example.
|
|
||||||||||||||||||
|
|
||||||||||||||||||
note that the node and element numbers used in the examples, both here and in the sensitivites page, follow the mathcad convention so that that node and element numbers start with a zero; the book uses the normal finite element convention with node and element numbers starting with a "1".
Static Framework Example
This is one of the problems used in chapter 2 of the book "A Practical Guide to Reliable Finite Element Analysis" and is shown in the figure to the right of this text. It involves the analysis of a pin-jointed framework modelled by 2-noded bar elements subjected to the two loads and fixed to a rigid boundary at the nodes 0, 5 and 8. In this example all the bars have the same density and Modulus of Elasticity but these values can be varied so that each element could have different properties. The data defining the problem is presented to the MathCad program through a set of vectors and matrices described in the example. By modifying these data files any 2-dimensional structural analysis problem involving a structure that can be modelled by 2-dimensional bars which is subject to static loads that generates a linear elastic response can be solved.
Deflated Stiffness Matrix
In this case the boundary conditions are set up on the assumption that the structure is rigidly fixed at specified nodal positions. The data is presented to the MathCad program through a set of vectors and matrices described in an initial 'Initial Information' block in the program. The solution process takes into account the boundary conditions by deflating the stiffness matrix through removal of appropriate rows and columns from the global stiffness matrix. This is the simplest way of accommodating boundary conditions but works only in the case where the boundary displacements are specified as having zero values.
In order to see this example click on the Framework symbol below. If you have a version of MathCad working on your computer the example will open and you can then copy the contents of the MathCad file and paste it into your own MathCad folder so that you can modify it and work on it at your leisure to incorporate your own data for a different framework analysis problem. If you do not have MathCad available then follow the instructions given in the Introduction section above to get a trial version. Once you have opened the MathCad example you can return to this page by clicking on the back button at the top of the MathCad web page.
Penalty Method
This example is the same problem as that in the Deflated Stiffness Matrix section above and again the relevant data is presented to the MathCad program through a set of vectors and matrices described in the program. As above the boundary conditions are set-up on the assumption that the structure has displacements specified at set of nodal boundary positions but, because the solution process employs the penalty method solution procedure, these specified displacements need not have zero value. Although the penalty method is not used in the book it is introduced here to present a more general approach for problems involving 2-dimensional pin-jointed structures.
Dynamic Framework Example
This second problem employs the same basic framework as that in the static example and is also used in chapter 2 of the book. The problem is to find the natural frequencies, mode shapes, kinetic energy and strain energy of the 18-bar frame shown on the right. The numbering of the nodes has been selected to both reduce the bandwidth and to make applying the boundary conditions relatively easy. The bar is fully restrained at the nodes 0, 1 and 2 as shown. As in the static case all the bars in this example have the same density and Modulus of Elasticity but these values can be varied so that each element could have different values for these quantities.
In order to see this example click on the Dynamic symbol below. As explained above if you have a version of MathCad working on your computer the example will open and you can then copy the contents of the MathCad file and paste it into your own MathCad folder so that you can modify it to setup and run a vibration analysis for your own 2-dimensional pin-jointed framework. After having opened and copied the progam you can return to this page by clicking on the back button at the top of the MathCad web page.
Static Beam Example
This section illustrates the analysis of the loaded beam shown in the diagram which is used in chapter 8 of the book. This problem can be found in Roark's Formulas for Stress & Strain 6th Edition on page 103 as example 2c. The beam is propped at the end maked "A" and encastre at "B" the load "W" is 12000N/mtr, the overall beam length "l" is 2 metres and the distributed load range "a" is 1 metre. The full details of the structural properties and the element layout employed are found in the MathCad example. In the book the primary purpose is to demonstrate the sensitivity of a specific behavioural parameter to changes in the stiffness of a group of elements. However, in this section, our purpose is to show how the MathCad finite element analysis is able to generate the displacements, rotations, shear forces and bending moments of the complete structure. The sensitivity calculation can be found in the sensitvities page.
As with the other problems in this section the MathCad program can be called into play by clicking on the "Beam" symbol shown above. After having opened and copied the progam to your own files you can return to this page by clicking on the back button at the top of the MathCad web page. This finite element analysis problem can be used to solve other beam analysis problems involving different loading configurations, support conditions or structural properties by making approriate changes to the input data.