12 Rosemarie Wagner Length Lx=Ly=10m Radius R=L80 pretension Gr=10 kN/m Elastic stiffness E-d=750 kN/m Uniformly distributed load p=1,0 kN/m2 2f 5Ed 4Ed +△o1 3Ed 3R 2R R 0 System,load and material Increase in tension stress related to the pretension 0 0 0 0 R R Ed 2R E.d 2R 3R 3R T+△f 3Ed 3Ed -△/G 4Ed 4Ed 5Ed 5Ed Vertical displacements Decrease in tension stress related to the pretension Fig.15.Influence of curvature and elastic stiffness to the stresses and deformation The change of the stresses caused by external loads is: [oag+△aa]·[bag+△bag]=p3 (2) Multiplication of the terms and neglecting terms of high order result in: △ga9bag+△baBg8=p3 (3) Linear elastic behaviour of the material lead to △aaB=naB61.△E67 (4) The deviation of the curvature can be approximately seen as the displacement in vertical direction △bag=uia。B (5) The change in the elastic strain is approximately multiplication of the curvature with the vertical displacement: △e6=bi1·u3 (6) (6)substituted (4)and with (5)is the change of the stresses na.b5·p3 nagy.ba8·b6y12 Rosemarie Wagner Fig. 15. Influence of curvature and elastic stiffness to the stresses and deformation The change of the stresses caused by external loads is: [σαβ + ∆σαβ] · [bαβ + ∆bαβ] = p3 (2) Multiplication of the terms and neglecting terms of high order result in: ∆σαβbαβ + ∆bαβσαβ = p3 (3) Linear elastic behaviour of the material lead to ∆σαβ = nαβδγ · ∆εδγ (4) The deviation of the curvature can be approximately seen as the displacement in vertical direction ∆bαβ = u3 |α,β (5) The change in the elastic strain is approximately multiplication of the curvature with the vertical displacement: ∆εδγ = bδγ · u3 (6) (6) substituted (4) and with (5) is the change of the stresses ∆σαβ = nαβδγ · bδγ nαβδγ · bαβ · bδγ · p3 Length Lx = Ly = 10 m Radius R = L²/(8f) pretension VT = 10 kN/m Elastic stiffness Ed = 750 kN/m Uniformly distributed load p = 1,0 kN/m² System, load and material Increase in tension stress related to the pretension Vertical displacements Decrease in tension stress related to the pretension Ly Lx 2 f 5 Ed 4Ed 3Ed 2Ed d 0 3R 2R R 0 0 Ed 2Ed 3Ed 4Ed 5Ed 3R 2R R 0 +'f -'VVT 0 Ed 2Ed 3Ed 4Ed 5Ed 3R 2R R 0