CHAPTER 5 - SOLUTION OF SIMULTANEOUS EQUATIONS

Abstract

This chapter discusses the solution of simultaneous equations. The solution of simultaneous linear equations is perhaps the most common operation in any modeling of physical systems. Structural engineers are generally concerned with frameworks of infinite variety whose stress analysis ultimately requires the solution of either equilibrium or compatibility equations. Symmetrical and banded arrays of coefficients occur almost universally in linear structural analysis when the equations of equilibrium of the nodes in a structure are generated automatically by computer. The basis of the so-called compact elimination methods is the possibility that the original array may be decomposed into the product of a lower and an upper triangular array. The advantage of compact elimination methods over Gaussian Elimination is that the right-hand sides of the equations are not involved in the decomposition process. This is an important consideration in linear-elastic structural analysis where alternative loading arrangements give rise to multiple right-hand sides in any analysis. In Crout's method, symmetry in the initial array is not required, and it is possible to hold the upper and lower triangular forms in little more space than occupied by the initial coefficients.