Stateczność Mostu Jednodźwigarowego - Dąbrowski, R. (Engineering Transactions, 1957)
The problem of loss of stability in a structure consisting of a single main girder (a truss or a tied arch) in the axis of the bridge carrying the vertical load, and a stiffening girder (usually of closed cross-section) securing the torsional rigidity of the structure. The stiffening girder is connected with the upper cord of the structure by means of rigid hangers. In the first part of the paper (Arts. 2-4) consideration is given to the case of loss of stability under full symmetric load of the bridge, assuming that 1) the hangers are uniformly distributed along the bridge; 2) they are rigidly connected with the upper cord; 3) the upper cord buckles in the torsional-flexural form characterized by a horizontal displacement u and a rotation P, the stiffening girder being subjected to torsion 0 only (Fig. 4). First, three differential equations are derived from the three equations of equilibrium in the case of buckling of a straight cord [Eqs. (2.4.1), (2.4.2) and (2.4.3)], where the action of the hangers (struts) is determined by means of the elastic constants K1, ..., K. (Fig. 5). Next, differential equations are derived for buckling of a thin-walled curved bar of monosymmetric cross-section uniformly loaded in the plane of the curvature, Eq. (12). Now, the influence of the curvature determined by terms containing the curvature radius R is taken into consideration in the Eqs. (3.9.1) and (3.9.2) concerning the buckling of bridges of arched girder either circular with radial hangers. In the latter hangers or parabolic with vertical case, constant normal force and constant mean radius of curvature are assumed. The differential equations are solved with the simplifying assumption of constant parameters Kr(i=1, ..., 6) and sinusoidal buckling mode. The equations of equilibrium reduce to the Eq. (4.7) and the Eq. (4.8) for the critical force P. (For a main girder having parallel cords, the assumption of constant rigidities KI may be considered to be satisfied. For an arch girder, a simplification providing increased safety is introduced if in the computation the values K; are assumed as for the most flexible middle hanger. Using the previous equations (Art. 2 and 3), the case of side load of a bridge is considered as a non-linear stress problem in Art. 6. In Art. 7, such dynamic problem as that of computation of frequency of free vibration and buckling under pulsating force are considered. In these problems, the structure is simplified to that of a straight cord, and hinged joints between the hangers and the cord. Next, in Art. 5, a general solution of the stability problem óf a one- garnered arch structure is obtained on the basis of the energy method. In particular, certain influences disregarded above are now taken into consideration, for instance variable rigidity in the hangers, their finite number and the fact that they are not at right angles to the arch axis. The solution reduces to the determination of the parameter u, by which the values assumed for the rigidities must be multiplied in order that buckling may occur for the force vP, where y is the safety factor required. In an appendix, among other quantities the values K; are given for constant moment of inertia of the hangers. The influence of the deflection of the stiffening beam is taken into account in a simple way. A numerical example shows the computation of stability according to the Eq. (4.8), together with comparative computations for the appraisal of the influence of the torsion of the stiffening beam, the rigid joints between hangers and the upper cord, the curvature of the upper cord, and the stabilizing action of the horizontal component of the normal force in a hanger. By means of a simplified calculation based on the energy method, the influence of a variable hanger length is determined. In addition, the approximate frequency of free vibration of the structure is calculated. All the above solutions apply also to a double girder bridge without top bracing.