Bsi bs en 01991 1 4 2005 + a1 2010 (2011)

BRITISH STANDARD Eurocode 1: Actions on structures — Part 1-4: General actions — Wind actions ICS 91.010.30 12&23 L then K may be obtained from the curve for two span bridges, neglecting the shortest side span and treating the largest side span as the main span of an equivalent two-span bridge d) For symmetrical four-span continuous bridges (i.e bridges symmetrical about the central support): K may be obtained from the curve for two-span bridges in Figure F.2 treating each half of the bridge as an equivalent two-span bridge e) For unsymmetrical four-span continuous bridges and continuous bridges with more than four spans: K may be obtained from Figure F.2 using the appropriate curve for three-span bridges, choosing the main span as the greatest internal span 138 BS EN 1991-1-4:2005+A1:2010 EN 1991-1-4:2005+A1:2010 (E) NOTE If the value of EI b at the support exceeds twice the value at mid-span, or is less than 80 % of m the mid-span value, then the Expression (F.6) should not be used unless very approximate values are sufficient NOTE A consistent set should be used to give n1,B in cycles per second (6) The fundamental torsional frequency of plate girder bridges is equal to the fundamental bending frequency calculated from Expression (F.6), provided the average longitudinal bending inertia per unit width is not less than 100 times the average transverse bending inertia per unit length (7) The fundamental torsional frequency of a box girder bridge may be approximately derived from Expression (F.7): n1,T = n1,B ⋅ P1 ⋅ (P2 + P3 ) (F.7) with: P1 = P2 = Š P3 = m ⋅ b2 Ip ∑r j (F.8) ⋅Ij (F.9) b ⋅ Ip L2 ⋅ ∑J j ⋅ K ⋅ b ⋅ I p ⋅ (1 + ν ) 2 (F.10) ‹ where: n1,B is the fundamental bending frequency in Hz b is the total width of the bridge m is the mass per unit length defined in F.2 (5) ν is Poisson´s ratio of girder material rj is the distance of individual box centre-line from centre-line of bridge Ij is the second moment of mass per unit length of individual box for vertical bending at midspan, including an associated effective width of deck Ip is the second moment of mass per unit length of cross-section at mid-span It is described by Expression (F.11) Ip = md ⋅ b + ∑ (I pj + m j ⋅ r j2 ) 12 (F.11) where: md is the mass per unit length of the deck only, at mid-span Ipj is the mass moment of inertia of individual box at mid-span mj is the mass per unit length of individual box only, at mid-span, without associated portion of deck 139 BS EN 1991-1-4:2005+A1:2010 EN 1991-1-4:2005+A1:2010 (E) Jj is the torsion constant of individual box at mid-span It is described by Expression (F.12) Jj = ⋅ Aj2 ds #∫ t (F.12) where: Aj #∫ is the enclosed cell area at mid-span ds t is the integral around box perimeter of the ratio length/thickness for each portion of box wall at mid-span NOTE Slight loss of accuracy may occur if the proposed Expression (F.12) is applied to multibox bridges whose plan aspect ratio (=span/width) exceeds 140 BS EN 1991-1-4:2005+A1:2010 EN 1991-1-4:2005+A1:2010 (E) Figure F.2 — Factor K used for the derivation of fundamental bending frequency F.3 Fundamental mode shape (1) The fundamental flexural mode Φ1(z) of buildings, towers and chimneys cantilevered from the ground may be estimated using Expression (F.13), see Figure F.3 z h ζ Φ ( z) =   (F.13) where: ζ = 0,6 for slender frame structures with non load-sharing walling or cladding 141 BS EN 1991-1-4:2005+A1:2010 EN 1991-1-4:2005+A1:2010 (E) ζ = 1,0 for buildings with a central core plus peripheral columns or larger columns plus shear bracings ζ = 1,5 for slender cantilever buildings and buildings supported by central reinforced concrete cores ζ = 2,0 for towers and chimneys ζ = 2,5 for lattice steel towers Figure F.3— Fundamental flexural mode shape for buildings, towers and chimneys cantilevered from the ground (2) The fundamental flexural vertical mode Φ1(s) of bridges may be estimated as shown in Table F.1 Table F.1 — Fundamental flexural vertical mode shape for simple supported and clamped structures and structural elements Scheme Mode shape Φ1(s)  s sin π ⋅   !  s   ⋅ 1 − cos ⋅ π ⋅   !   142 BS EN 1991-1-4:2005+A1:2010 EN 1991-1-4:2005+A1:2010 (E) F.4 Equivalent mass (1) The equivalent mass per unit length me of the fundamental mode is given by Expression (F.14) ! me = ∫ m(s ) ⋅ Φ (s ) ds ! ∫Φ (F.14) (s ) ds where: m is the mass per unit length ! is the height or span of the structure or the structural element i=1 is the mode number (2) For cantilevered structures with a varying mass distribution me may be approximated by the average value of m over the upper third of the structure h3 (see Figure F.1) (3) For structures supported at both ends of span ! with a varying distribution of the mass per unit length me may be approximated by the average value of m over a length of !/3 centred at the point in the structure in which Φ(s) is maximum (see Table F.1) F.5 Logarithmic decrement of damping (1) The logarithmic decrement of damping δ for fundamental bending mode may be estimated by Expression (F.15) δ = δs + δa + δd (F.15) where: δs is the logarithmic decrement of structural damping δa is the logarithmic decrement of aerodynamic damping for the fundamental mode δd is the logarithmic decrement of damping due to special devices (tuned mass dampers, sloshing tanks etc.) (2) Approximate values of logarithmic decrement of structural damping, δs, are given in Table F.2 (3) The logarithmic decrement of aerodynamic damping δa, for the fundamental bending mode of alongwind vibrations may be estimated by Expression (F.16) δa = cf ⋅ ρ ⋅ v m ( zs ) ⋅ n1 ⋅ µe (F.16) where: cf is the force coefficient for wind action in the wind direction stated in Section µe is the equivalent mass per unit area of the structure which for rectangular areas given by Expression (F.17) 143 BS EN 1991-1-4:2005+A1:2010 EN 1991-1-4:2005+A1:2010 (E) h b µe = ∫ ∫ µ(y, z) ⋅ Φ ( y , z ) dydz 0 h b ∫ ∫Φ (F.17) ( y , z ) dydz 0 where µ(y,z) is the mass per unit area of the structure Φ1(y,z) is the mode shape The mass per unit area of the structure at the point of the largest amplitude of the mode shape is normally a good approximation to µe (4) In most cases the modal deflections Φ(y,z) are constant for each height z and instead of Expression (F.16) the logarithmic decrement of aerodynamic damping δa, for alongwind vibrations may be estimated by Expression (F.18) δa = cf ⋅ ρ ⋅ b ⋅ v m ( zs ) ⋅ n1 ⋅ me (F.18) (5) If special dissipative devices are added to the structure, δd should be calculated by suitable theoretical or experimental techniques 144 BS EN 1991-1-4:2005+A1:2010 EN 1991-1-4:2005+A1:2010 (E) Table F.2 —Approximate values of logarithmic decrement of structural damping in the fundamental mode, δs structural damping, Structural type δs reinforced concrete buildings 0,10 steel buildings 0,05 mixed structures concrete + steel 0,08 reinforced concrete towers and chimneys 0,03 unlined welded steel stacks without external thermal insulation 0,012 unlined welded steel stack with external thermal insulation 0,020 steel stack with one liner with external thermal insulationa steel stack with two or more liners with external thermal insulation a h/b < 18 0,020 20≤h/b