Methods of Structural Analysis by Kotur Raghavan

 Structural analysis has been practiced for over two hundred years now. It has been an evolutionary process and a long journey. In the figure major approaches of structural analyses have been presented.

Analytical methods or closed form solutions involve solutions of differential equation of equilibrium. Solutions have been obtained for Bernoulli-Euler beams and for circular and rectangular plates based on Kirchhoff theory. They were, however, for very simple edge conditions. For rectangular plates, for instance, solutions are available only when all the edges are simply supported (Navier’s solution). For more complex combinations of edge conditions engineers resorted to series solutions. They involved assumed displacement functions and energy principles. Rayleigh-Ritz, Galerkin methods are worth mentioning in this category. These were the only methods in vogue till mid-fifties and solutions were obtained through hand calculations or mechanical calculators.


In mid-fifties digital computers started making their appearance and engineers started focusing on numerical methods. Earliest methods focused on numerical solution of differential equations. Either numerical integration or finite difference approximations were used. The difficulty was that structural analysis problems are mostly boundary value problems and numerical solutions approach always pose challenges. In addition the methods were not easily adaptable to problems with multiply connected regions. Procedures also could not be generalized.

Almost simultaneous with finite difference approach researchers set their sights on Matrix Methods. The method was not conceptually new. I was being applied for space frames. No method was yet developed for continuum. Two approaches, namely Force Method and Displacement Method, were possible. Limitations of the former were soon identified. Consequently Displacement Method became the sole survivor. Big breakthrough was achieved in 1956 at University of California, Berkley. Professor Clough and associates came out with the first ever discrete element for continuum. It was the Constant Strain Triangle (CST) for plane stress problems.

At some point of time Matrix Displacement Method came to be known as Finite Element Method (FEM). CST has been widely regarded as “Mother of All Finite Elements”.  After 1956, FEM has not looked back and it has been a success story all along for nearly six and half decades.


  1. Dear Dr. Raghavan,
    Nice of you to take this great initiative. Your preliminary remarks look excellent and informative.
    I would like to suggest a point here.
    In the hierarchy of solutions, the term 'Differential Equations' is put only in the numerical solution category. But, most analytical solutions also come from differential equations. Therefore Diff. Eq. may also be a part of analytical methods category.
    There are also one or two minor spelling errors which may be corrected. ( eg. 'I' instead of 'It' in a paragraph; 'continuum' if intended in the plural, may be 'continua'.
    Looking forward to further expositions on the website

  2. I think exact solution should be replaced by Closed Form Solution

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