PSI - Issue 17

Ivan Baláž et al. / Procedia Structural Integrity 17 (2019) 734 – 741 Ivan Baláž, Yvona Koleková, Lýdia Moroczová/ Structural Integrity Procedia 00 ( 2019) 000 – 000

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1. Introduction The paper describes behaviour of simply supported compression member at its ends and supported by ( n – 1) intermediate elastic springs. The span of the member L = n ℓ , where ℓ is the length of the field which equals to the distance between the neighbouring elastic springs. The investigated members have various numbers of the fields n = 2, 3, 4, 5, 6, 7 and ∞. The results of the cases n = 5 and 6 are published in Baláž et al (2019). Only cases n = 2, 3, 4, 7 and ∞ are presented here. The early investigations of the behaviour of the column transversely supported by elastic springs was reported by Engesser (1894). He assumed the equally spaced elastic supports having the same spring stiffness. The problem was also investigated by F. S. Jasinsky (1892, 1894, 1902), S. P. Timoshenko (1910, 1936), I. G. Bubnov (1912), F. Bleich (1924), J. Ratzersdorfer (1936), P. F. Papkovitch (1940), N. K. Snitko (1952), S. D. Lejtes (1954) and others. They used different methods: Jasinsky (1894) and Papkovitch (1940) used the Ritz`s method, Bubnov (1912) used the force method, Bleich (1924) used the mixed method (the unknown quantities were a) force factors – moments and b) displacement factors – displacement of the elastic springs), etc. We use the procedure described in Lejtes (1954). The solution of this problem attracted a lot of famous scientists because it has important application in design of the truss bridges. Where the bridge spans are short, and underslung trusses are not possible, the semi-through trusses may be used. Bracing between the top chords is not possible and restraint to the compression members has to be provided by U-frames. In the case of semi-through truss bridges, the top chord is supported laterally by the diagonals and/or by the verticals and behaves as a strut supported on elastic springs. Stability of the compressed chord is reduced to one of buckling member with hinged ends, supported laterally by a continuous medium and axially loaded by a continuous load, that intensity of which is proportional to the distance from the member middle. In this form the problem was first discussed by Jasinsky (1894, 1902). Some corrections to the important Jasinsky's results have been discussed by Timoshenko (1910) by using the energy method. The contributions to the solution of this problem may be found also in Engesser (1893), Bleich, F. and Bleich H. (1937), Chwala (1929, 1939), Chladný (1955, 1974). The method of determination of the effective length of the compressed chord is given in the appropriate bridge codes. E.g. in DIN 4114 (1952), ČSN 73 6205 (1969), Eurocode EN 1993-2 (2006), for design of steel bridges. The solutions below are scientific backgrounds for application in bridge engineering of the simply supported compression member on the intermediate elastic springs and on the elastic foundation. For stability of the beams on elastic foundation we use the procedure given in Hetényi (1971). Hetényi's (1971) results were taken from Ratzersdorfer (1936). 2. Critical force and buckling length of simply supported member on ( n -1) intermediate elastic springs Solution of the problem leads to the relation between the relative non-dimensional spring stiffness and the dimensionless member parameter ε .

3 

EI N = 

C C w w =

,



(1)

EI

where C w is the spring stiffness [kN/m], EI is the bending stiffness of the member [kNm 2 ], N is the compression axial force [kN], ℓ is the length of the field, distance between the springs [m], w C is the relative non-dimensional spring stiffness [-], ε is the dimensionless member parameter [-].