ANALYSIS OF STRUCTURES USING SYMBOLIC MATHEMATICS

Presented By: AMAR SHUKLAFaculty Sponsor: DR. JAVED ALAM

CIVIL & ENVIRONMENTAL ENGINEERING DEPT.

ABSTRACT

Structural Analysis is an integral component of Structural Design. It is used to analyze structures in the area of Civil, Mechanical and Aeronautical Engineering.

Structural Analysis requires high level computations.

The conventional approach to these computations is numerical in the past due to the unavailability of symbolic mathematics algorithms.

We discuss the capabilities of a freely available open source software package Maxima that is capable of performing symbolic mathematics calculations.

This work also presents the use of symbolic mathematics to develop algebraic solutions to the several problems in structural analysis for indeterminate structures.

INTRODUCTION The responses of the structural systems to the transient loading is idealized in

the form of systems of both single degree and multi degree freedom systems.

The solution to these idealized structural problems involves higher level of mathematics including advanced calculus, matrix operations and second order homogeneous and non-homogeneous differential equations.

  In their original form these equations are solved by using symbolic algebra or

numerical techniques.

There are many commercially available software packages such as Mathematica, Maple and Mathcad that can be used to solve the resulting equations. However, these packages are not widely available at all the Universities. Also, once the students leave the university they may not have access to these packages at their workplace.

Therefore, the use of an open source software program MAXIMA that is widely available for Windows and Unix operating systems at zero cost to users by a simple download from the INTERNET is explored in solving problems related to vibration analysis in symbolic algebra mode.

MAXIMA, A COMPUTER ALGEBRA SYSTEM AND ITS USE IN VIBRATION ANALYSIS

MAXIMA is a software package that allows solution of complex algebra problems symbolically as well as numerically.

It originated from the original Macysma system that was developed at MIT in 1960. The software is available free of cost on the INTERNET under GPL license.

The software has extensive set of modules that allows the solution of very complex algebraic problems.

For structural analysis the capabilities of symbolic differentiation, indefinite and definite integrals, matrix operations and the solution of second order homogenous and non-homogeneous differential equations are the most relevant. In addition to that numerical solution can also be obtained by using the programming capabilities built into the program.

Six representative problems from structural analysis were chosen to demonstrate the capabilities and the ease of use for MAXIMA program.

SOLVING DETERMINATE STRUCTURE

SOLVING A TRUSS

FIND THE VALE OF REACTION IN A CANTILEVER BEAM

REACTIONS AND MOMENTS IN A FRAME STRUCTURE

CONCLUSION

The solution of six representative structural analysis problems shows that MAXIMA program is capable of handling the solution to these types of problems. It can produce results in symbolic algebraic form as well as evaluate them numerically.

Problems can easily be solved with a few appropriate MAXIMA commands that would require a larger effort if the solutions were done manually.

The MAXIMA is a free program and it is distributed on the INTERNET under GPL license. Therefore, it is available to everyone without the licensing fee requirement of the equivalent commercial software.

The adoption of MAXIMA software and its cost free nature presents a way for the instructors at the economically challenged institutions to offer close to similar teaching and learning experience to their students that is possible at other institutions of higher learning.

The choice of MAXIMA program as an algebraic and numerical compute engine to teach structural analysis will enable the lecturers, students and the engineers to work in an efficient computing environment that allows a reduction in the cost of computing along with the full fledged capabilities to perform structural dynamics simulation studies.

REFERENCES

Paz, M, “Structural Dynamics”. Van Nostrand Rheinhold, New York, 1980

Wester, M. The review of Computer Algebra System http://www.math.unm.edu/~wester/cas_review.html

Maxima Website http://maxima.sourceforge.net/ GNU General Public License

http://www.gnu.org/copyleft/gpl.html MAXIMA reference Manual

http://maxima.sourceforge.net/docs.shtml