Calculating Elastic Foundation Beam Using Castigliano's Second Theorem

The calculation of elastic foundation beams with Castigliano Second Theorem is studied in this paper, and calculation methods based on the model of Winkler Foundation, of Riftin Foundation, and of Pasternak Foundation are worked out. The calculation of changing section foundation beams is also studied, and the calculation formulae are worked out.It was as early as in 1867 and in 1873 that E.winkler put forward the hypothesis of Winkler Foundation and Castigliano put forward Castigliano Second Theorem respectively. Nevertheless, the two ones haven't been combined with each other in calculations of elastic foundation beams because Castigliano Second Theorem couldn't be used for the calculation of foundation beams having translation. But according to the author's finding, the vertical displacement can be regarded as "the sum of both vertical displacement produced by beams as rigid bodies moving as a whole and the deflection produced by beams under external force". Thus, Castigliano Second Theorem can be used for the calculation of deflection of foundation beams. And numerical value methods of calculating foundation beams can be worked out after the integral equation of deflection formula is processed with numerical value. Accordingly, the various elastic foundation beams models and the formulae of foundation beams with changing section can be worked out to develop a calculation system.In order to have these research findings beneficial to ordinary engineers, in the calculation methods worked out in this paper, the first-cut work is processed by EXCEL and the rest calculation is finished by MATLAB reading in EXCEL table. For the whole calculation can be finished with less than ten MATLAB instructions and without programming, even the engineers without learning MATLAB can also use it easily. In addition, owning to the powerful function of MATLAB at plotting, and easy to link FORTRAN language and C language, it has much more room for further research and commercial development.Through a great deal calculation and comparing the calculation resultswith ones worked out by ANSYS and ones worked out by Direct Difference, it shows that such calculation methods have advantages of more precise result, rapid calculation speed, wide application, and being easy for computers to make calculations.The methods of calculating elastic foundation beams with Castigliano Second Theorem bring new cut-in point, new vitality, and open a new field for the research on elastic foundation beams calculations.