An aerodynamical analysis of four existing light footbridges of different measures of making span of footbridges stiff enough (i.e. inclined cable system, horizontal truss, horizontal arches) is presented. A model of wind load adopted in agreement with a quasi-steady concept takes into consideration not only unsteady air onflow but also motion of the structure itself. Wind load caused by vortices is neglected. In addition, it is assumed, that considered structures behave in a linear elastic way. Flaga A. Bosak G. Michalowski T. 2 2 / ) ' ( 2 s u u q + = ρ e e e α e α α η α ξ − + − + − − + − + = * * * )] ' /( )] sin( ) cos( [ ) ' /( ) cos sin ' ( s G G s s u u Y X u u v W Theoretical approach to a problem An analysis of four existing light footbridges is made at assumptions as follows : • Wind action on pylons and cable systems of footbridges is of static-type; • Wind action on spans of footbridges is considered in accordance with a quasi-steady theory [1, 5]. • Linearised small vibrations of footbridges around their mean (static) position are considered. Quasi-steady theory enables to consider several aerodynamic phenomena, such as galloping, special type of flutter, buffeting, and divergence. Three components of wind load include static and dynamic wind actions on span footbridges. These three components of wind load, i.e. drag force, lift force and aerodynamic moment, can be given by : (1) (2) wx = q H [Cx + Cxy W] (3) wy = q H [Cy + Cyx W] (4) wm = q H [Cm + Cmm W] (5) where: ρ density of air; u, u’, v’, α, η, ξ, e – as in fig. 1; H – characteristic diameter of cross-section; XG and YG – coordinate of geometric centre of the outline curve of cros-ssection; Cx , Cxy , Cy , Cyx , Cm, Cmm – aerodynamic coe-fficients; – time averaged value; ∗ = d /dt; ’ – fluctuations; s – spatially averaged value. Fig. 1 Systems of coordinates and relations between different geometrical quantities in the case of a moving structure and an unsteady air onflow 3 EECWE, May, 2002,Kiev, Ukraine 3 Mathematical model of the htree degrees of freedom system as a substitutional system of existing structure Spans of the considered footbridges should be treated as a slender structures, which can be divided into m elements. Each of them is represented by the inner point k of the cross-section (see fig. 2). Fig. 2 A sector of span of bridge under wind load Formulas (1-5), which are related to all points of span next are transformed to the global system of coordinates. It is assumed, that the structures behave in a linear elastic way. Their motion can be described by a matrix equation : [M]{r} + [C]{r} + [K]{r}={w} (6) The Bubnow-Galerkin’s method is adopted. As a result, a system of many degrees of freedom is substituted by a representative system of three degrees of freedom. Motion of the substitutional system under wind load described by the formulas (1-6) can be expressed by : (7) {A(t)}j = [CxWK(t), CxyWK(t)ak(t), CxyWK(t)bk(t), -CxyWK(t)ck(t), CxyWK(t)dk(t), CxyWK(t)ek(t), -CxyWK(t)] (8) {A(t)}j = [CyWK(t), CyxWk(t)ak(t), CyxWk(t)bk(t), CyxWk(t)ck(t), CyxWk(t)dk(t), CyxWk(t)ek(t), CyxWk(t),] (9) {A(t)}j = [HCmWK(t), HCmmWK(t)ak(t), HCmmWK(t)bk(t), -HCmmWK(t)ck(t), HCmmWK(t)dk(t), HCmmWK(t)ek(t), -HCmmWK(t)] (10) ∑ 〉 Φ + + 〈 = = Ψ + Ψ + Ψ m k k T k j T k j T k j j j j j j j t t A t A Y t A X t K t C t M )] ( [ )} ( { )} ( { )} ( { ) ( ) ( ) ( * * *