𝑎𝑎𝑜𝐴𝐴𝑠𝑎𝐴𝑠𝑎𝑜𝐴𝑐𝐴𝑐𝑚𝐴𝑔𝐴ℎ𝐴𝑠𝑏𝐴𝑠𝑏𝑜𝐴𝑠𝑚𝐴𝑠𝑚𝑜𝐴𝑡𝑏𝐵𝑐𝑐𝑐𝑚𝑐𝑓𝑐ℎ𝑐𝑠𝑐𝑠𝑡𝑐𝑡𝐶𝐶𝐻𝐶𝐻𝑜𝐶𝑜, 𝐶𝑓𝑑𝐸𝐸𝑐𝐸𝑒𝑓𝑓𝐸𝑠𝑒𝑐𝐸𝑡𝑎𝑛𝑓𝑓ℎ𝑓𝑜𝑓𝑝𝑓𝑦𝐹𝑐𝐹ℎ𝐹ℎ𝑥, 𝐹ℎ𝑦, 𝐹ℎ𝑧𝐹𝑠𝐹𝑝𝑔ℎℎ𝐵ℎ𝑜ℎ𝑇ℎ𝑇𝑅 ℎ𝑇: 𝐿𝑚ℎ𝑇𝑅𝑜ℎ𝑥, ℎ𝑦, ℎ𝑧𝐻𝐻𝑐𝑚, 𝐻𝑐𝑠𝐻𝑆𝑅𝐼𝑔𝐼𝑡𝐾𝐾𝑟𝐿𝑐𝐿𝑐𝑚 _1𝐿𝑐𝑚_2𝐿𝑐𝑜𝐿ℎ𝐿𝑚𝐿𝑚𝑜𝐿𝑝𝐿𝑅 𝐿ℎ: 𝐿𝑚𝐿𝑠𝐿𝑆𝑅 𝐿𝑠: 𝐿𝑚𝐿𝑆𝑅𝑜𝑀𝑀𝑆𝑂𝑛𝑁𝑣𝑁𝑓𝑝, 𝑞𝑄𝑄𝑐𝑚, 𝑄𝑐𝑠𝑄𝑓𝑄𝐹𝑎𝑛, 𝑄𝐻𝑎𝑟𝑝𝑄ℎ𝑄𝑠𝑄𝑠𝑡𝑄𝑠𝑡𝑚 , 𝑄𝑠𝑡𝑠𝑄𝑡𝑅𝑝𝑅𝑥, 𝑅𝑦, 𝑅𝑧𝑆𝑅 𝑓: 𝐿𝑚𝑇𝑇𝑐𝑠𝑇𝐴, 𝑇𝐵𝑈𝑆𝐿𝑈𝑆𝐿𝑐 , 𝑈𝑆𝐿𝑝𝑈𝑆𝐿𝑜, 𝑈𝑆𝐿𝑓𝑉𝐴, 𝑉𝐵𝑉𝑐𝑚,𝑉𝑐𝑠𝑉𝑝𝑓𝑉𝑠𝑉𝑇𝑉𝑂𝐿𝑠𝑎𝑊𝑡𝑥𝑝𝑥𝑝𝑖𝑦𝐵𝑦𝑐 , 𝑦𝑝𝑧𝛼𝑝𝛼𝑠𝑑𝑙𝛽𝑡𝛾𝑐𝛾𝑐𝑚𝛾𝑠𝛾𝑠𝑡𝛾𝑡𝛿𝛿𝑆𝑂(𝛿)𝐹𝐿, (𝛿)𝑃𝐿𝛿𝑐𝛿𝑒𝛿ℎ, 𝛿𝑣𝛿𝑟𝛿𝑠𝑓𝛿𝑎𝛿𝐶𝛿𝑓𝛿ℎ𝛿ℎ𝑥, 𝛿ℎ𝑦, 𝛿ℎ𝑧𝛿𝐿𝑐𝛿𝜃𝑐𝛿𝜎𝑐𝛥𝑒𝛥𝑒𝑜 , 𝛥𝑒𝑓𝛥𝑥, 𝛥𝑦, 𝛥𝑧𝛥𝑋𝑇 , 𝛥𝑌𝑇 , 𝛥𝑍𝑇𝛥𝜔𝑠𝜖𝑐𝜂𝜃𝐴, 𝜃𝐵𝜃𝑐𝜃𝑥, 𝜃𝑦 , 𝜃𝑧𝜆Λμ𝜉𝜌𝑐𝑚, 𝜌𝑐𝑠𝜌ℎ𝜌𝑠𝑡𝜌𝑡𝜎𝑎𝑙𝑙𝑜𝑤𝜎𝑎,𝐷𝐿, 𝜎𝑎,𝐿𝐿𝜎𝑏,𝐿𝐿𝜎𝑐 𝐹𝑐 𝐴𝑐⁄𝜎𝑐𝑜, 𝜎𝑐𝑓𝜎𝑐𝑚, 𝜎𝑐𝑠𝜎ℎ𝜎𝑠𝑡𝜎𝑠𝜎𝑡𝛴𝐻𝑠𝑡𝑠, 𝛴𝑉𝑠𝑡𝑠𝛹𝑐 𝜔𝑐𝑎/𝐻𝛹𝑐𝑝 𝜔𝑐𝑝𝑎/𝐻?̅?𝑝 (𝜔𝑐 + 𝜔𝑠)𝐿𝑚/𝐻𝜔𝑐𝜔𝑐𝑚𝜔𝑐𝑝𝜔𝑝𝜔𝑠𝜔𝑠𝑚, 𝜔𝑠𝑠𝜔𝑠𝑜𝜔𝑅 𝜔𝑝: 𝜔𝑠𝜔𝑅𝑚 𝜔𝑝: 𝜔𝑠𝑚𝜔𝑅𝑜 𝜔𝑝: 𝜔𝑠𝑜𝛺Superstructure Tower Stay Cable Anchor Cable Anchor Pier C Tower L C Tower L Superstructure Tower Cable Anchorage Hanger Anchorage Suspension Cable 38.4 35.8 35.8 45.0 45.0 0.30 2.17 28.6 29.2 24.5 7.9 32.4 Bridge Elevation Superstructure Cross Section Cross Section at Towers 𝜔𝑐 𝑓𝐻𝑑2𝑦𝑑𝑥2=𝜔𝑐𝐻√1+ (𝑑𝑦𝑑𝑥)2𝜃𝐵 𝜃𝐴 𝑓 𝒚 𝒙 𝑎2⁄ 𝑎2⁄ ℎ 𝑉𝐴 𝑉𝐵 𝑇𝐵 𝑇𝐴 𝐻 𝐻 𝐹𝑐 𝐹𝑐 Ψ𝑐 =𝜔𝑐𝑎𝐻Ω =ℎ𝑎𝑑𝑦𝑑𝑥= 𝑠𝑖𝑛ℎ (Ψ𝑐𝑥𝑎+ 𝐴) 𝑦 =𝑎Ψ𝑐𝑐𝑜𝑠ℎ (Ψ𝑐𝑥𝑎+ 𝐴) + 𝐵𝑦(𝑥 = 0) = 0 𝑦(𝑥 = 𝑎) = ℎ 𝐴 = 𝑎𝑠𝑖𝑛ℎ [Ψ𝑐Ω2𝑠𝑖𝑛ℎ (Ψ𝑐2 )]−Ψ𝑐2𝐵 = −𝑎Ψ𝑐𝑐𝑜𝑠ℎ (𝐴)𝑉𝐴 = 𝐻𝑑𝑦𝑑𝑥|𝑥=0=𝜔𝑐𝑎Ψ𝑐𝑠𝑖𝑛ℎ (𝐴)𝑉𝐵 = 𝐻𝑑𝑦𝑑𝑥|𝑥=𝑎=𝜔𝑐𝑎Ψ𝑐𝑠𝑖𝑛ℎ(Ψ𝑐 + 𝐴)ℎ 𝑇𝐴 𝑇𝐵ℎ𝑇𝐴 = √𝐻2 + 𝑉𝐴2 =𝜔𝑐𝑎Ψ𝑐𝑐𝑜𝑠ℎ(𝐴)𝑇𝐵 = √𝐻2 + 𝑉𝐵2 =𝜔𝑐𝑎Ψ𝑐𝑐𝑜𝑠ℎ(Ψ𝑐 + 𝐴)𝜃𝐴 = 𝑎𝑡𝑎𝑛𝑑𝑦𝑑𝑥|𝑥=0= 𝑎𝑡𝑎𝑛[𝑠𝑖𝑛ℎ(𝐴)]𝜃𝐵 = 𝑎𝑡𝑎𝑛𝑑𝑦𝑑𝑥|𝑥=𝑎= 𝑎𝑡𝑎𝑛[𝑠𝑖𝑛ℎ(Ψ𝑐 +𝐴)]𝐶𝐶 = ∫ √1 + (𝑑𝑦𝑑𝑥)2𝑑𝑥𝑎0=𝑎Ψ𝑐[𝑠𝑖𝑛ℎ(Ψ𝑐 + 𝐴) − 𝑠𝑖𝑛ℎ (𝐴)]𝑈𝑆𝐿 Δ𝑒𝐶 = 𝑈𝑆𝐿 + Δ𝑒∆𝑒 =𝐻𝐸𝑐𝐴𝑐∫ [1 + (𝑑𝑦𝑑𝑥)2] 𝑑𝑥𝑎0=𝛾𝑐𝑎2Ψ𝑐𝐸𝑐[Ψ𝑐Ω22𝑐𝑜𝑡ℎ (Ψ𝑐2) +12+12Ψ𝑐𝑠𝑖𝑛ℎ(Ψ𝑐)]𝐴𝑐 𝐸𝑐 𝛾𝑐𝐶 Δ𝑒𝑈𝑆𝐿 = 𝐶 − Δ𝑒Ψ𝑐𝐹𝑐𝐹𝑐 = 𝐻𝐿𝑐𝑎= 𝐻√1 + Ω2Ψ𝑐Ψ𝑐 =𝜔𝑐𝑎𝐹𝑐√1 + Ω2𝜎𝑐Ψ𝑐 =𝛾𝑐𝑎𝜎𝑐√1 + Ω2𝜔𝑐𝑝 𝜔𝑐𝑝 = 𝜔𝑐√1+ Ω2Ω𝑑2𝑦𝑑𝑥2=𝜔𝑐𝑝𝐻𝑑𝑦𝑑𝑥=Ψ𝑝2𝑎(2𝑥 − 𝑎) + Ω𝑦 =Ψ𝑝𝑥2𝑎(𝑥 − 𝑎) + Ω𝑥𝜔𝑐 𝜔𝑐 𝜔𝑐𝑝 Ψ𝜔𝑐𝑝 Ψ𝑝 =𝜔𝑐𝑝𝑎𝐻𝑉𝐴 = 𝐻𝑑𝑦𝑑𝑥|𝑥=0=𝜔𝑐𝑝𝑎Ψ𝑝(Ω −Ψ𝑝2)𝑉𝐵 = 𝐻𝑑𝑦𝑑𝑥|𝑥=𝑎=𝜔𝑐𝑝𝑎Ψ𝑝(Ω +Ψ𝑝2)𝑇𝐴 = √𝐻2 + 𝑉𝐴2 =𝜔𝑐𝑝𝑎Ψ𝑝√1 + (Ω −Ψ𝑝2)2𝑇𝐵 = √𝐻2 + 𝑉𝐵2 =𝜔𝑐𝑝𝑎Ψ𝑝√1 + (Ω +Ψ𝑝2)2𝜃𝐴 = 𝑎𝑡𝑎𝑛𝑑𝑦𝑑𝑥|𝑥=0= 𝑎𝑡𝑎𝑛(Ω −Ψ𝑝2)𝜃𝐵 = 𝑎𝑡𝑎𝑛𝑑𝑦𝑑𝑥|𝑥=𝑎= 𝑎𝑡𝑎𝑛(Ω +Ψ𝑝2)𝐶 = ∫ √[1 + (𝑑𝑦𝑑𝑥)2] 𝑑𝑥𝑎0𝐶 =𝑎2Ψ𝑝[(Ω +Ψ𝑝2)√1 + Ω2 + ΩΨ𝑝 +Ψ𝑝24− (Ω −Ψ𝑝2)√1 + Ω2 − ΩΨ𝑝 +Ψ𝑝24+ 𝑎𝑠𝑖𝑛ℎ (Ω +Ψ𝑝2)− 𝑎𝑠𝑖𝑛ℎ (Ω −Ψ𝑝2)]√1 + 𝑘 𝑘√1 + 𝑘 = ∑(−1)𝑚(2𝑚)!(1 − 2𝑚)(𝑚!)2(4𝑚)𝑘𝑚∞𝑚=0𝑘 = 0 |𝑘| ≤ 1ΩΩ = 0 𝐶𝑎𝑝𝑝𝑟𝑜𝑥|Ω=0 = ∫ [1 +12(𝑑𝑦𝑑𝑥)2]𝑑𝑥 = 𝐿𝑐 [1 +124(𝜔𝑐𝐿𝑐𝐻ℎ𝑜𝑟𝑖𝑧)2]𝐿𝑐0𝐻ℎ𝑜𝑟𝑖𝑧 𝐿𝑐Ω𝐶𝑠𝑖𝑚𝑝𝑙𝑖𝑓𝑖𝑒𝑑 = 𝑎√1 + Ω2 [1 +124(Ψ𝑝1 + Ω2)2]∆𝑒 =𝐻𝐸𝑐𝐴𝑐∫ [1 + (𝑑𝑦𝑑𝑥)2] 𝑑𝑥𝑎0=𝛾𝑐𝑎2√1 + Ω2Ψ𝑝𝐸𝑐[1 + Ω2 +112Ψ𝑝2]Ψ𝑝Ψ𝑝 =𝜔𝑐𝑎𝐹𝑐(1 + Ω2)Ψ𝑝 =𝛾𝑐𝑎𝜎𝑐(1 + Ω2) 𝐿𝑐 𝑓ℎ𝑜𝑟𝑖𝑧 𝑎2⁄ 𝑎2⁄ 𝐻ℎ𝑜𝑟𝑖𝑧 =𝜔𝑐𝐿𝑐28𝑓ℎ𝑜𝑟𝑖𝑧 𝐻ℎ𝑜𝑟𝑖𝑧 𝐻 𝐻 =𝜔𝑐𝑎28𝑓ℎ𝑜𝑟𝑖𝑧= 𝐻ℎ𝑜𝑟𝑖𝑧 (𝑎𝐿𝑐)2 Ψ𝑝𝑓𝑝Ψ𝑝 =8𝑓𝑝𝑎𝜎𝑐𝑓𝑝𝛀 = 𝟎 𝛀 = 𝟎. 𝟓 𝛀 = 𝟏 𝛀 = 𝟏. 𝟓 𝛀 = 𝟐0 0.02 0.04 0.06 0.08 0.1 0.12 0.1401 1042 1043 1044 1045 104𝜸𝒄𝒂𝝈𝒄 𝑶𝑹 𝟖𝒇𝒑𝒂(𝟏 + 𝜴𝟐) |(𝒚𝒄 − 𝒚𝒑)𝒎𝒂𝒙|𝒂 𝜎𝑐 𝑎 𝑓𝑝 𝜎𝑐 𝑎 < 𝐶𝛾𝑐 𝑎 𝛺 𝜎𝑐𝝈𝒄 𝒂 𝝈𝒄 𝝈𝒄 𝒂 𝝈𝒄 (𝒚𝒄 − 𝒚𝒑)𝒎𝒂𝒙 (𝒚𝒄 − 𝒚𝒑)𝒎𝒂𝒙 Ω𝑈𝑆𝐿𝑝𝑈𝑆𝐿𝑐𝐸𝑐 𝐿𝑚 𝛾𝑐 𝜎𝑐𝐿𝑚 1.0 0.5 0.0 𝐿𝑚4 1.0 0.5 0.0 𝛀 = 𝟎𝛀 = 𝟎. 𝟓𝛀 = 𝟏𝛀 = 𝟏. 𝟓𝛀 = 𝟐𝐸𝑐0 0.02 0.04 0.06 0.08 0.1 0.12 0.1402 1064 1066 1068 1060 0.02 0.04 0.06 0.08 0.1 0.12 0.1402 1064 1066 1068 106|𝑼𝑺𝑳𝒄 − 𝑼𝑺𝑳𝒑|𝒂 𝜸𝒄𝒂𝝈𝒄 𝑶𝑹 𝟖𝒇𝒑𝒂(𝟏 + 𝜴𝟐) 𝜎𝑐 𝑎 𝜎𝑐 𝑓𝑝 𝜎𝑐𝑜 𝜎𝑐𝑓𝐶𝑜 = 𝐿𝑐𝑜 [1 +124(1 + Ω2)(𝛾𝑐𝐿𝑐𝑜𝜎𝑐𝑜)2]Δ𝑒𝑜 =𝐿𝑐𝑜𝜎𝑐𝑜𝐸𝑐+112𝐸𝑐(𝛾𝑐2𝐿𝑐𝑜3𝜎𝑐𝑜)𝐶𝑓 = 𝐿𝑐𝑜 + δ𝐿𝑐 +124(1 + Ω2)(𝛾𝑐𝜎𝑐𝑓)2(𝐿𝑐𝑜 + 𝛿𝐿𝑐)3Δ𝑒𝑓 =(𝐿𝑐𝑜 + δ𝐿𝑐)𝜎𝑐𝑓𝐸𝑐++112𝐸𝑐𝛾𝑐2𝜎𝑐𝑓(𝐿𝑐𝑜 + 𝛿𝐿𝑐)3 𝑓𝑜 𝑓𝑜 − 𝛿𝑓 𝑎𝑜 𝜎𝑐𝑓 = 𝜎𝑐𝑜 + 𝛿𝜎𝑐 + 𝜎𝑐𝑜 𝜎𝑐𝑜 𝜎𝑐𝑓 ℎ𝑜 ℎ𝑜 + 𝛿ℎ 𝑎𝑜 + 𝛿𝑎 𝑈𝑆𝐿𝑜 = 𝑈𝑆𝐿𝑓δ𝐿𝑐𝐿𝑐𝑜−δ𝐿𝑐𝐿𝑐𝑜𝜎𝑐𝑓𝐸𝑐=(𝜎𝑐𝑓 − 𝜎𝑐𝑜)𝐸𝑐−𝛾𝑐2𝑎224[1𝜎𝑐𝑓2(𝐿𝑐𝑜 + δ𝐿𝑐)3𝐿𝑐𝑜3 −1𝜎𝑐𝑜2]+𝛾𝑐2𝐿𝑐𝑜212𝐸𝑐𝜎𝑐𝑓[(𝐿𝑐𝑜 + δ𝐿𝑐)3𝐿𝑐𝑜3 −𝜎𝑐𝑓𝜎𝑐𝑜](𝛿𝐿𝑐 ≪ 𝐿𝑐𝑜)δ𝐿𝑐𝐿𝑐𝑜=(𝜎𝑐𝑓 − 𝜎𝑐𝑜)𝐸𝑐+𝛾𝑐2𝑎224(1𝜎𝑐𝑜2−1𝜎𝑐𝑓2)Linear Term Nonlinear Term ~1~0~0𝐸𝑐𝐸𝑠𝑒𝑐𝐸𝑠𝑒𝑐 = (𝜎𝑐𝑓 − 𝜎𝑐𝑜)𝐿𝑐𝑜δ𝐿𝑐𝐸𝑠𝑒𝑐 =11𝐸𝑐+𝛾𝑐2𝑎2(𝜎𝑐𝑓 + 𝜎𝑐𝑜)24𝜎𝑐𝑓2𝜎𝑐𝑜2 𝝈𝒄𝒇 (0,0) 𝐿𝑐𝑜 𝜹𝑳𝒄𝑳𝒄𝒐 𝜎𝑐𝑜 𝑬𝒄 𝟏 𝛿𝜎𝑐 𝛿𝜎𝑐𝛿𝜎𝑐 𝛿𝜎𝑐𝛿𝜎𝑐𝛿𝜎𝑐 ≪ 𝜎𝑐𝑜𝜎𝑐𝑓 = 𝜎𝑐𝑜 𝜎𝑐𝑓 𝜎𝑐𝑜𝐸𝑡𝑎𝑛 =11𝐸𝑐+𝛾𝑐2𝑎212𝜎𝑐𝑜3 𝛿𝐿𝑐𝐿𝑐𝑜 𝜎𝑐𝑜 𝑬𝒔𝒆𝒄 𝟏 𝜎𝑐𝑓 𝛿𝜎𝑐 𝜎𝑐𝑜𝐸𝑐𝛔𝒄𝒐 = 𝟏𝟎𝟎𝑴𝑷𝒂𝛔𝒄𝒐 = 𝟐𝟎𝟎𝑴𝑷𝒂𝛔𝒄𝒐 = 𝟑𝟎𝟎𝑴𝑷𝒂𝛔𝒄𝒐 = 𝟒𝟎𝟎𝑴𝑷𝒂𝛔𝒄𝒐 = 𝟓𝟎𝟎𝑴𝑷𝒂𝛔𝒄𝒐 = 𝟔𝟎𝟎𝑴𝑷𝒂𝛔𝒄𝒐 = 𝟕𝟎𝟎𝑴𝑷𝒂𝛔𝒄𝒐 = 𝟖𝟎𝟎𝑴𝑷𝒂𝛔𝒄𝒐 = 𝟗𝟎𝟎𝑴𝑷𝒂𝛔𝒄𝒐 = 𝟏𝟎𝟎𝟎𝑴𝑷𝒂𝐸𝑐 𝛾𝑐𝒂 (𝒎) 𝑬𝒕𝒂𝒏𝑬𝒄 𝜎𝑐𝑜 𝜎𝑐𝑜 𝑎 𝑃𝛿𝜃𝑐 ≪ 𝜃𝑐δ𝑐 = 𝜖𝑐𝐿𝑐 =𝜎𝑐𝐸𝑒𝑓𝑓𝑎𝑐𝑜𝑠𝜃𝑐𝐸𝑒𝑓𝑓 𝐸𝑠𝑒𝑐 𝐸𝑡𝑎𝑛𝛿𝑣 =δ𝑐𝑠𝑖𝑛𝜃𝑐=𝜎𝑐𝐸𝑒𝑓𝑓(1𝑠𝑖𝑛𝜃𝑐1𝑐𝑜𝑠𝜃𝑐)𝑎𝑎𝜃𝑐 = 45°𝜃𝑐 = 45°21.5° ≤ 𝜃𝑐 ≤ 26.5°𝜃𝑐 = 26.5°‘𝑎’ 0 7.5 15 22.5 30 37.5 45 52.5 60 67.5 75 82.5 9000.511.522.533.544.55𝛿𝑣 𝑎 ℎ 𝑃 Conventional Inclination of Longest Stay in a Cable-Stayed Bridge 𝜽𝒄 (𝒅𝒆𝒈𝒓𝒆𝒆𝒔) 𝜹?̂? 𝛿𝜃𝑐 𝜃𝑐 𝑖 𝑦𝑖 =𝐻𝜔𝑐𝑐𝑜𝑠ℎ (𝜔𝑐𝑥𝑖𝐻+ 𝐴𝑖) + 𝐵𝑖 𝑁1…𝑛 𝑇𝑚𝑎𝑥𝑆1…𝑛 𝐻𝐹ℎ𝑖 {𝑥𝑖 , 𝑦𝑖}𝜔𝑐 Δ𝑥𝑖𝑓 Δy𝑖𝐿𝑚𝑓 𝐿𝑚2 𝛥𝑥𝑖 𝛥𝑦𝑖 𝑦𝑖 𝐹ℎ𝑖 𝑥𝑖 𝐻 𝑇𝑚𝑎𝑥 𝜔𝑐 𝜔𝑐 𝐻𝑑𝑦𝑖𝑑𝑥𝑖|𝑥𝑖=𝛥𝑥𝑖 𝐻 𝐻𝑑𝑦𝑖𝑑𝑥𝑖|𝑥𝑖=0 𝐻 𝐹ℎ1 𝐹ℎ𝑖−1 𝐹ℎ𝑖+1 𝐹ℎ𝑛 𝑆1 𝑆𝑖−1 𝑆𝑖 𝑆𝑖+1 𝑆𝑛 𝑁1 𝑁𝑖−1 𝑁𝑖 𝑁𝑖+1 𝑁𝑛 Free Body Diagram of Segment i 𝐴𝑖 = 𝑎𝑠𝑖𝑛ℎ [𝜔𝑐Δ𝑦𝑖2𝐻𝑠𝑖𝑛ℎ (𝜔𝑐Δ𝑥𝑖2𝐻 )]−𝜔𝑐Δ𝑥𝑖2𝐻𝐵𝑖 = −𝐻𝜔𝑐𝑐𝑜𝑠ℎ (𝐴𝑖) 𝑛 + 1 Δ𝑦1…𝑛 𝐻𝐻 Δ𝑦1…𝑛𝑖 = 12𝐻𝑑𝑦1𝑑𝑥1|𝑥1=0= 𝐹ℎ1𝑟𝑒𝑑𝑢𝑐𝑒𝑠 𝑡𝑜→ 𝐻𝑠𝑖𝑛ℎ(𝐴1) =𝐹ℎ12𝑖 = 2…𝑛 𝐻𝑑𝑦𝑖𝑑𝑥𝑖|𝑥𝑖=0= 𝐻𝑑𝑦𝑖−1𝑑𝑥𝑖−1|𝑥𝑖−1=Δ𝑥𝑖−1+ 𝐹ℎ𝑖𝑟𝑒𝑑𝑢𝑐𝑒𝑠 𝑡𝑜→ 𝐻 [𝑠𝑖𝑛ℎ(𝐴𝑖) − 𝑠𝑖𝑛ℎ (𝜔𝑐Δ𝑥𝑖−1𝐻+ 𝐴𝑖−1)] = 𝐹ℎ𝑖Δ𝑦1…𝑛 𝑓𝐻 𝑓 =∑Δy𝑖𝑛𝑖=1𝐻 𝜔𝑐 = 𝛾𝑐𝐴𝑐 Δ𝑥1…𝑛 𝐹ℎ1…𝑛 𝑓𝑇𝑇𝑂𝐿 𝐻𝐴 Δ𝑦1…𝑛 𝐻𝐴 𝑓𝐴 𝐻𝐴Δ𝑦1…𝑛 𝑓𝐸 = 𝑓𝑇 − 𝑓𝐴 𝑓𝐸 > 𝑇𝑂𝐿 𝑓𝐸 ≤ 𝑇𝑂𝐿 𝑑𝐻𝐴𝐵𝑑𝑓𝐴𝐵= (𝐻𝐴−𝐻𝐵𝑓𝐴−𝑓𝐵) 𝐻𝐵 = (1 − 𝑇𝑂𝐿)𝐻𝐴 Δ𝑦1…𝑛 𝐻𝐵 𝑓𝐵 𝐻𝐵Δ𝑦1…𝑛 𝐻𝐴 𝑁𝐸𝑊 = 𝐻𝐴 𝑂𝐿𝐷 + 𝑓𝐸 (𝑑𝐻𝐴𝐵𝑑𝑓𝐴𝐵) 𝐻𝐴Δ𝑦1…𝑛 Δ𝑥1…𝑛 𝐹ℎ1…𝑛𝐹ℎ1 Δ𝑥1 𝐻𝐻𝐴 𝐻𝐴 𝑁𝐸𝑊𝐻𝐴 𝑁𝐸𝑊 ≥ 0.5𝐻𝐴 𝑂𝐿𝐷 𝑈𝑆𝐿 = 2 [∑𝐶𝑖 − Δ𝑖𝑛𝑖=1] 𝐶𝑖 =𝐻𝜔𝑐[𝑠𝑖𝑛ℎ (𝜔𝑐Δ𝑥𝑖𝐻+ 𝐴𝑖) − 𝑠𝑖𝑛ℎ (𝐴𝑖)] Δ𝑒𝑖 =𝐻Δ𝑥𝑖𝐸𝑐𝐴𝑐[𝜔𝑐Δ𝑦𝑖22𝐻Δ𝑥𝑖𝑐𝑜𝑡ℎ (𝜔𝑐Δ𝑥𝑖2𝐻) +12+𝐻2𝜔𝑐Δ𝑥𝑖𝑠𝑖𝑛ℎ (𝜔𝑐Δ𝑥𝑖𝐻)] 𝑆𝑚,𝑦(𝑥)|𝑆𝑚 =1𝐻[∑ 𝐹ℎ𝑗 (𝑥 −∑𝜆𝑖𝑗𝑖=1)+𝜔𝑐𝑥22− 𝑉𝑠𝑥𝑚−1𝑗=1]𝑥 𝑦𝑉𝑠 =12∑ 𝐹ℎ𝑖 +𝜔𝑐𝐿𝑚2𝑁ℎ𝑎𝑛𝑔𝑒𝑟𝑠𝑖=1𝐻 =1𝑓[𝑉𝑠𝐿𝑚2− 𝜔𝑐𝐿𝑚28−∑𝐹ℎ𝑗 (𝐿𝑚2−∑𝜆𝑖𝑗𝑖=1)𝑁∗𝑗=1]𝑁ℎ𝑎𝑛𝑔𝑒𝑟𝑠−12, 𝑁∗ =𝑁ℎ𝑎𝑛𝑔𝑒𝑟𝑠2 𝜔𝑐𝜆 𝜔𝑠 𝑁1… 𝐿𝑚𝑆1… 𝐻𝑁ℎ𝑎𝑛𝑔𝑒𝑟𝑠 𝑉𝑠𝜔𝑐 λ𝑚𝑓 𝐹ℎ𝑚𝑦 𝐹ℎ𝑚+1 𝑥 𝐻 𝐹ℎ1 𝐹ℎ𝑚 𝐹ℎ𝑚−2 𝑉𝑠 𝐹ℎ𝑚−1 𝜆1 𝜆𝑚−2 𝜆𝑚−1 𝜆𝑚 𝜆𝑚+1 𝑁1 𝑁𝑚−2 𝑁𝑚−1 𝑁𝑚 𝑁𝑚+1 𝐻 𝐿𝑚2 𝐹ℎ2 𝜆2 𝑓 𝑁2 𝑆1 𝑆2 𝑆𝑚−2 𝑆𝑚−1 𝑆𝑚 𝑆𝑚+1 (𝜔𝑐+𝜔𝑠)𝐿𝑚28𝑓, 𝐻 =(𝜔𝑐+𝜔𝑠)𝐿𝑚28𝑓−𝜔𝑠𝜆28𝑓,𝑦(𝑥) =(𝜔𝑐 +𝜔𝑠)𝑥2𝐻[𝑥 − 𝐿𝑚]𝑥𝐶𝑚 = 𝜆𝑚√1+ Ω𝑚2 [1 +124Ψ𝑐𝑚2(1 + Ω𝑚2)]∆𝑒𝑚 =𝐻𝜆𝑚𝐸𝑐𝐴𝑐[1 + Ω𝑚2 +112Ψ𝑐𝑚2(1 + Ω𝑚2)]Ψ𝑐𝑚 =𝜔𝑐𝜆𝑚𝐻Ω𝑚 =ℎ𝑚𝜆𝑚 𝜆𝑚𝜆 𝐻ℎ𝑚ℎ𝑚 = 𝑦𝑚 − 𝑦𝑚−1𝑦𝑚 𝑚𝑦𝑚 𝐻𝑈𝑆𝐿𝑇𝑜𝑡𝑎𝑙 = ∑ (𝑈𝑆𝐿𝑚)𝑁ℎ𝑎𝑛𝑔𝑒𝑟𝑠+1𝑚=1 = ∑ (𝐶𝑚 − Δ𝑒𝑚)𝑁ℎ𝑎𝑛𝑔𝑒𝑟𝑠+1𝑚=1𝑥𝑦𝑐 𝑦𝑝λ 𝐿𝑚 𝜔𝑐 𝜔𝑠𝜔𝑝𝜎𝑎𝑙𝑙𝑜𝑤 0 0.1 0.2 0.3 0.4 0.505 1051 1041.5 1042 104𝝎𝒄𝝎𝒔⁄ (𝒚𝒑 − 𝒚𝒄)𝒎𝒂𝒙𝑳𝒎 𝑓 =𝐿𝑚10𝑈𝑆𝐿𝑐𝑈𝑆𝐿𝑝 𝛾𝑐 𝜎𝑎𝑙𝑙𝑜𝑤 𝜔𝑝 𝜔𝑠250 500 750 1000 1250 1500 1750 200005 1051 1041.5 1042 104(𝒚𝒑 − 𝒚𝒄)𝒎𝒂𝒙𝑳𝒎 𝑳𝒎(𝒎) 𝑓 =𝐿𝑚10𝜔𝑝 𝜔𝑠𝐶 = ∫ √[1 + (𝑑𝑦𝑑𝑥)2] 𝑑𝑥𝐿𝑚0=𝐿𝑚2Ψ̅𝑝[Ψ̅𝑝√1+Ψ̅𝑝24+ 2𝑠𝑖𝑛ℎ (Ψ̅𝑝2)]𝛾𝑐 𝜎𝑎𝑙𝑙𝑜𝑤 𝜔𝑝 𝜔𝑠 𝐸𝑐𝑳𝒎 (𝒎) (𝑼𝑺𝑳𝒄 − 𝑼𝑺𝑳𝒑)𝑳𝒎 𝑓 =𝐿𝑚10Δ𝑒 =𝐻𝐸𝑐𝐴𝑐∫ [1 + (𝑑𝑦𝑑𝑥)2] 𝑑𝑥𝐿𝑚0=𝐻𝐿𝑚𝐸𝑐𝐴𝑐[1 +112Ψ̅𝑝2]Ψ̅𝑝 =(𝜔𝑐 +𝜔𝑠)𝐿𝑚𝐻= 8(𝑓𝐿𝑚) 𝜔𝑐 , 𝜔𝑠 𝜔𝑝 𝜎𝑎𝑙𝑙𝑜𝑤ω𝑐 =(ω𝑠 +ω𝑝) 𝜉 √1 + 16 𝑆𝑅21 − 𝜉 √1 + 16 𝑆𝑅2𝑆𝑅 𝑓: 𝐿𝑚𝜉𝜉 =𝛾𝑐 𝐿𝑚8 𝜎𝑎𝑙𝑙𝑜𝑤 𝑆𝑅𝜔𝑠 𝜔𝑝𝜔𝑝:𝜔𝑠 𝛾𝑐 𝜎𝑎𝑙𝑙𝑜𝑤0 250 500 750 1000 1250 1500 1750 200000.050.10.150.20.250.30.350.40.450.5𝑳𝒎 (𝒎) 𝝎𝒄𝝎𝒔 +𝝎𝒑 𝑆𝑅 = 0.20 𝑆𝑅 = 0.15 𝑆𝑅 = 0.10 𝜔𝑝:𝜔𝑠 = 0.6𝑥𝑝 𝐿𝑝𝐿𝑝2 ≤ 𝑥𝑝 ≤ 𝐿𝑚 −𝐿𝑝2 𝐿𝑚 𝛾𝑐 𝐸𝑐 𝜎𝑎𝑙𝑙𝑜𝑤 𝜔𝑝 𝜔𝑠 𝑆𝑅𝜔𝑠 𝜔𝑝 𝑥𝑝 𝐿𝑝 𝛿𝑚𝑎𝑥 𝐿𝑚 𝑳𝒑𝑳𝒎⁄ 𝒙𝒑𝑳𝒎⁄ 𝜹𝒎𝒂𝒙𝑳𝒎 𝑥𝑝 𝐿𝑚⁄ = 0.21 & 0.79𝐿𝑝 𝐿𝑚⁄ = 0.4𝑥𝑝 𝐿𝑚⁄ = 0.5𝐿𝑝 𝐿𝑚⁄ = 0.38 𝐿𝑚 𝛾𝑐 𝐸𝑐 𝜎𝑎𝑙𝑙𝑜𝑤 𝜔𝑝 𝜔𝑠 𝑆𝑅𝑳𝒑𝑳𝒎⁄ 𝒙𝒑𝑳𝒎⁄ 𝑥𝑝 𝐿𝑝 𝛿𝑚𝑎𝑥 𝐿𝑚 𝜔𝑠 𝜔𝑝 𝜹𝒎𝒂𝒙𝑳𝒎 δ𝑠𝑓 𝐿𝑚 𝛾𝑐 𝐸𝑐 𝜎𝑎𝑙𝑙𝑜𝑤 𝜔𝑝 𝜔𝑠 𝑆𝑅𝜔𝑠 𝜔𝑝 𝑥𝑝 𝐿𝑝 𝛿𝑚𝑎𝑥 𝐿𝑚 𝑳𝒑𝑳𝒎⁄ 𝒙𝒑𝑳𝒎⁄ 𝜹𝒔𝒇𝜹𝒎𝒂𝒙 (%) 𝛿𝑠𝑓 𝐸𝑐𝐴𝑐 𝐸𝑐𝐴𝑐 = ∞ 𝜔𝑝 𝜔𝑠 𝐿𝑚 𝛾𝑐 𝐸𝑐 𝜎𝑎𝑙𝑙𝑜𝑤 𝜔𝑝 𝜔𝑠 𝑆𝑅𝝎𝒔 𝜔𝑝 𝑥𝑝 𝐿𝑝 𝛿𝑚𝑎𝑥 𝐿𝑚 𝑳𝒑𝑳𝒎⁄ 𝒙𝒑𝑳𝒎⁄ 𝜹𝒎𝒂𝒙𝑳𝒎 𝟐𝝎𝒔 𝜔𝑝 𝑥𝑝 𝐿𝑝 𝛿𝑚𝑎𝑥 𝐿𝑚 Suspension Cables Superstructure Hangers Transitory Load 0 100 200 300 400 500 6001020304050607080𝑺𝒑𝒂𝒏− 𝒕𝒐 − 𝑫𝒆𝒑𝒕𝒉 𝑹𝒂𝒕𝒊𝒐 𝑺𝒑𝒂𝒏−𝒕𝒐−𝑾𝒊𝒅𝒕𝒉 𝑹𝒂𝒕𝒊𝒐 𝒙𝒑 𝑳𝒎⁄ = 𝟎. 𝟖, 𝑳𝒑 𝑳𝒎⁄ = 𝟎. 𝟒 𝒙𝒑 𝑳𝒎⁄ = 𝟎. 𝟓, 𝑳𝒑 𝑳𝒎⁄ = 𝟎. 𝟒 𝒙𝒑 𝑳𝒎⁄ = 𝟎. 𝟓, 𝑳𝒑 𝑳𝒎⁄ = 𝟏. 𝟎 𝛾𝑐 𝐸𝑐 𝜎𝑎𝑙𝑙𝑜𝑤 𝜔𝑝 𝜔𝑠 𝑆𝑅250 500 750 100000.0010.0020.0030.0040.0050.0060.007𝜹𝒎𝒂𝒙𝑳𝒎 𝑳𝒎 (𝒎) LS3𝛿𝑚𝑎𝑥 𝜔𝑝 𝜔𝑠 𝛿𝑚𝑎𝑥 𝜔𝑝 𝜔𝑠 𝛿𝑚𝑎𝑥 𝜔𝑝 𝜔𝑠 LS2LS1𝛼𝑝 𝛼𝑝 𝒙𝒑 𝑳𝒎⁄ = 𝟎. 𝟖, 𝑳𝒑 𝑳𝒎⁄ = 𝟎. 𝟒 𝒙𝒑 𝑳𝒎⁄ = 𝟎. 𝟓, 𝑳𝒑 𝑳𝒎⁄ = 𝟎. 𝟒 𝒙𝒑 𝑳𝒎⁄ = 𝟎. 𝟓, 𝑳𝒑 𝑳𝒎⁄ = 𝟏. 𝟎 𝐿𝑚 𝛾𝑐 𝐸𝑐 𝜎𝑎𝑙𝑙𝑜𝑤 𝑆𝑅 𝜔𝑝 𝜔𝑠𝛼𝑝 = 10 0.002 0.004 0.006 0.008 0.01 0.012 0.01400.250.50.7511.251.51.752𝜶𝒑 𝜹𝒎𝒂𝒙𝑳𝒎 𝛿𝑚𝑎𝑥 𝛼𝑝𝜔𝑝 𝜔𝑠 𝛿𝑚𝑎𝑥 𝛼𝑝𝜔𝑝 𝜔𝑠 𝛿𝑚𝑎𝑥 𝛼𝑝𝜔𝑝 𝜔𝑠 LS1 LS2 LS3𝛿𝑚𝑎𝑥 SO (𝐿𝑚 = 250𝑚) (𝐿𝑚 = 500𝑚) (𝐿𝑚 = 750𝑚) (𝐿𝑚 = 1000𝑚)𝛾𝑐 𝐸𝑐 𝜎𝑎𝑙𝑙𝑜𝑤 𝑆𝑅 𝜔𝑝 𝜔𝑠0 0.25 0.5 0.75 1 1.25 1.5 1.75 211.11.21.31.41.51.60 0.25 0.5 0.75 1 1.25 1.5 1.75 211.11.21.31.41.51.60 0.25 0.5 0.75 1 1.25 1.5 1.75 211.11.21.31.41.51.60 0.25 0.5 0.75 1 1.25 1.5 1.75 211.11.21.31.41.51.6𝜹𝒎𝒂𝒙 𝑺𝑶𝜹𝒎𝒂𝒙 𝜹𝒎𝒂𝒙 𝑺𝑶𝜹𝒎𝒂𝒙 𝜶𝒑 𝜶𝒑 𝜶𝒑 𝜶𝒑 LS1LS2LS3LS1LS2LS3LS1LS2LS3LS1LS2LS3𝜔𝑝: 𝜔𝑐𝜔𝑝: (𝜔𝑐 +𝜔𝑠 +𝜔𝑝)𝜔𝑐 = 𝜔𝑠 (1 + 𝜔𝑅) · 𝛫ω𝑅 𝜔𝑝: 𝜔𝑠Κ𝛫 = 𝜉 √1 + 16 𝑆𝑅21 − 𝜉 √1 + 16 𝑆𝑅2𝜉 =𝛾𝑐 𝐿𝑚8 𝜎𝑎𝑙𝑙𝑜𝑤 𝑆𝑅𝜔𝑝𝜔𝑝𝜔𝑐=𝜔𝑅 (1 + 𝜔𝑅) · 𝛫𝛿𝑒2𝛿𝑒1≅ (𝜔𝑅2 1 + 𝜔𝑅2) (1 + 𝜔𝑅1 𝜔𝑅1)𝛿𝑒1 𝛿𝑒2𝜔𝑝𝜔𝑐 +𝜔𝑠 +𝜔𝑝=𝜔𝑅 (1 + 𝜔𝑅)(𝛫 + 1)𝛿𝑠𝑓2𝛿𝑠𝑓1≅ (𝜔𝑅2 1 + 𝜔𝑅2) (1 + 𝜔𝑅1 𝜔𝑅1)𝛿𝑚𝑎𝑥2𝛿𝑚𝑎𝑥1=𝛿𝑠𝑓2 + 𝛿𝑒2𝛿𝑠𝑓1 + 𝛿𝑒1𝛿𝑚𝑎𝑥2𝛿𝑚𝑎𝑥1≅𝛿𝑠𝑓2𝛿𝑠𝑓1≅𝛿𝑒2𝛿𝑒1≅ (𝜔𝑅2 1 + 𝜔𝑅2) (1 + 𝜔𝑅1 𝜔𝑅1)𝛿𝑚𝑎𝑥 𝛿𝑠𝑓𝑥𝑝𝐿𝑝𝑥𝑝 𝐿𝑚 𝐿𝑝 𝐿𝑚𝛼𝑝 = 1𝛚𝑹𝟐 = 𝟎.𝟐 𝛚𝑹𝟐 = 𝟎.𝟑 𝛚𝑹𝟐 = 𝟎.𝟒 𝛚𝑹𝟐 = 𝟎. 𝟓 𝛚𝑹𝟐 = 𝟎. 𝟔 𝐿𝑚 𝛾𝑐 𝐸𝑐 𝜎𝑎𝑙𝑙𝑜𝑤 𝑆𝑅0.2 0.3 0.4 0.5 0.600.250.50.7511.251.51.7522.252.50.2 0.3 0.4 0.5 0.600.250.50.7511.251.51.7522.252.5𝝎𝑹𝟏 𝝎𝑹𝟏 𝜹𝒆𝟐𝜹𝒆𝟏 𝜹𝒔𝒇𝟐𝜹𝒔𝒇𝟏 (𝐿𝑚 = 250𝑚) (𝐿𝑚 = 1000𝑚)𝛾𝑐 𝐸𝑐 𝜎𝑎𝑙𝑙𝑜𝑤 𝑆𝑅0 0.1 0.2 0.3 0.4 0.5 0.611.051.11.151.21.251.30 0.1 0.2 0.3 0.4 0.5 0.611.051.11.151.21.251.3𝜹𝒎𝒂𝒙 𝑺𝑶𝜹𝒎𝒂𝒙 𝝎𝑹 𝝎𝑹 LS1LS2LS3LS1LS2LS3 𝒙𝒑 𝑳𝒎⁄ = 𝟎. 𝟖, 𝑳𝒑 𝑳𝒎⁄ = 𝟎. 𝟒 𝒙𝒑 𝑳𝒎⁄ = 𝟎. 𝟓, 𝑳𝒑 𝑳𝒎⁄ = 𝟎. 𝟒 𝒙𝒑 𝑳𝒎⁄ = 𝟎. 𝟓, 𝑳𝒑 𝑳𝒎⁄ = 𝟏. 𝟎 𝐿𝑚 𝛾𝑐 𝐸𝑐 𝜎𝑎𝑙𝑙𝑜𝑤 𝜔𝑝 𝜔𝑠𝐶𝑎𝑝𝑝𝑟𝑜𝑥 = 𝐿𝑚 [1 +83(𝑓𝐿𝑚)2−325(𝑓𝐿𝑚)4+⋯ ]𝑓0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.200.0010.0020.0030.0040.0050.0060.0070.0080.0090.01Typical Sag Ratio for a Suspension Bridge 𝜹𝒎𝒂𝒙𝑳𝒎 𝑺𝑹 𝛿𝑚𝑎𝑥 𝜔𝑝 𝜔𝑠 𝜔𝑠 𝛿𝑚𝑎𝑥 𝜔𝑝 𝛿𝑚𝑎𝑥 𝜔𝑝 𝜔𝑠 LS1LS3LS2𝛿𝑓 = [1516 𝑆𝑅 (5 − 24𝑆𝑅2)]δ𝐶𝑎𝑝𝑝𝑟𝑜𝑥𝛿𝐶𝛿𝑓 Δ𝑒1 = Δ𝑒2 = 0.0325 · 𝐿𝑚1 (𝑁𝑜𝑡𝑒: 𝑇ℎ𝑒 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑒𝑙𝑎𝑠𝑡𝑖𝑐 𝑒𝑙𝑜𝑛𝑔𝑎𝑡𝑖𝑜𝑛 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑖𝑛 𝑏𝑜𝑡ℎ 𝑐𝑎𝑏𝑙𝑒𝑠)𝛿𝑚𝑎𝑥2 > 𝛿𝑚𝑎𝑥1 𝛿𝑚𝑎𝑥1 𝑺𝑹𝟏 = 𝟎.𝟏 𝑺𝑹𝟐 = 𝟎. 𝟎𝟓 𝐷𝑒𝑓𝑜𝑟𝑚𝑒𝑑 𝑃𝑜𝑠𝑖𝑡𝑖𝑜𝑛 𝐿𝑚1 𝐿𝑚2 = 𝐿𝑚1 𝐿𝑚 𝐸𝐴𝑔 𝐸𝐼𝑔 𝑓 𝜔𝑝 𝜔𝑠 𝜔𝑝𝑓 𝐸𝐴𝑔, 𝐸𝐼𝑔 𝐿𝑚 𝑥𝑝 𝐿𝑝 𝑅𝑜𝑙𝑙𝑒𝑟 𝑆𝑢𝑝𝑝𝑜𝑟𝑡 (TYP) 𝜔𝑝 𝜔𝑠 𝐿𝑚 50⁄ (𝑇𝑌𝑃) 𝑅𝑖𝑔𝑖𝑑 𝑇𝑟𝑢𝑠𝑠 𝐸𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝐿𝑚 𝐸𝐴𝑔,𝐸𝐼𝑔 𝑓 𝑅𝑖𝑔𝑖𝑑 𝑇𝑟𝑢𝑠𝑠 𝐸𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝐿𝑝 𝑥𝑝 𝑃𝑖𝑛 𝑆𝑢𝑝𝑝𝑜𝑟𝑡 𝜔𝑝 𝜔𝑠 𝐿𝑝 𝐿𝑝𝐿𝑚𝐸𝑎𝑟𝑡ℎ − 𝐴𝑛𝑐ℎ𝑜𝑟𝑒𝑑 𝑆𝑦𝑠𝑡𝑒𝑚 𝑆𝑒𝑙𝑓 − 𝐴𝑛𝑐ℎ𝑜𝑟𝑒𝑑 𝑆𝑦𝑠𝑡𝑒𝑚 0 2 4 6012340 2 4 601234trace 1111trace 2trace 30 2 4 602550751000 2 4 60255075100trace 1111trace 2trace 3𝑬𝑰𝒈 (𝑴𝑵𝒎𝟐 × 𝟏𝟎𝟓) 𝜹𝒑𝒆𝒂𝒌 (𝒎) 𝑺𝑹 = 𝟎. 𝟏𝟎 𝑺𝑹 = 𝟎. 𝟏𝟓 𝑺𝑹 = 𝟎. 𝟐𝟎 𝑬𝑰𝒈 (𝑴𝑵𝒎𝟐 × 𝟏𝟎𝟓) 𝑴𝒑𝒆𝒂𝒌 (𝑴𝑵𝒎) 𝑬𝑰𝒈 (𝑴𝑵𝒎𝟐 × 𝟏𝟎𝟓) 𝑬𝑰𝒈 (𝑴𝑵𝒎𝟐 × 𝟏𝟎𝟓) 𝐿𝑝𝐿𝑚𝑥𝑝𝐿𝑚𝑺𝑹 = 𝟎. 𝟏𝟎 𝑺𝑹 = 𝟎. 𝟏𝟓 𝑺𝑹 = 𝟎. 𝟐𝟎 𝐿𝑝𝐿𝑚𝑥𝑝𝐿𝑚𝐸𝑎𝑟𝑡ℎ − 𝐴𝑛𝑐ℎ𝑜𝑟𝑒𝑑 𝑆𝑦𝑠𝑡𝑒𝑚 𝑆𝑒𝑙𝑓 − 𝐴𝑛𝑐ℎ𝑜𝑟𝑒𝑑 𝑆𝑦𝑠𝑡𝑒𝑚 0 2 4 611.051.11.151.21.250 2 4 611.051.11.151.21.25trace 1111trace 2trace 30 2 4 611.051.11.151.21.250 2 4 611.051.11.151.21.25trace 1111trace 2trace 3𝑬𝑰𝒈 (𝑴𝑵𝒎𝟐 × 𝟏𝟎𝟓) 𝜹𝒑𝒆𝒂𝒌 𝑺𝑶𝜹𝒑𝒆𝒂𝒌 𝑺𝑹 = 𝟎. 𝟏𝟎 𝑺𝑹 = 𝟎. 𝟏𝟓 𝑺𝑹 = 𝟎. 𝟐𝟎 𝑬𝑰𝒈 (𝑴𝑵𝒎𝟐 × 𝟏𝟎𝟓) 𝑴𝒑𝒆𝒂𝒌 𝑺𝑶𝑴𝒑𝒆𝒂𝒌 𝑬𝑰𝒈 (𝑴𝑵𝒎𝟐 × 𝟏𝟎𝟓) 𝑬𝑰𝒈 (𝑴𝑵𝒎𝟐 × 𝟏𝟎𝟓) 𝑺𝑹 = 𝟎. 𝟏𝟎 𝑺𝑹 = 𝟎. 𝟏𝟓 𝑺𝑹 = 𝟎. 𝟐𝟎 ᴇᴇ 𝐿𝑚 𝜔𝑝 𝜔𝑠 𝜔𝑝0.1 0.15 0.211.21.41.61.80 0.5 1 1.5 211.21.41.61.80.1 0.15 0.21821.52528.532𝑓 𝐿𝑚 𝑓 𝐿𝑚 𝑺𝑹 = 𝒇 𝑳𝒎⁄ 𝜹𝒑𝒆𝒂𝒌 (𝒎) 𝐸𝐼𝑔 = 25000𝑀𝑁𝑚2 𝐸𝐴𝑔 = 200000𝑀𝑁 𝑹𝒐𝒍𝒍𝒆𝒓 𝑺𝒖𝒑𝒑𝒐𝒓𝒕 (TYP) 𝐸𝐴𝑔, 𝐸𝐼𝑔 𝑷𝒊𝒏 𝑺𝒖𝒑𝒑𝒐𝒓𝒕 𝐸𝐴𝑔, 𝐸𝐼𝑔 𝑬𝑨𝒈 (𝑴𝑵 × 𝟏𝟎𝟓) 𝐸𝐼𝑔 = 25000𝑀𝑁𝑚2 𝑆𝑅 = 0.2 𝑺𝑹 = 𝒇 𝑳𝒎⁄ 𝑴𝒑𝒆𝒂𝒌 (𝑴𝑵𝒎) 𝐸𝐼𝑔 = 25000𝑀𝑁𝑚2 𝐸𝐴𝑔 = 200000𝑀𝑁 𝐿𝑚 𝜔𝑝 𝜔𝑠 𝜔𝑝 𝐸𝐴𝑔0.1 0.15 0.20.911.11.21.31.41.50.1 0.15 0.20.911.11.21.31.41.5𝜹𝒑𝒆𝒂𝒌(𝜹𝒑𝒆𝒂𝒌)𝑹 𝑴𝒑𝒆𝒂𝒌(𝑴𝒑𝒆𝒂𝒌)𝑹 𝑺𝑹 = 𝒇 𝑳𝒎⁄ 𝑺𝑹 = 𝒇 𝑳𝒎⁄ Flexible Superstructure (𝐸𝐼𝑔 = 25000𝑀𝑁𝑚2) Stiff Superstructure (𝐸𝐼𝑔 = 600000𝑀𝑁𝑚2) Sag Ratio = 0.1 Sag Ratio = 0.2 𝐿𝑚 𝐸𝐼𝑔 ᴇ 𝜔𝑝 𝜔𝑠 𝜔𝑝4.531.501.534.54.531.501.534.5300200100010020030030020010001002003006040200204060604020020406030020010001002003003002001000100200300Deflection Envelope (m) Bending Moment Envelope (MN·m) Percent Change (%) Percent Change (%) System 2 System 1 𝐿ℎ 𝑓ℎ𝐿𝑚 𝑓𝑓 𝑓ℎ𝑓ℎ 𝐿ℎ 𝐿𝑚 𝑓𝐿ℎ 𝑓ℎ𝐿𝑚 2⁄ 𝐿ℎ 𝑓ℎ 𝑓 Suspended Region Stayed Region Stayed Region 𝐿𝑚 2⁄ 𝑁1…𝑛 𝐿ℎ𝑆1…𝑛 𝑇𝑚𝑎𝑥𝐹ℎ𝑖 𝐻𝜔𝑐 {𝑥𝑖 , 𝑦𝑖}𝑓 Δ𝑥𝑖𝐿𝑚 Δy𝑖Ɨ𝑓 𝐿𝑚 2⁄ 𝛥𝑥𝑖 𝛥𝑦𝑖 𝑦𝑖 𝐹ℎ𝑖 𝑥𝑖 𝐻 𝑇𝑚𝑎𝑥 𝜔𝑐 𝜔𝑐 𝐻𝑑𝑦𝑖𝑑𝑥𝑖|𝑥𝑖=𝛥𝑥𝑖 𝐻 𝐻𝑑𝑦𝑖𝑑𝑥𝑖|𝑥𝑖=0 𝐻 𝐹ℎ1 𝐹ℎ𝑖−1 𝐹ℎ𝑛 𝑆1 𝑆𝑖−1 𝑆𝑖 𝑆𝑛 𝑁1 𝑁𝑖−1 𝑁𝑖 𝑁𝑛 Free Body Diagram of Segment i ƗSame as in fully-laden suspension cable ƗSimilar for Segment n, except 𝛥𝑥𝑛 = 𝐿𝑚 − 𝐿ℎ 2 ⁄ 𝐿ℎ 2⁄ Stayed Region 𝐿𝑚 − 𝐿ℎ 2⁄ 𝑓𝜔𝑐𝑝𝜔𝑐𝑝 = 𝜔𝑐√1+ Ω2 ΩΩ =2 y𝐵𝐿𝑚 − 𝐿ℎ 𝐿𝑚 − 𝐿ℎ 2⁄ 𝐿ℎ 𝑓ℎ 𝑓 𝐴 𝐵 𝐶 𝐷 𝜔𝑐𝑝 𝜔𝑐𝑝 𝜔𝑐 𝑦𝐵 𝐿𝑚 − 𝐿ℎ 2⁄ 𝑭𝒐𝒓 𝟎 ≤ 𝒙 < 𝑳𝒎 − 𝑳𝒉 𝟐⁄𝑦 𝑥 =1𝐻[𝜔𝑐𝑝𝑥22− 𝑉𝑠𝑥]𝑭𝒐𝒓 𝑳𝒎 − 𝑳𝒉 𝟐⁄ ≤ 𝒙 ≤ 𝑳𝒎 + 𝑳𝒉 𝟐⁄𝑦 𝑥 |𝑆𝑚 =1𝐻[∑ 𝐹ℎ𝑗 (𝑥 − ∑𝜆𝑖𝑗𝑖=1) +𝜔𝑐𝑝𝜆12 2𝑥 − 𝜆1 +𝜔𝑐2 𝑥 − 𝜆1 2 − 𝑉𝑠𝑥𝑚−1𝑗=1]𝑭𝒐𝒓 𝑳𝒎 + 𝑳𝒉 𝟐⁄ ≤ 𝒙 < 𝑳𝒎𝑦 𝑥 =1𝐻[𝜔𝑐𝑝2 2𝜆1 + 𝐿ℎ − 𝑥 2 − 𝑉𝑠𝑥]𝑥 𝑦𝑉𝑠 =12∑ 𝐹ℎ𝑖 + 𝜔𝑐𝑝𝜆1 +𝜔𝑐𝐿ℎ2𝑁ℎ𝑎𝑛𝑔𝑒𝑟𝑠𝑖=1𝐻 =1𝑓[𝑉𝑠𝐿𝑚2−𝜔𝑐𝑝𝜆12 𝐿𝑚 − 𝜆1 −𝜔𝑐𝐿ℎ28− ∑𝐹ℎ𝑗 (𝐿𝑚2− ∑𝜆𝑖𝑗𝑖=1)𝑁∗𝑗=1]𝐻 =1𝑓ℎ[(𝑉𝑠 − 𝜔𝑐𝑝𝜆1)𝐿ℎ2−𝜔𝑐𝐿ℎ28− ∑𝐹ℎ𝑗 (𝐿𝑚2− ∑𝜆𝑖𝑗𝑖=1)𝑁∗𝑗=1]𝑁ℎ𝑎𝑛𝑔𝑒𝑟𝑠−12, 𝑁∗ =𝑁ℎ𝑎𝑛𝑔𝑒𝑟𝑠2 𝜔𝑐 𝑁1… 𝐿𝑚𝑆1… 𝐿ℎ𝑁ℎ𝑎𝑛𝑔𝑒𝑟𝑠 Ɨ 𝐻𝜔𝑐 Ɨ 𝑉𝑠𝑓 λ𝑚𝑓ℎ Ɨ 𝐹ℎ𝑚Ɨ𝑦 𝐹ℎ𝑚+1 𝑥 𝐻 𝐹ℎ1 𝐹ℎ𝑚 𝑉𝑠 𝐹ℎ𝑚−1 𝜆1 = 𝐿𝑚 − 𝐿ℎ 2⁄ 𝜆𝑚−1 𝜆𝑚 𝜆𝑚+1 𝑁1 𝑁𝑚−1 𝑁𝑚 𝑁𝑚+1 𝐻 𝐿𝑚 2⁄ 𝐹ℎ2 𝜆2 𝑓 𝑁2 𝑆1 𝑆2 𝑆𝑚−1 𝑆𝑚 𝑆𝑚+1 𝐿ℎ 2⁄ 𝑑0𝑑0 = 0𝜆 𝜔𝑠𝜔𝑐𝑝 𝐿𝑚−𝐿ℎ 2+𝐿ℎ 2𝐿𝑚−𝐿ℎ 𝜔𝑐+𝜔𝑠 8𝑓,𝐻 =𝜔𝑐𝑝 𝐿𝑚−𝐿ℎ 2+𝐿ℎ 2𝐿𝑚−𝐿ℎ 𝜔𝑐+𝜔𝑠 −𝜔𝑠𝜆28𝑓, 𝜔𝑐+𝜔𝑠 𝐿ℎ28𝑓ℎ,𝐻 = 𝜔𝑐+𝜔𝑠 𝐿ℎ2−𝜔𝑠𝜆28𝑓ℎ, 𝐿𝑚 2⁄ 𝐿𝑚 − 𝐿ℎ 2⁄ 𝑑0 𝐿ℎ 2⁄ Stay Cables 𝑥[𝜔𝑐𝑝 𝑥−𝐿𝑚 −𝐿ℎ(𝜔𝑐+𝜔𝑠−𝜔𝑐𝑝)]2𝐻, 𝑓𝑜𝑟 0 ≤ 𝑥 < 𝐿𝑚 − 𝐿ℎ 2⁄𝑦 𝑥 =[ 𝐿𝑚−𝐿ℎ 2(𝜔𝑐+𝜔𝑠−𝜔𝑐𝑝)+4𝑥 𝜔𝑐+𝜔𝑠 𝑥−𝐿𝑚 ]8𝐻, 𝑓𝑜𝑟 𝐿𝑚 − 𝐿ℎ 2⁄ ≤ 𝑥 ≤ 𝐿𝑚 + 𝐿ℎ 2⁄ 𝑥−𝐿𝑚 [𝜔𝑐𝑝 𝑥 +𝐿ℎ(𝜔𝑐+𝜔𝑠−𝜔𝑐𝑝)]2𝐻, 𝑓𝑜𝑟 𝐿𝑚 + 𝐿ℎ 2⁄ ≤ 𝑥 < 𝐿𝑚 𝐿𝑚 − 𝐿ℎ 2⁄ ≤ 𝑥 ≤ 𝐿𝑚 + 𝐿ℎ 2⁄ 𝑥𝑑0 < 0𝐶𝑚 = 𝜆𝑚√1+ 𝛺𝑚2 [1 +124𝛹𝑐𝑚2(1 + 𝛺𝑚2)]∆𝑒𝑚 =𝐻𝜆𝑚𝐸𝑐𝐴𝑐[1 + 𝛺𝑚2 +112𝛹𝑐𝑚2(1 + 𝛺𝑚2)]𝛹𝑐𝑚 =𝜔𝑐𝜆𝑚𝐻𝛺𝑚 =ℎ𝑚𝜆𝑚ℎ𝑚ℎ𝑚 = 𝑦𝑚 − 𝑦𝑚−1𝑦𝑚 𝑚𝐻𝑈𝑆𝐿𝑇𝑜𝑡𝑎𝑙 = ∑ 𝑈𝑆𝐿𝑚 𝑁ℎ𝑎𝑛𝑔𝑒𝑟𝑠+1𝑚=1 = ∑ (𝐶𝑚 − 𝛥𝑒𝑚)𝑁ℎ𝑎𝑛𝑔𝑒𝑟𝑠+1𝑚=1𝜔𝑐𝑝 = 𝜔𝑐√1 + (2 𝑦𝐵𝐿𝑚 − 𝐿ℎ)2𝑦𝐵 = 𝑦 (𝑥 =𝐿𝑚 − 𝐿ℎ2)𝜔𝑐𝑝 𝑦𝐵𝜔𝑐𝑝𝜔𝑐 𝜔𝑐𝑝 𝑦𝐵𝜔𝑐𝑝𝜔𝑐𝑝 ≅ 𝜔𝑐√1+ 4𝑓2 [𝜔𝑐 𝐿𝑚 + 𝐿ℎ + 2𝜔𝑠𝐿ℎ𝜔𝑐𝐿𝑚2 + 𝜔𝑠𝐿ℎ 2𝐿𝑚 − 𝐿ℎ ]2𝑓ℎ = 𝑓 − 𝑦𝐵𝑦𝐵𝑓ℎ𝐿ℎ≅ 𝐿𝑅 (𝑓𝐿𝑚) [𝜔𝑐 + 𝜔𝑠𝜔𝑐𝑝 1 − 𝐿𝑅 2 + 𝐿𝑅 𝜔𝑐 + 𝜔𝑠 2 − 𝐿𝑅 ]𝐿𝑅𝐿𝑅 =𝐿ℎ𝐿𝑚 𝜔𝑐:𝜔𝑠 𝛚𝒄 𝛚𝒔⁄ = 𝟎. 𝟎𝟎𝛚𝒄 𝛚𝒔⁄ = 𝟎. 𝟎𝟓 𝐋𝐄𝐆𝐄𝐍𝐃𝛚𝒄 𝛚𝒔⁄ = 𝟎. 𝟏𝟎𝛚𝒄 𝛚𝒔⁄ = 𝟎. 𝟏𝟓0.1 0.15 0.2 0.25 0.30.040.060.080.10.120.140.160.180.20.1 0.15 0.2 0.25 0.30.040.060.080.10.120.140.160.180.20.1 0.15 0.2 0.25 0.30.040.060.080.10.120.140.160.180.20.1 0.15 0.2 0.25 0.30.040.060.080.10.120.140.160.180.20.1 0.15 0.2 0.25 0.30.040.060.080.10.120.140.160.180.2𝑳𝒉 = 𝟎. 𝟐 𝑳𝒎 𝑳𝒉 = 𝟎. 𝟑 𝑳𝒎 𝑳𝒉 = 𝟎. 𝟒 𝑳𝒎 𝑳𝒉 = 𝟎. 𝟓 𝑳𝒎 𝒇 𝑳𝒎⁄ 𝒇𝒉 𝑳𝒉⁄ 𝒇 𝑳𝒎⁄ 𝒇 𝑳𝒎⁄ 𝒇 𝑳𝒎⁄ 𝒇𝒉 𝑳𝒉⁄ 𝒇𝒉 𝑳𝒉⁄ 𝒇𝒉 𝑳𝒉⁄ 𝑳𝒉 = 𝟎. 𝟔 𝑳𝒎 𝒇 𝑳𝒎⁄ 𝒇𝒉 𝑳𝒉⁄ 𝒇 𝒇𝒉 𝑳𝒉 𝑳𝒎 𝑦𝑐 𝑦𝑝𝑥𝑥𝜔𝑐: 𝜔𝑠 𝑳𝒉 = 𝟎. 𝟐 𝑳𝒎𝑳𝒉 = 𝟎. 𝟒 𝑳𝒎 𝐋𝐄𝐆𝐄𝐍𝐃𝑳𝒉 = 𝟎. 𝟔 𝑳𝒎𝑳𝒉 = 𝟏. 𝟎 𝑳𝒎𝜔𝑐: 𝜔𝑠0 0.05 0.1 0.15 0.201.25 1052.5 1053.75 1055 1050 0.05 0.1 0.15 0.205 1051 1041.5 1042 1040 0.05 0.1 0.15 0.202 1044 1046 1048 104𝝎𝒄 = 𝑢𝑛𝑖𝑓𝑜𝑟𝑚 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑠𝑢𝑠𝑝𝑒𝑛𝑠𝑖𝑜𝑛 𝑐𝑎𝑏𝑙𝑒 𝝎𝒔 = 𝑢𝑛𝑖𝑓𝑜𝑟𝑚 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑠𝑢𝑝𝑒𝑟𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒 𝝎𝒄 𝝎𝒔⁄ 𝝎𝒄 𝝎𝒔⁄ 𝒇 = 𝟎. 𝟑 𝑳𝒎 𝝎𝒄 𝝎𝒔⁄ |(𝒚𝒑 − 𝒚𝒄)𝒎𝒂𝒙|𝑳𝒎 𝒇 𝒇𝒉 𝑳𝒉 𝑳𝒎 𝒇 = 𝟎. 𝟏 𝑳𝒎 𝒇 = 𝟎. 𝟐 𝑳𝒎 |(𝒚𝒑 − 𝒚𝒄)𝒎𝒂𝒙|𝑳𝒎 𝑳𝒉 = 𝟎. 𝟐 𝑳𝒎𝑳𝒉 = 𝟎. 𝟒 𝑳𝒎 𝐋𝐄𝐆𝐄𝐍𝐃𝑳𝒉 = 𝟎. 𝟔 𝑳𝒎𝑳𝒉 = 𝟏. 𝟎 𝑳𝒎𝛾𝑐 𝜎𝑎𝑙𝑙𝑜𝑤 𝜔𝑝 𝜔𝑠200 400 600 800 100001.25 1052.5 1053.75 1055 105200 400 600 800 100005 1051 1041.5 1042 104200 400 600 800 100002 1044 1046 1048 104𝑳𝒎 𝒎 𝑳𝒎 𝒎 𝒇 = 𝟎. 𝟑 𝑳𝒎 𝑳𝒎 𝒎 |(𝒚𝒑 − 𝒚𝒄)𝒎𝒂𝒙|𝑳𝒎 𝒇 𝒇𝒉 𝑳𝒉 𝑳𝒎 𝒇 = 𝟎. 𝟏 𝑳𝒎 𝒇 = 𝟎. 𝟐 𝑳𝒎 |(𝒚𝒑 − 𝒚𝒄)𝒎𝒂𝒙|𝑳𝒎 𝜔𝑝: 𝜔𝑠 𝑳𝒉 = 𝟎. 𝟐 𝑳𝒎𝑳𝒉 = 𝟎. 𝟒 𝑳𝒎 𝐋𝐄𝐆𝐄𝐍𝐃𝑳𝒉 = 𝟎. 𝟔 𝑳𝒎𝑳𝒉 = 𝟏. 𝟎 𝑳𝒎𝛾𝑐 𝜎𝑎𝑙𝑙𝑜𝑤 𝜔𝑝 𝜔𝑠 𝐸𝑐 𝜆 ·𝐿𝑚200 400 600 800 100002 1064 1066 1068 1061 105200 400 600 800 100001 1052 1053 1054 1055 105200 400 600 800 100005 1051 1041.5 1042 1042.5 104𝑳𝒎 𝒎 𝑳𝒎 𝒎 𝒇 = 𝟎. 𝟑 𝑳𝒎 𝑳𝒎 𝒎 |(𝑼𝑺𝑳𝒄 − 𝑼𝑺𝑳𝒑)|𝑳𝒎 𝒇 𝒇𝒉 𝑳𝒉 𝑳𝒎 𝒇 = 𝟎. 𝟏 𝑳𝒎 𝒇 = 𝟎. 𝟐 𝑳𝒎 |(𝑼𝑺𝑳𝒄 − 𝑼𝑺𝑳𝒑)|𝑳𝒎 𝒇𝑳𝒎𝑳𝒉𝑳𝒎𝝎𝒑𝝎𝒔𝑳𝒎 𝒎 |(𝑦𝑝 − 𝑦𝑐)𝑚𝑎𝑥| 𝐿𝑚⁄𝛾𝑐 𝜎𝑎𝑙𝑙𝑜𝑤𝒇𝑳𝒎𝑳𝒉𝑳𝒎𝝎𝒑𝝎𝒔𝑳𝒎 𝒎 |(𝑈𝑆𝐿𝑐 − 𝑈𝑆𝐿𝑝)𝑚𝑎𝑥| 𝐿𝑚⁄ 𝛾𝑐 𝜎𝑎𝑙𝑙𝑜𝑤 𝐸𝑐 𝜆 ·𝐿𝑚 𝜔𝑐 , 𝜔𝑠 𝜔𝑝 𝑇𝑚𝑎𝑥 = 𝐻√1 + (𝑑𝑦𝑑𝑥|𝑥=0)2𝐴𝑐_𝑟𝑒𝑞 =𝑇𝑚𝑎𝑥𝜎𝑎𝑙𝑙𝑜𝑤=𝐻𝜎𝑎𝑙𝑙𝑜𝑤√1 + (𝑑𝑦𝑑𝑥|𝑥=0)2𝐴𝑐_𝑟𝑒𝑞𝜔𝑐 =𝛾𝑐𝐻𝜎𝑎𝑙𝑙𝑜𝑤√1+ (𝑑𝑦𝑑𝑥|𝑥=0)2𝜔𝑐 =𝛾𝑐𝐿𝑚[𝜔𝑐 + 𝜔𝑠𝐿𝑅 2 − 𝐿𝑅 ]8 𝑆𝑅𝜎𝑎𝑙𝑙𝑜𝑤√1 + 16𝑆𝑅2 (𝜔𝑐 + 𝜔𝑠𝐿𝑅𝜔𝑐 − 𝜔𝑠𝐿𝑅2 + 2𝜔𝑠𝐿𝑅)2𝑆𝑅𝑓 𝐿𝑚 𝐿𝑅 𝐿ℎ 𝐿𝑚𝜔𝑐𝐿𝑅𝜔𝑐𝜔𝑐𝜔𝑐 =𝛾𝑐𝐿𝑚[𝜔𝑐 + 𝜔𝑠𝐿𝑅 2 − 𝐿𝑅 ]8 𝑆𝑅𝜎𝑎𝑙𝑙𝑜𝑤√1 + 16(𝑆𝑅2 − 𝐿𝑅)2𝜔𝑠 (𝜔𝑠 + 𝜔𝑝)𝜔𝑐𝜔𝑐 =(𝜔𝑠 + 𝜔𝑝) 𝜉 𝐿𝑅 2 − 𝐿𝑅 √1 + 16(𝑆𝑅2 − 𝐿𝑅)21 − 𝜉 √1 + 16 (𝑆𝑅2 − 𝐿𝑅)2ξ𝜉 =𝛾𝑐 𝐿𝑚8 𝜎𝑎𝑙𝑙𝑜𝑤 𝑆𝑅𝐿𝑅𝐿𝑅𝑆𝑅 𝐿𝑚Suspension Ratio = 0.2 Suspension Ratio = 0.4 Suspension Ratio = 0.6 𝛾𝑐 𝜎𝑎𝑙𝑙𝑜𝑤0 125 250 375 500 625 750 875 100000.030.060.090.120.150 125 250 375 500 625 750 875 100000.030.060.090.120.150 125 250 375 500 625 750 875 100000.030.060.090.120.15𝑳𝒎 𝒎 𝑳𝒎 𝒎 𝑳𝒎 𝒎 𝝎𝒄𝝎𝒔 +𝝎𝒑 𝝎𝒄𝝎𝒔 +𝝎𝒑 𝝎𝒄𝝎𝒔 +𝝎𝒑 𝑆𝑅 = 0.3 𝑆𝑅 = 0.2 𝑆𝑅 = 0.1 𝑓ℎ 𝐿ℎ 𝛿𝑚𝑎𝑥 𝐹𝐿 𝛿𝑚𝑎𝑥 𝑃𝐿(𝛿𝑝𝑒𝑎𝑘)𝐹𝐿𝑘𝐸𝑘𝐺 𝐿𝑚 𝛾𝑐 𝐸𝑐 𝜎𝑎𝑙𝑙𝑜𝑤 𝜔𝑝 𝜔𝑠𝑥𝑝 𝐿𝑝 𝐿ℎ = 0.2 · 𝐿𝑚 0.1 · 𝐿ℎ 𝑥𝑝 𝐿𝑝 𝜔𝑠 𝜔𝑝 𝐿𝑝 𝑥𝑝 𝜔𝑝 𝜔𝑠 𝜔𝑠 𝜔𝑝 𝜹𝒎𝒂𝒙𝑳𝒎 𝜹𝒎𝒂𝒙𝑳𝒎 𝜹𝒎𝒂𝒙𝑳𝒎 𝑳𝒑 𝑳𝒉⁄ 𝑳𝒑 𝑳𝒉⁄ 𝑳𝒑 𝑳𝒉⁄ 𝒙𝒑𝑳𝒉⁄ 𝒙𝒑𝑳𝒉⁄ 𝒙𝒑𝑳𝒉⁄ 0.1 · 𝐿ℎ 𝐿ℎ = 0.6 · 𝐿𝑚 0.1 · 𝐿ℎ 𝐿ℎ = 1.0 · 𝐿𝑚 Suspension Ratio = 0.2 Suspension Ratio = 0.6 Suspension Ratio = 1.0 𝐿𝑚 𝛾𝑐 𝐸𝑐 𝜎𝑎𝑙𝑙𝑜𝑤 𝜔𝑝 𝜔𝑠 𝜹𝒎𝒂𝒙 𝑷𝑳 − 𝜹𝒎𝒂𝒙 𝑭𝑳(𝜹𝒑𝒆𝒂𝒌)𝑭𝑳 𝜹𝒎𝒂𝒙 𝑷𝑳 − 𝜹𝒎𝒂𝒙 𝑭𝑳(𝜹𝒑𝒆𝒂𝒌)𝑭𝑳 0.1 · 𝐿ℎ 0.1 · 𝐿ℎ 0.1 · 𝐿ℎ 0.1 · 𝐿ℎ 𝐿ℎ = 0.6 · 𝐿𝑚 𝐿𝑚 𝐿ℎ 𝑳𝒑 𝑳𝒉⁄ 𝑳𝒑 𝑳𝒉⁄ 𝒙𝒑𝑳𝒉⁄ 𝒙𝒑𝑳𝒉⁄ 𝐿ℎ = 0.2 · 𝐿𝑚 𝐿𝑚 𝐿ℎ 𝑥𝑝 𝐿𝑝 𝜔𝑠 𝜔𝑝 𝐿𝑝 𝑥𝑝 𝜔𝑝 𝜔𝑠 Suspension Ratio = 0.6 Suspension Ratio = 0.2 𝐿ℎ 𝑓ℎ 𝑘𝑠𝑝𝑟𝑖𝑛𝑔 = 𝑘𝐸 + 𝑘𝐺 𝐿𝑚 𝛾𝑐 𝐸𝑐 𝜎𝑎𝑙𝑙𝑜𝑤 𝜔𝑝 𝜔𝑠𝐿ℎ 𝐿𝑚 𝛾𝑐 𝐸𝑐 𝜎𝑎𝑙𝑙𝑜𝑤 𝜔𝑝 𝜔𝑠0.2 0.4 0.6 0.8 100.0050.010.0150.020.2 0.4 0.6 0.8 100.010.020.030.04𝑳𝑹 = 𝑳𝒉 𝑳𝒎⁄ 𝑆𝑅 = 0.3 𝑆𝑅 = 0.2 𝑆𝑅 = 0.1 𝑳𝑹 = 𝑳𝒉 𝑳𝒎⁄ 𝜹𝒑𝒆𝒂𝒌𝑳𝒎 𝜹𝒑𝒆𝒂𝒌𝑳𝒉 𝑆𝑅 = 0.3 𝑆𝑅 = 0.2 𝑆𝑅 = 0.1 𝐿𝑚 500⁄ 𝐿𝑚 50⁄ 𝑇𝑌𝑃 𝐿𝑚 𝐸𝐴𝑔, 𝐸𝐼𝑔 𝑓 𝑅𝑖𝑔𝑖𝑑 𝑇𝑟𝑢𝑠𝑠 𝐸𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑅𝑖𝑔𝑖𝑑 𝑇𝑟𝑢𝑠𝑠 𝐸𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑃𝑖𝑛 𝑆𝑢𝑝𝑝𝑜𝑟𝑡 𝑅𝑜𝑙𝑙𝑒𝑟 𝑆𝑢𝑝𝑝𝑜𝑟𝑡 (TYP) 𝐿ℎ 𝐿𝑚 − 𝐿ℎ 2⁄ 𝐿𝑚 − 𝐿ℎ 2⁄ 𝐿𝑚 500⁄ 𝜔𝑝 𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝜔𝑠 𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝐿𝑚 100⁄ᴇ𝑆𝑅 𝑓 𝐿𝑚⁄ 𝐿𝑚 𝐸𝐼𝑔 ᴇ 𝐸𝐴𝑔 ᴇ 𝜔𝑝 𝜔𝑠 𝜔𝑝Suspension Ratio = 0.6 Suspension Ratio = 0.4 Suspension Ratio = 0.2 Deflection Envelope (m) Bending Moment Envelope (MN·m) Axial Force Envelope (MN) 𝑆𝑅 = 0.1 𝑆𝑅 = 0.2 𝑆𝑅 = 0.3 𝐿𝑚 𝐸𝐼𝑔 ᴇ6 𝐸𝐴𝑔 ᴇ 𝜔𝑝 𝜔𝑠 𝜔𝑝0.2 0.3 0.4 0.5 0.69633690.1 0.15 0.2 0.25 0.39633690.2 0.3 0.4 0.5 0.65253501751753505250.1 0.15 0.2 0.25 0.35253501751753505250.2 0.3 0.4 0.5 0.63002001001002003000.1 0.15 0.2 0.25 0.3300200100100200300 𝑆𝑅 = 0.1 𝑆𝑅 = 0.2 𝑆𝑅 = 0.3 𝐿𝑅 = 0.2 𝐿𝑅 = 0.4 𝐿𝑅 = 0.6𝐿𝑅 𝑆𝑅Peak Deflections (m) Peak Bending Moments (MN·m) Peak Axial Force (MN) 𝐹𝑝𝑒𝑎𝑘 ≅ ∑𝐻𝑠𝑡𝑠 + 𝐻𝑐𝑠 =(𝜔𝑠 + 𝜔𝑝)[𝐿𝑚 1 − 𝐿𝑅 ]28 ℎ𝑇+ 𝐻𝑐𝑚𝐻𝑐𝑚𝜔𝑐𝑝 ≈ 𝜔𝑐ℎ𝑇 ≈ 𝑓𝐹𝑝𝑒𝑎𝑘 ≅(𝛾𝑐𝑚𝐴𝑐𝑚 + 𝜔𝑠 + 𝜔𝑝)𝐿𝑚8 𝑆𝑅𝛾𝑐𝑚 𝐴𝑐𝑚𝐻𝑐𝑚/ 𝑉𝑐𝑚 = Horizontal/Vertical reaction from suspension cable 𝐻𝑐𝑠/ 𝑉𝑐𝑠 = Horizontal/Vertical reaction from anchor cable ∑𝐻𝑠𝑡𝑠/ ∑𝑉𝑠𝑡𝑠= Cumulative horizontal/vertical reaction from stay cables in side span ℎ𝑇 = Tower height 𝑉𝑇𝑏𝑎𝑠𝑒= Reaction at tower base 𝜔𝑠/ 𝜔𝑝 = Dead/Live loading 𝐿𝑅 = Suspension ratio 𝐿𝑚 = Main span length 𝐿𝑚 1 − 𝐿𝑅 2⁄ ℎ𝑇 𝐻𝑐𝑚 (𝜔𝑠 + 𝜔𝑝) 𝐻𝑐𝑚 ∑𝐻𝑠𝑡𝑠 + 𝐻𝑐𝑠 ∑𝑉𝑠𝑡𝑠 +𝑉𝑐𝑠 + 𝑉𝑐𝑚 𝑉𝑇𝑏𝑎𝑠𝑒 Stay Cables ∑𝐻𝑠𝑡𝑠 + 𝐻𝑐𝑠 Main Span Side Span C Tower L Suspension Ratio = 0.2 Suspension Ratio = 0.4 Suspension Ratio = 0.6 𝐿𝑚 𝐸𝐴𝑔 ᴇ5 𝜔𝑝 𝜔𝑠 𝜔𝑝0 5 10 15 20 25 3096303690 5 10 15 20 25 3096303690 5 10 15 20 25 3096303690 5 10 15 20 25 30180012006000600120018000 5 10 15 20 25 30180012006000600120018000 5 10 15 20 25 3018001200600060012001800 𝑆𝑅 = 0.1 𝑆𝑅 = 0.2 𝑆𝑅 = 0.3Peak Bending Moments (MN·m) Peak Deflections (m) 𝐸𝐼𝑔 𝑀𝑁𝑚 𝑥 106 𝐸𝐼𝑔 𝑀𝑁𝑚 𝑥 106 𝐸𝐼𝑔 𝑀𝑁𝑚 𝑥 106 Suspension Ratio = 0.2 Suspension Ratio = 0.4 Suspension Ratio = 0.6 𝐿𝑚 𝐸𝐴𝑔 ᴇ5 𝜔𝑝 𝜔𝑠 𝜔𝑝0 5 10 15 20 25 300.850.90.9511.051.11.150 5 10 15 20 25 300.850.90.9511.051.11.150 5 10 15 20 25 300.850.90.9511.051.11.150 5 10 15 20 25 300.850.90.9511.051.11.150 5 10 15 20 25 300.850.90.9511.051.11.150 5 10 15 20 25 300.850.90.9511.051.11.150 5 10 15 20 25 300.850.90.9511.051.11.150 5 10 15 20 25 300.850.90.9511.051.11.150 5 10 15 20 25 300.850.90.9511.051.11.15 𝑆𝑅 = 0.1 𝑆𝑅 = 0.2 𝑆𝑅 = 0.3𝜹𝒑𝒆𝒂𝒌𝑺𝑶𝜹𝒑𝒆𝒂𝒌−− 𝑴𝒑𝒆𝒂𝒌𝑺𝑶𝑴𝒑𝒆𝒂𝒌++ 𝑴𝒑𝒆𝒂𝒌𝑺𝑶𝑴𝒑𝒆𝒂𝒌−− 𝐸𝐼𝑔 𝑀𝑁𝑚 𝑥 106 𝐸𝐼𝑔 𝑀𝑁𝑚 𝑥 106 𝐸𝐼𝑔 𝑀𝑁𝑚 𝑥 106 𝛿𝑣1 =𝜎𝑐−𝑓𝐸𝑒𝑓𝑓−𝑓(1𝑠𝑖𝑛𝜃𝑐−𝑓1𝑐𝑜𝑠𝜃𝑐−𝑓)𝐿𝑚2𝛿ℎ =𝜎𝑐−𝑎𝐸𝑒𝑓𝑓−𝑎(1𝑐𝑜𝑠𝜃𝑐−𝑎1𝑠𝑖𝑛𝜃𝑐−𝑎) ℎ𝑇𝛿𝑣2 =𝜎𝑐−𝑎𝐸𝑒𝑓𝑓−𝑎(1𝑐𝑜𝑠𝜃𝑐−𝑎1𝑠𝑖𝑛𝜃𝑐−𝑎)𝐿𝑚2𝜎𝑐−𝑓 𝐸𝑒𝑓𝑓−𝑓 𝜎𝑐−𝑎 𝐸𝑒𝑓𝑓−𝑎𝜎𝑐−𝑓 = (𝜔𝑅1 + 𝜔𝑅) 𝜎𝑎𝑙𝑙𝑜𝑤𝜎𝑐−𝑎 = (𝜔𝑅1 + 𝜔𝑅 − 4𝐿𝑆𝑅2)𝜎𝑎𝑙𝑙𝑜𝑤𝜔𝑅 𝜔𝑝: 𝜔𝑠 𝐿𝑆𝑅𝐿𝑠: 𝐿𝑚 𝛿ℎ 𝐿𝑠 𝐿𝑚 2⁄ ℎ𝑇 𝜃𝑐−𝑓 𝜃𝑐−𝑎 𝜔𝑠 𝜔𝑝 𝛿𝑣1 𝛿𝑣2 (From Elongation of Forestay) (From Elongation of Anchor Cable) Forestay Anchor Cable 𝛿𝑣1 + 𝛿𝑣2𝛿𝑣1= 1 + (1 + 𝜔𝑅1 + 𝜔𝑅 − 4𝐿𝑆𝑅2) (𝑠𝑖𝑛𝜃𝑐−𝑓𝑠𝑖𝑛𝜃𝑐−𝑎𝑐𝑜𝑠𝜃𝑐−𝑓𝑐𝑜𝑠𝜃𝑐−𝑎)𝜃𝑐−𝑓 𝜃𝑐−𝑎𝛿𝑣1 + 𝛿𝑣2𝛿𝑣1= 1 + (1 + 𝜔𝑅1 + 𝜔𝑅 − 4𝐿𝑆𝑅2) (2𝐿𝑆𝑅) (ℎ𝑇𝑅 + 𝐿𝑆𝑅24ℎ𝑇𝑅2 + 1)ℎ𝑇𝑅 ℎ𝑇: 𝐿𝑚 𝜔𝑅0.1 0.2 0.3 0.4 0.512340.1 0.15 0.2 0.25 0.31234 ℎ𝑇𝑅 = 0.1 ℎ𝑇𝑅 = 0.2 ℎ𝑇𝑅 = 0.3 𝐿𝑆𝑅 = 0.35 𝐿𝑆𝑅 = 0.40 𝐿𝑆𝑅 = 0.45𝐿𝑆𝑅 ℎ𝑇𝑅𝛿𝑣1 + 𝛿𝑣2𝛿𝑣1Typical Range |𝛿𝑒|𝑝𝑒𝑎𝑘|𝛿𝑟|𝑝𝑒𝑎𝑘 𝐿𝑚 𝐿𝑆𝑅 𝐸𝐼𝑔 ᴇ 𝐸𝐴𝑔 ᴇ 𝜔𝑝 𝜔𝑠 𝜔𝑝Suspension Ratio = 0.6 Suspension Ratio = 0.4 Suspension Ratio = 0.2 Deflection Envelope (m) Bending Moment Envelope (MN·m) Axial Force Envelope (MN) 𝑆𝑅 = 0.1 𝑆𝑅 = 0.2 𝑆𝑅 = 0.3 𝐿𝑚 𝐿𝑆𝑅 𝐸𝐼𝑔 ᴇ6 𝐸𝐴𝑔 ᴇ 𝜔𝑝 𝜔𝑠 𝜔𝑝 𝐿𝑚 𝐿𝑆𝑅 𝐸𝐼𝑔 ᴇ6 𝐸𝐴𝑔 ᴇ 𝜔𝑝 𝜔𝑠 𝜔𝑝0.1 0.15 0.2 0.25 0.312340.2 0.3 0.4 0.5 0.69633690.1 0.15 0.2 0.25 0.39633690.2 0.3 0.4 0.5 0.65253501751753505250.1 0.15 0.2 0.25 0.3525350175175350525𝑆𝑅|𝛿𝑒|𝑝𝑒𝑎𝑘|𝛿𝑟|𝑝𝑒𝑎𝑘𝐿𝑅 = 0.6 𝐿𝑅 = 0.4 𝐿𝑅 = 0.2 𝑆𝑅 = 0.1 𝑆𝑅 = 0.2 𝑆𝑅 = 0.3 𝐿𝑅 = 0.2 𝐿𝑅 = 0.4 𝐿𝑅 = 0.6𝐿𝑅 𝑆𝑅Peak Deflections (m) Peak Bending Moments (MN·m) Suspension Ratio = 0.2 Suspension Ratio = 0.4 Suspension Ratio = 0.6 𝐿𝑚 𝐿𝑆𝑅 𝐸𝐴𝑔 ᴇ5 𝜔𝑝 𝜔𝑠 𝜔𝑝0 5 10 15 20 25 3096303690 5 10 15 20 25 3096303690 5 10 15 20 25 3096303690 5 10 15 20 25 30180012006000600120018000 5 10 15 20 25 30180012006000600120018000 5 10 15 20 25 3018001200600060012001800 𝑆𝑅 = 0.1 𝑆𝑅 = 0.2 𝑆𝑅 = 0.3Peak Bending Moments (MN·m) Peak Deflections (m) 𝐸𝐼𝑔 𝑀𝑁𝑚 𝑥 106 𝐸𝐼𝑔 𝑀𝑁𝑚 𝑥 106 𝐸𝐼𝑔 𝑀𝑁𝑚 𝑥 106 𝑑 ∝ 𝐿𝑚𝛿𝑝𝑒𝑎𝑘 =5𝜔𝑝𝐿𝑚4384𝐸𝐼𝑔𝐼𝑔 ∝ 𝐿𝑚3Suspension Ratio = 0.2 Suspension Ratio = 0.4 Suspension Ratio = 0.6 𝐸𝐼𝑔 𝐿𝑚3 𝐸𝐴𝑔 ᴇ5 𝜔𝑝 𝜔𝑠 𝜔𝑝250 500 750 1000963369250 500 750 1000963369250 500 750 1000963369250 500 750 10000.60.40.20.20.40.6250 500 750 10000.60.40.20.20.40.6250 500 750 10000.60.40.20.20.40.6 𝑆𝑅 = 0.1 𝑆𝑅 = 0.2 𝑆𝑅 = 0.3𝐿𝑚 𝑚 𝐿𝑚 𝑚 𝐿𝑚 𝑚 𝜹𝒑𝒆𝒂𝒌𝑳𝒎 𝑴𝒑𝒆𝒂𝒌𝑳𝒎𝟐 [𝑴𝑵𝒎] (𝜎𝑏,𝐿𝐿)𝑝𝑒𝑎𝑘 =𝑀𝑝𝑒𝑎𝑘𝑐𝐼𝑔𝑐𝑐 𝑑𝑐 = 𝑘𝑑𝑘 𝑑𝐿𝑚 𝐼𝑔 𝐿𝑚3(𝜎𝑏,𝐿𝐿)𝑝𝑒𝑎𝑘 ∝𝑀𝑝𝑒𝑎𝑘𝐿𝑚2𝑑 ∝ 𝐿𝑚 𝐼𝑔 ∝ 𝐿𝑚3(𝜎𝑎,𝐿𝐿)𝑝𝑒𝑎𝑘 ≅𝜔𝑝𝐿𝑚8 𝑆𝑅𝐴𝑔𝐴𝑔𝐼𝑔 𝑑𝑑𝐼𝑔 ≅𝐴𝑔𝑑24𝑑 ∝ 𝐿𝑚 𝐼𝑔 ∝ 𝐿𝑚3𝐴𝑔 ∝ 𝐿𝑚𝑑 ∝ 𝐿𝑚𝐼𝑔 ∝ 𝐿𝑚3(𝜎𝑎,𝐷𝐿)𝑝𝑒𝑎𝑘 ≅ 𝛾𝑐𝑚𝐴𝑐𝑚 + 𝜔𝑠 𝐿𝑚8 𝑆𝑅𝐴𝑔(𝜎𝑎,𝐷𝐿)𝑝𝑒𝑎𝑘 ≅(𝛾𝑠𝐴𝑔𝛼𝑠𝑑𝑙)𝐿𝑚8 𝑆𝑅𝐴𝑔 𝑑 𝐴𝑔 2⁄ 𝐴𝑔 2⁄ 𝐴𝑔 Transverse Stiffener Longitudinal Stiffener (TYP) Top Plate Bottom Plate Idealized Top Plate Idealized Bottom Plate 𝛼𝑠𝑑𝑙𝐴𝑔𝜔𝑅 = 𝜔𝑝 𝜔𝑠⁄𝐸𝐼𝑔 ᴇ 𝐿𝑚 𝐿𝑆𝑅 𝑆𝑅 𝐿𝑅 𝐸𝐼𝑔 ᴇ 𝐸𝐴𝑔 𝜔𝑝32101237550250255075Deflection Envelope (m) Bending Moment Envelope (MN·m) 𝜔𝑅 = 0.6𝜔𝑅 = 0.4𝜔𝑅 = 0.2𝑆𝑅 = 0.2𝐿𝑅 = 0.4𝛿𝑝𝑒𝑎𝑘2𝛿𝑝𝑒𝑎𝑘1≅(𝜔𝑅21 + 𝜔𝑅2)(𝜔𝑅11 + 𝜔𝑅1)𝛿𝑝𝑒𝑎𝑘2𝛿𝑝𝑒𝑎𝑘1≅(𝜔𝑅21 + 𝜔𝑅2) + (𝜔𝑅21 + 𝜔𝑅2 − 4𝐿𝑆𝑅2)𝐾𝑟(𝜔𝑅11 + 𝜔𝑅1) + (𝜔𝑅11 + 𝜔𝑅1 − 4𝐿𝑆𝑅2)𝐾𝑟𝐾𝑟𝐾𝑟𝐾𝑟 = (2𝐿𝑆𝑅)(ℎ𝑇𝑅 + 𝐿𝑆𝑅24ℎ𝑇𝑅2 + 1) 𝐿𝑚 𝐿𝑆𝑅 𝑆𝑅 𝐿𝑅 𝐸𝐼𝑔 ᴇ 𝐸𝐴𝑔 𝜔𝑝0.2 0.3 0.4 0.5 0.60.40.50.60.70.80.910.2 0.3 0.4 0.5 0.60.40.50.60.70.80.910.2 0.3 0.4 0.5 0.60.40.50.60.70.80.91𝜔𝑅 𝜔𝑅Peak Negative Deflections Peak Negative Bending Moments Peak Positive Bending Moments 154 Chapter 5 OPTIMUM PROPORTIONS The optimum proportions of cable-stayed and suspension bridges have long been established. The optimum proportions are commonly expressed in terms of a number of ratios which are frequently employed during conceptual design. The two most important design ratios for conventional cable bridges include the tower height-to-span ratio and the side-to-main span length ratio. This chapter examines the optimum values of these ratios with respect to self-anchored discontinuous hybrid cable bridges. Specific to discontinuous hybrid cable bridges, the optimum suspension ratio is also studied. The optimum design ratios depend upon the cable arrangement employed. Therefore, the first section in this chapter focuses on investigating the optimum cable arrangement. The other requisite for evaluating the optimum design ratios is the determination of expressions which can be used to estimate the volumetric quantities for major bridge components such as the cables, towers, and superstructure. These expressions are derived in the second section of this chapter and are later used to evaluate the optimum design ratios from a cost perspective. Notwithstanding, in the process, other factors relating to structural efficiency and aesthetics are also considered. In addition, the overall economic attributes of self-anchored discontinuous hybrid bridges are discussed. In a generalized study of this nature, a number of simplifications are necessary because different bridges are subject to unique loading, market, and site conditions. Therefore, the intent is not to produce precise optimum values for the design ratios; this can only be achieved through rigorous case specific optimization studies. Instead, the intent is to provide a range of optimal values for each design ratio to be used during conceptual design. It is then expected that engineering judgment be employed to determine the appropriate value for each design ratio given the specific nature of the bridge project; nevertheless, guidance is provided to aid designers in making that determination. 5.1 Cable Arrangement 155 5.1 Cable Arrangement An array of cables may be configured into numerous different longitudinal arrangements; however, from a practical standpoint the choices are limited as there are only a few options which are advantageous from both a form, and function perspective. This section discusses the positive and negative attributes of the more conventional stay cable and hanger arrangements which have used throughout history. 5.1.1 Stay Cables Traditionally, stay cables have been arranged in a harp, or a fan type of arrangement. Each is pictured in Figure 5.1 below. The selection of the longitudinal cable arrangement is a subject which has been extensively discussed in literature by a number of different authors (Podolny & Scalzi, 1976; Leonhardt & Zellner, 1980; Troitsky, 1988; Gimsing & Georgakis, 2012; Svensson, 2012). The following sections summarize the key aspects of arrangement selection. 5.1.1.1 Cost For stay cables symmetrically arranged about the centre-line of the towers, the following formulas obtained from Podolny & Scalzi (1976) provide an estimate of the cable steel quantity in a conventional cable-stayed bridge, (Harp Arrangement) (Fan Arrangement) Figure 5.1: Traditional Longitudinal Arrangements for Stay Cables 5.1 Cable Arrangement 156 ܳி ≅ߩ௦௧ߪ௦௧ ൫߱௦ + ߱൯ܮଶ 2 ൬ℎ்ܮ൰ +16 ൬ܮℎ்൰൨ (5.1) ܳு ≅ߩ௦௧ߪ௦௧ ൫߱௦ + ߱൯ܮଶ ൬ℎ்ܮ൰ +14 ൬ܮℎ்൰൨ (5.2) ߩ௦௧ and ߪ௦௧ are the density and design stress of the stay cable material; ߱ ௦ and ߱ are the magnitude of the uniformly distributed dead and live load; ܮ is the main span length; and ℎ் is the tower height above deck. A plot of these equations is provided in Figure 5.2 as a function of the tower height-to-span ratio (ℎ்: ܮ). Figure 5.2: Variation of Stay Cable Quantity in Harp and Fan Arrangements Since cost is directly related to the quantity of cable steel, the fan arrangement clearly yields the lowest cost for most practical cases; the harp arrangement being only advantageous from a cost perspective when the tower height-to-span ratio exceeds a value of approximately 0.3. Moreover, because Equation (5.1) and Equation (5.2) have the same coefficients the above result is independent of the specified loading, cable material, and span length. 5.1.1.2 Structural Efficiency The structural efficiency of a structure is often measured in terms of its strength-to-weight ratio or stiffness-to-weight ratio. In both respects the fan arrangement is more efficient. This is because in a harp arrangement, the bending stiffness of the towers and/or the superstructure needs to be activated in order for the bridge to remain stable when live loads are positioned asymmetrically with respect to the center-line of the towers. This is demonstrated in Figure 5.3 wherein hinges have been placed at all cable anchorage locations in order to inhibit the bending ability of the towers and superstructure. 0.00 0.10 0.20 0.30 0.40 0.500.01.02.03.04.0ܪܽݎܨܽ݊ ⁄ ൫࣓ + ࣓൯ 5.1 Cable Arrangement 157 From the figure it can clearly be observed that the harp arrangement is unstable under the applied loading scenario. Consequently, more material is required in the superstructure or towers to provide the necessary stability. For the same reason, unless additional anchor piers are provided in the side spans, the dead load distribution of moments in the superstructure cannot be as greatly optimized when utilizing the harp arrangement. In addition, there is less flexibility in configuring the side-to-main span ratio. Figure 5.3: Unstable Model of a Harp Arrangement Figure adapted from (Schüller, 1998) In contrast, in a fan arrangement there is a direct load path between the main span cables and the anchorage cables and thus stability can still be obtained even when the superstructure and towers are devoid of bending stiffness. This characteristic has led some authors to conclude that the fan arrangement is advantageous from not only a static, but also an aerodynamic perspective (Gimsing & Georgakis, 2012). However, there is a trade-off. Because the anchor cables stabilize unbalanced loading in the main span, and in the side span, they are subject to a greater stress range during service. Consequently, the side span length must be restricted, or alternatively, the area of the anchor cable must be increased to avoid any possibility of fatigue (more information is provided in Section 5.3.2). It should be noted though that the same fatigue concerns exist in a harp arrangement if used in conjunction with relatively stiff towers. 5.1.1.3 Aesthetics Aesthetics is one aspect where the harp arrangement triumphs. When multiple planes of cables are used and the bridge is viewed at a skewed angle, a fan arrangement gives rise to the optical effect of cables crossing each other which can be displeasing to the viewer depending on the angle of observation. However, the effect does become less pronounced with increasing span length. In contrast, this phenomenon does not occur when a harp arrangement is used because in a harp arrangement all cables have the same inclination. 5.1.1.4 Additional Considerations Due to the relatively low inclination of the stays in a harp arrangement, inclined cable planes are not possible as their use would interfere with vehicular clearance requirements. Consequently, a harp 5.1 Cable Arrangement 158 arrangement cannot be used in conjunction with A-frame, diamond shaped, modified-diamond shaped, or inverted Y-shaped towers whenever multiple cable planes are desired. In addition to affecting the tower layout, this also affects the foundation design and the amount of torsional stiffness which can be achieved by the cable system alone. In regard to construction, the first cable tower anchorage point in a harp arrangement is located much closer to the deck in comparison to in a fan arrangement. As a result, by using a harp arrangement, cantilever construction can theoretically commence at an earlier date and then subsequently proceed in unison with the construction of the towers. 5.1.1.5 Concluding Remarks Considering all of the above aspects, cost and structural efficiency are generally the most heavily weighted, and in regard to these aspects, the fan arrangement is undoubtedly superior. This finding is reflected in the current state of design: upon surveying the one hundred longest spanning cable-stayed bridges, less than five percent possess what can be classified as harp arrangements. On these grounds, the harp configuration will not be considered in deriving the optimum proportions for a self-anchored discontinuous hybrid cable bridge. Despite the advantages of the fan arrangement, in modern cable bridges fan arrangements are impractical because there is not enough anchorage space at each tower to allow for the axes of all adjoining cables to converge at a common point. This has led to the adoption of what is commonly referred to as the ‘semi-fan’ configuration which is picture in Figure 5.4. The only difference relative to the fan configuration is that the anchorage zone at the towers is extended downwards. Figure 5.4: The Semi-Fan Arrangement 5.1.2 Hangers In addition to the conventional vertical arrangement of hangers, a diagonal arrangement of hangers (Figure 5.5) has also been employed. Notwithstanding, the diagonal arrangement has only been applied in three vehicular bridges: the Severn Bridge (1966), the Bosporus Bridge (1973), and the Humber Bridge (1981). All three were designed by the same engineering firm, each serving as a model for the latter. (Kawada, 2010) 5.1 Cable Arrangement 159 The primary advantage of diagonal hangers is that they form a truss-like structure between the suspension cables and the superstructure which significantly suppresses strain-free deformations. Consequently, the stiffness of the cable system, and thus the bridge as a whole, is greatly increased. Nevertheless, diagonal hangers were first employed in the Severn Bridge for a different reason, as explained below. The suppression of strain-free deformations from live, wind, and other forms of loading causes a cyclic variation of forces in the hangers. Prior to the Severn Bridge, heavy space trusses were used exclusively in suspension bridges due to ongoing concerns regarding aerodynamic stability following the Tacoma Narrows Disaster in 1940. The Severn Bridge was the first bridge to use to a lightweight, streamlined box girder for the superstructure. Because of its reduced gravity stiffness, attention was directed towards enhancing structural damping. (Kawada, 2010) This was achieved by using diagonal hangers in conjunction with helical cables. The helical cables have a unique hysteresis which is activated by the cyclic variation of wind forces in the hangers. Unfortunately, it was not until near the opening of the Humber Bridge, that severe structural problems started to emerge with the hangers in the Severn Bridge. There were many contributing factors. Poor penetration of the hanger socketing material was observed, in addition to a lack of axial and angular alignment along the hanger pin centre axes (Flint & Smith, 1992). It also became evident that live load demands had reached levels close to three times the original design estimate (Bradley, 2010). Nevertheless, ironically, the foremost cause of the problems can be linked to the rationale behind the employment of the diagonal arrangement. Although advantageous from a structural damping perspective, the cyclic variation of forces in the hangers from live load severely reduced their fatigue life. Moreover, the slackening of the hangers made them particularly susceptible to wind-induced vibration which further exacerbated their fatigue. Prior to the Severn Bridge, wind-(Vertical Arrangement) (Diagonal Arrangement) Figure 5.5: Types of Longitudinal Arrangements for Hangers 5.2 Derivation of Bridge Quantities 160 induced vibration of hangers had never been observed. (Kawada, 2010) When studies were undertaken to replace the hanger system, it was found that by using a vertical arrangement the combined bending and axial stress range in the hangers could be reduced by roughly 50% at mid-span, and 85% at the quarter span. However, replacing the original hangers with a vertical arrangement would have required repositioning of the hanger clamps or, alternatively, the installation of new deck attachment stools. Furthermore, the deck and tower would have had to be substantially strengthened. For these reasons, the diagonal arrangement was retained. (Flint & Smith, 1992) Notwithstanding, following the Humber Bridge, the diagonal arrangement was completely abandoned in vehicular bridges. The reasons that led to the abandonment of the diagonal arrangement in suspension bridges are no less valid in regard to hybrid cable bridges. Consequently, the diagonal arrangement will not be considered when deriving the optimum proportions for a self-anchored discontinuous hybrid cable bridge. 5.2 Derivation of Bridge Quantities As previously mentioned, the accurate calculation of quantities required in principal bridge components is an exceedingly complex task. Numerous assumptions are required in order to generalize and simplify the calculations. In addition, extreme caution must be exercised in differentiating parameters which are arbitrary from those which have a prominent effect. Bearing this in mind, the estimation approach utilized in Gimsing & Georgakis (2012) for conventional cable bridges is adapted herein for the hybrid system. Inherent in the adopted approach are several general assumptions: 1. The governing loading scenarios for the principal bridge components are assumed to occur when either the entire bridge, or the entire main span, is loaded with dead and live load. Under these loading scenarios the majority of the applied load is transferred through the cable system and, on that account, the bending stiffness of the superstructure can be neglected. This assumption is particularly valid in modern cable bridges where the superstructure is often of slender construction. It allows principal bridge components to be considered isolated from the rest of the bridge when deriving their internal forces which greatly simplifies the calculations involved. 2. Secondary forms of loading (i.e. wind, temperature, earthquake, etc…) are not directly 5.2 Derivation of Bridge Quantities 161 considered as their effects are site specific. Still, the impacts of secondary loads are indirectly considered in the assigned design stress of the principal components. 3. The quantity of cable steel in a single array of cables is approximated by assuming that the cables act as a continuous membrane. This assumption derives from the fact that in modern cable bridges the spacing of cables along the superstructure is relatively small in comparison to the span length of the bridge. 4. Out-of-plane effects are not directly considered in the derivation of quantities. Accordingly, only two dimensions are considered. In addition, since the overall goal is to determine the optimum proportions of the hybrid system, it is appropriate to neglect the difference between the tower height above deck and the global sag of the suspension cable. A large difference between these two parameters only serves to reduce the efficiency of the system. Furthermore, to simplify some of the expressions, it is sufficiently accurate in this context to assume that the self-weight of the suspension cable acts uniformly along its projected length. 5.2.1 Stay Cable Quantity When determining the area required for each stay cable it is appropriate to assume that each cable is effectively anchored at its respective tower connection point since the governing loading scenario producing the maximum tension will occur when live load is balanced on both sides of the tower. In this regard, Figure 5.6 shows an idealized array of stay cables. ℎ் is the height of the towers above deck; ܾ is the height above deck of the first tower connection; and ܽ is the length of the array. Figure 5.6: Idealized Array of Stay Cables ܽℎ்ܾ ൫߱௦ + ߱൯ݔC TowerL ݀ݔ(ℎ் − ܾ)ܽ ݔ 5.2 Derivation of Bridge Quantities 162 The shaded area in Figure 5.6 is meant to represent an infinitesimal segment of the array. The tension in the infinitesimal segment due to uniformly distributed dead and live load (߱௦ + ߱) acting over a tributary length of ݀ݔ, is given by the following expression: ݀ܶ = ൫߱௦ + ߱൯ටቂቀℎ் − ܾܽ ቁ ݔ + ܾቃଶ+ ݔଶቂቀℎ் − ܾܽ ቁ ݔ + ܾቃ݀ݔ + 12݀ܶߪ௦௧ ߛ௦௧ඨ൬ℎ் − ܾܽ ൰ݔ + ܾ൨ଶ+ ݔଶ (5.3) where ߪ௦௧ and ߛ௦௧ are the design stress and unit weight of the stay cable material. The first term in Equation (5.3) is due to the applied loading, whereas the second term is due to self-weight. If ݀ܶ is isolated, the following equation is obtained, ݀ܶ =൫߱௦ + ߱൯ටቂቀℎ் − ܾܽ ቁ ݔ + ܾቃଶ+ ݔଶቂቀℎ் − ܾܽ ቁ ݔ + ܾቃ ቈ1 −12ߛ௦௧ߪ௦௧ ටቂቀℎ் − ܾܽ ቁ ݔ + ܾቃଶ+ ݔଶ݀ݔ (5.4) The quantity of cable steel in the infinitesimal segment, ݀ܳ, will then be given by the density of the stay cable material, ߩ௦௧, multiplied by the cable area required, multiplied by the length of the segment: ݀ܳ = ߩ௦௧݀ܶߪ௦௧ඨ൬ℎ் − ܾܽ ൰ݔ + ܾ൨ଶ+ ݔଶ (5.5) When Equation (5.4) is substituted in for ݀ܶ, the quantity becomes, ݀ܳ = ൫߱௦ + ߱൯ߩ௦௧ߪ௦௧ቂቀℎ் − ܾܽ ቁ ݔ + ܾቃଶ+ ݔଶቂቀℎ் − ܾܽ ቁ ݔ + ܾቃ ቈ1 −12ߛ௦௧ߪ௦௧ ටቂቀℎ் − ܾܽ ቁ ݔ + ܾቃଶ+ ݔଶ݀ݔ (5.6) Thereupon, the total quantity of cable steel in the array can be obtained by integrating ݀ܳ over the length of the array, ܳ = ൫߱௦ + ߱൯ߩ௦௧ߪ௦௧ නቂቀℎ் − ܾܽ ቁ ݔ + ܾቃଶ+ ݔଶቂቀℎ் − ܾܽ ቁ ݔ + ܾቃ ቈ1 −12ߛ௦௧ߪ௦௧ ටቂቀℎ் − ܾܽ ቁ ݔ + ܾቃଶ+ ݔଶ݀ݔ (5.7) 5.2 Derivation of Bridge Quantities 163 Figure 5.7 plots the quantity of cable steel in a single array of cables for a fan arrangement (ܾ = ℎ்). The quantity for a semi-fan arrangement is also included assuming that the tower anchorages for the cables are distributed over the top quarter of the tower (ܾ = 3/4ℎ்). In accordance with the remarks in Section 5.1.1, the harp arrangement is not considered. As can be seen from the Figure 5.7, there is little difference in quantity between the two, particularly in the range of tower height-to-span ratios commonly employed for cable-stayed bridges. Furthermore, for conventional cable types, the plots do not greatly depend on the unit weight and design stress of the cable material. Figure 5.7: Variation of Stay Cable Quantity in Fan and Semi-Fan Arrangements Parameters: ܽ = 500m, ߪ௪ = 800MPa, ߛ௦௧௬ = 0.09MN/m3 Since the quantity for a fan arrangement is mathematically more convenient, its formulation will be used in place of the formulation for a semi-fan arrangement. Still, even for the fan arrangement the symbolical evaluation of Equation (5.7) produces a highly complex expression. However, a simple approximate expression can be obtained by neglecting the contribution from the self-weight of the cables. This is equivalent to setting ߛ௦௧௬ equal to zero, at which point, the expression becomes, ܳ௫ =ߩ௦௧ߪ௦௧ ܽଶ൫߱௦ + ߱൯ ൬ܽ3ℎ்൰ ቈ1 + 3 ൬ℎ்ܽ ൰ଶ (5.8) Note that if ܽ is replaced by ܮ/2 and the whole expression is multiplied by 4, then Equation (5.1) is obtained. Also, the error produced by neglecting the self-weight of the cable is plotted in Figure 5.8 for various lengths of cable arrays. Even for long spanning arrays, it can be seen that the error is marginal. 0.00 0.10 0.20 0.30 0.40 0.500.01.02.03.04.0 ܵ݁݉݅ − ܨܽ݊ܨܽ݊ ()⁄ ൫࣓ + ࣓൯ 5.2 Derivation of Bridge Quantities 164 Figure 5.8: The Influence of Self-Weight on Cable Steel Quantity in a Fan Arrangement Parameters: ߪ௦௧ = 800MPa, ߛ௦௧ = 0.09MN/m3 The final step to arrive at an expression for the total quantity of stay cable steel in a discontinuous hybrid cable bridge involves summing the contributions from the stay cable arrays in the main span and side spans. Denoting ܮோ as the suspension ratio, ℎ்ோ as the tower height-to-main span ratio, ܮௌோ as the side-to-main span length ratio, and ܮ as the main span length, the final expression may be written as, ܳ௦௧ =ߩ௦௧ߪ௦௧ ܮଶ൫߱௦ + ߱൯ ቈ(1 − ܮோ)ଷ12ℎ்ோ + (1 − ܮோ)ℎ்ோ +23ܮௌோଷℎ்ோ + 2ℎ்ோܮௌோ (5.9)5.2.2 Suspension Cable Quantity In deriving the suspension cable quantity, it is assumed that the cable area required for the suspension cable is independent to the cable area required for the anchor cables. Accordingly, an expression for the area of the suspension cable has already been derived. From Equation (4.21), the required area is, ܣ =1ߛ൫߱௦ + ߱൯ ߦ ܮோ (2 − ܮோ)ඨ1 + 16 ቀ ℎ்ோ2 − ܮோቁଶ1 − ߦ ඨ1 + 16 ቀ ℎ்ோ2 − ܮோቁଶ (5.10) where as before, ߦ = ߛ ܮ8 ߪℎ்ோ Using Equation (5.10), the quantity of cable steel can then be expressed as, 0.00 0.10 0.20 0.30 0.40 0.500.960.970.980.991.00 ()⁄ ܽ = 500݉ ܽ = 400݉ ܽ = 300݉ ܽ = 200݉ ܽ = 100݉ 5.2 Derivation of Bridge Quantities 165 ܳ = ߩܣൣ2ܮ_ଵ + ܮ_ଶ൧ (5.11) where ߩ and ܣ are the density and area of the cable material. In addition, ܮ_ଵ is the length of the cable in the stayed region (the value is multiplied by two because there are two stayed regions) and ܮ_ଶ is the length of the cable in the suspended region. There will be a negligible difference in the total cable quantity if the elastic elongation of the cable is ignored. In view of that, the stressed length can be used in place of the unstressed length when computing ܮ_ଵ and ܮ_ଶ. The expression for the cable quantity then becomes, ܳ = ߩܣ 2න ඨ1 + ൬݀ݕ(ݔ)݀ݔ ൰ଶ݀ݔ + න ඨ1 + ൬݀ݕ(ݔ)݀ݔ ൰ଶ݀ݔ(ଵାೃ)ଶ(ଵିೃ)ଶ(ଵିೃ)ଶ (5.12) The ordinates of the cable curve, represented by ݕ(ݔ), can be approximated by Equation (4.11); however, additional simplifications must still be made in order to arrive at a straightforward solution. For the stayed region, it can be assumed that the sag of the cable in the stayed region is small relative to the length of the stayed region so that the length can be approximated as, ܮ_ଵ_௫ = ඨቆܮ(1 − ܮோ)2 ቇଶ+ ݕଶ (5.13) where ݕ is the vertical distance from the tower anchorage point to the start of the suspended region (Figure 4.3). If it is further assumed that the weight of the cable has a negligible effect on ݕ, then after substitution of Equations (4.9) and (4.11), Equation (5.13) reduces to, ܮ_ଵ_௫ =ܮ(1 − ܮோ)2 ඨ1 + ℎ்ோଶ ൬ 42 − ܮோ൰ଶ (5.14) For the length of the cable in the suspended region, a Maclaurin series expansion can be used to eliminate the radical in the expression for the stressed length. The integral then becomes, ܮ_ଶ_௫ = න ቈ1 +12 ൬݀ݕ(ݔ)݀ݔ ൰ଶ ݀ݔ(ଵାೃ)ଶ(ଵିೃ)ଶ (5.15)Thereafter, neglecting the effect of the weight of the cable and substituting in Equation (4.11) yields, 5.2 Derivation of Bridge Quantities 166 ܮ_ଶ_௫ = ܮܮோ ቈ1 +83 ℎ்ோଶ ൬ 12 − ܮோ൰ଶ (5.16) The final expression for the approximate quantity can then be obtained by substituting Equations (5.14) and (5.16) into Equation (5.11), ܳ௫ = ߩܣܮ ൦(1 − ܮோ)ඨ1 + ൬4ℎ்ோ2 − ܮோ൰ଶ+ ܮோ ቈ1 +83 ൬ℎ்ோ2 − ܮோ൰ଶ൪ (5.17) To provide an indication of the error in the approximate expression for the stressed length, Figure 5.9 compares the approximate quantity obtained from Equation (5.17) to the quantity obtained from Equation (5.12). Figure 5.9: Error in Approximate Suspension Cable Quantity Parameters: ܮ = 1000m, ߪ = 800MPa, ߛ = 0.09MN/m3 Clearly, as the suspension ratio increases, the self-weight of the cable becomes more dominant and, consequently, the error increases. Nevertheless, for all practical cases Equation (5.17) provides sufficient accuracy for the purposes of this study. Therefore, upon substitution of the cable area, the approximate formula for the suspension cable can be given as, ܳ =1݃ ܮ൫߱௦ + ߱൯ܮோ(2 − ܮோ)ߦߟ1 − ߦߟ ቈ(1 − ܮோ)ߟ + ܮோ(ߟଶ + 5)6 (5.18) where ݃ is the standard acceleration due to gravity, and the additional dimensionless parameter, ߟ, is defined as, 0.00 0.10 0.20 0.30 0.40 0.500.980.991.001.011.02 ܮோ = 0.2 ܮோ = 0.4 ܮோ = 0.65.2 Derivation of Bridge Quantities 167 ߟ = ඨ1 + 16 ൬ ℎ்ோ2 − ܮோ൰ଶ 5.2.3 Anchor Cable Quantity Anchor cables serve to balance loads positioned asymmetric to the centerline of the towers (refer to Section 4.2.1.5). The maximum force in the anchor cables will, therefore, occur when only the main span is loaded with live load. In accordance with Section 5.1.1, when deriving the required area for the anchor cables it will be assumed that the stay cables are arranged in a semi-fan configuration. Furthermore, it will also be assumed that the anchorage zone at the towers is relatively small and any unbalanced loading taken by the stay cables transfers directly to the anchor cables. These assumptions are reflected in the free body diagram shown in Figure 5.10. Figure 5.10: Idealized Free-Body Diagram for Maximum Anchor Pier Reaction In Figure 5.10, it is important to note that the superstructure has been ‘cut’ at the end of the stayed region, and the suspension cable has been ‘cut’ at the tower. Furthermore, for simplicity, for each array of stay cables, the centre of gravity for the overall stay cable weight is assumed to be consistent with that of a pure triangle. This is not entirely accurate because in a semi-fan arrangement the weight of the stay cables is not uniformly distributed throughout the array. Nevertheless, in practical cases the distance from the tower to the centre of gravity of a stay cable array will vary from 0.28 to 0.35 times the length of the array (Gimsing & Georgakis, 2012). It is, therefore, sufficient to fix the ܪ/ ܸ = Horizontal/Vertical reaction from suspension cable in main span ܳ = Suspension cable quantity ܳ௦ = Anchor cable quantity ܳ௦௧ = Main span stay cable quantity ܳ௦௧௦ = Side span stay cable quantity ݃= Standard acceleration due to gravity ℎ் = Tower height ܴ= Reaction at anchor pier ߱௦/ ߱ = Dead/Live loading ܮோ = Suspension ratio ܮ = Main span length ܮ௦ = Side span lengthܮ(1 − ܮோ) 6⁄ܮ(1 − ܮோ) 2⁄ܮ௦ ܮ௦ 3⁄ ℎ்ܪ ܳ௦݃4 + ܸ ߱௦ܴ߱ ܳ௦௧݃ 2⁄ܳ௦௧௦݃ 2⁄ ܳ௦݃4 Main SpanSide Span C Tower L ܪ5.2 Derivation of Bridge Quantities 168 distance at 1/3 times the length of the array, particularly given that the weight of the stay cables is minor in comparison to the weight of the applied loading. In accordance with the above assumptions, the reaction at the anchor pier can be obtained by taking moments about the base of the tower, ܴ =൫߱௦ + ߱൯ܮ(1 − ܮோ)ଶ8ܮௌோ + ܪℎ்ோܮௌோ +ܳ௦௧݃12ܮௌோ (1 − ܮோ) −ܳ௦௧௦݃6− 12߱௦ܮܮௌோ −ܳ௦݃4 (5.19) Vertical equilibrium at the anchor pier then gives the vertical component of the anchor cable chord tension, ܸ௦ =൫߱௦ + ߱൯ܮ(1 − ܮோ)ଶ8ܮௌோ + ܪℎ்ோܮௌோ +ܳ௦௧݃12ܮௌோ (1 − ܮோ) −ܳ௦௧௦݃6 −12߱௦ܮܮௌோ (5.20) Equation (4.9) can be substituted in for ܪ (during the substitution ߱௦ must be replaced by ߱௦ + ߱ to account for the applied live loading), and an expression for the tensile chord force in the anchor cable can then be found by combining the vertical and horizontal components of the tensile chord force, ܶ௦ =124ܮௌோ ൣ3ܮ൫߱௦ + ߱൯ + 3ܳ݃ + 2ܳ௦௧݃(1 − ܮோ) − 4ܳ௦௧௦݃(ܮௌோ)− 12ܮܮௌோଶ߱௦൧ටܮௌோଶ + ℎ்ோଶℎ்ோ (5.21) It then follows that the total quantity of cable steel in both anchor cables is, ܳ௦ = 2ߩ௦ ܶ௦ߪ௦ ܮටܮௌோଶ + ℎ்ோଶ (5.22) where ߩ௦ and ߪ௦ are the density and design stress of the anchor cable material. When expanded, Equation (5.22) becomes, ܳ௦ =112ߩ௦ߪ௦ ܮൣ3ܮ൫߱௦ + ߱൯ + 3ܳ݃ + 2ܳ௦௧݃(1 − ܮோ) − 4ܳ௦௧௦݃(ܮௌோ)− 12ܮܮௌோଶ߱௦൧ ൬ܮௌோℎ்ோ +ℎ்ோܮௌோ൰ (5.23)5.2 Derivation of Bridge Quantities 169 where from Equation (5.9), ܳ௦௧ =ߩ௦௧ߪ௦௧ ܮଶ൫߱௦ + ߱൯(1 − ܮோ)ℎ்ோ ቈ(1 − ܮோ)ଶ12ℎ்ோଶ+ 1 (5.24)and, ܳ௦௧௦ =ߩ௦௧ߪ௦௧ ܮଶ൫߱௦ + ߱൯ℎ்ோܮௌோ ቈ23 ൬ܮௌோℎ்ோ൰ଶ+ 2 (5.25) Note from Equation (5.23) that the contribution from ܳ ௦௧ and ܳ ௦௧௦ amount to zero when the stay cables are symmetrically arranged about the centre of the towers (i.e. when ܮௌோ = (1 − ܮோ)/2). Also, their contribution will be negligible when the length of the side span is less than the length of the stay cable array in the main span. This has mainly to do with the side span length. When the side span length is short, the anchorage force markedly increases due to the shortened lever arm. Consequently, the contribution from the stay cable weight becomes of little importance. As a final note, it is important to acknowledge that, for simplicity, the anchor cable quantity was derived based on the chord force in the anchor cable. The maximum force in the anchor cable, which occurs near the tower, is somewhat larger. In this respect, Equation (5.23) underestimates the quantity required. Nevertheless, for efficiently designed anchor cables, the difference between the chord force and the maximum cable force is minor (Podolny & Scalzi, 1976). 5.2.4 Hanger Quantity Figure 5.11 shows a reference diagram for the derivation of the hanger quantity. Similar to when estimating the quantity for the stay cables, the hanger area required for self-weight can be neglected. The error introduced as a result will be even less than in an array of stay cables due to the relatively shorter length and vertical inclination of the hangers. Figure 5.11: Diagram for Hanger Steel Quantity ൫߱௦ + ߱൯ݕ ݂ܮ(1 − ܮோ) 2⁄ܮ(1 − ܮோ) 2⁄ ܮܮݕ(ݔ) ݔ 5.2 Derivation of Bridge Quantities 170 In contrast to the derivation of the stay cable quantity, rather than integrating the quantity of cable steel in an infinitesimal segment of the hanger array, the solution can be obtained using a simpler approach. The total quantity in the hanger array can be obtained by multiplying together the density of the hanger material, the total hanger area required to carry the applied loads, and the average length of the hangers, ܳ = ߩ൫ܣ_௧௧൯൫ܮ_௩൯ (5.26) When expanded, the expression becomes, ܳ = ߩ൫߱௦ + ߱൯ܮߪ ൬ℎ் − ݕ3 ൰ (5.27) where ߪ is the design stress of the hangers. Substituting in Equations (4.9) and (4.11) then gives, ܳ = 13ߩߪ ܮଶ൫߱௦ + ߱൯ℎ்ோܮோଷ ൦߱௦ + ቀܳ݃ܮ ቁ߱௦ܮோ(2 − ܮோ) + ቀܳ݃ܮ ቁ൪ (5.28) However, a simpler expression can be obtained by neglecting the contribution from the self-weight of the suspension cable, ܳ௫ =13ߩߪ ܮଶ൫߱௦ + ߱൯ℎ்ோܮோଶ ൬12 − ܮோ൰ (5.29) Figure 5.12 shows a comparison of the quantities obtained from Equations (5.28) and (5.29) for an extreme main span length of 1000 metres. Although the error can be exorbitant when the tower height-to-span ratio is small it is important to recognize that, in those instances, the hanger quantity contributes very little to the overall cable steel quantity. This is demonstrated in Figure 5.13 where the hanger quantity is defined by ܳ (the design stress and unit weight of each cable type is set to 800MPa and 0.09MN/m3, respectively). Accordingly, the error introduced as a result of the use of Equation (5.29) will ultimately be insignificant. 5.2 Derivation of Bridge Quantities 171 Figure 5.12: Error in Approximate Hanger Cable Quantity Parameters: ܮ = 1000m, ߪ = 800MPa, ߛ = 0.09MN/m3, ߱ோ = 0.6 Figure 5.13: Hanger Cable Quantity in Relation to Total Cable Steel Quantity Parameters: ܮ = 1000m, ߱ோ = 0.6 5.2.5 Tower Quantity The derivation of the tower quantity is particularly challenging. The required quantity strongly depends on both in-plane, and out-of-plane loading. For the in-plane loading, the towers must be capable of sustaining considerable axial and bending demands; each governed by different loading scenarios. Furthermore, the magnitude of the bending demands will depend on the geometry of the tower section as well as the articulation scheme for the superstructure; two parameters which are very difficult to generalize. Likewise, for out-of-plane loading, bending demands are no less difficult to quantify in general terms. 0.00 0.10 0.20 0.30 0.40 0.501.001.051.101.150.00 0.10 0.20 0.30 0.40 0.500.0000.0200.040 ܮோ = 0.2 ܮோ = 0.4 ܮோ = 0.6 + + + ܮோ = 0.2 ܮோ = 0.4 ܮோ = 0.6 5.2 Derivation of Bridge Quantities 172 To facilitate the computations involved, Gimsing & Georgakis (2012) stipulate that the tower quantity should be based on the cross-sectional area required to support the maximum possible vertical load acting on the tower. This can be justified for in-plane forces because the governing loading scenario for the longitudinal bending demands produces comparatively less axial force in the tower. Therefore, sizing the towers based on the maximum possible vertical load ensures that there is some measure of reserved strength to handle the longitudinal bending demands. To account for coincidental out-of-plane loading, the design stress of the tower is reduced in proportion to the ratio of the out-of-plane bending and the in-plane axial demands. Thus, the design stress is in essence considered variable along the height of the towers. Nonetheless, it is sufficient in this context to assign an average value for the design stress. For efficiently designed towers, Gimsing & Georgakis (2012) cite that a reduction in design stress of anywhere from 20% to 40% is appropriate. Accordingly, a reduction of 30% will be assumed herein. Although reducing the design stress of the tower by a fixed percentage to account for out-of-plane loading may appear crude, it must be remembered that a precise estimate of the tower quantity is not the primary concern. Rather, the optimum proportions are ultimately influenced by the rate of change in the tower quantity. On that account, the use of a simplistic approach is justified. Nevertheless, the impact of varying the design stress of the tower / the reduction coefficient will be examined in Section 5.3.1. Conveniently, the free body diagram given in Figure 5.10 can be re-purposed to derive the maximum possible vertical load on the towers. The only change that needs to be considered is that in this case, the governing load scenario occurs when live load covers the entire bridge. Bearing this in mind, taking moments about the anchor piers results in the following expression for the axial force acting on the tower, ௩ܰ =ܳ௦௧݃2 ൬1 +1 − ܮோ6ܮௌோ ൰ +ܳ௦௧௦݃3 + ܸ + ܪ ൬ℎ்ோܮௌோ൰ + ൫߱௦ + ߱൯ܮܮௌோ2+ ൫߱௦ + ߱൯(1 − ܮோ)ܮ2 ൬1 +1 − ܮோ4ܮௌோ ൰ +ܳ௦݃4 (5.30) Substituting in the horizontal and vertical components of the suspension cable force then gives, ௩ܰ= 124ܮௌோ ൣ3ܮ൫߱௦ + ߱൯(2ܮௌோ + 1)ଶ + 2ܳ௦௧݃(6ܮௌோ − ܮோ + 1) + 8ܳ௦௧௦݃(ܮௌோ)+ 3ܳ݃(4ܮௌோ + 1) + 6ܳ௦݃(ܮௌோ)൧ (5.31)However, to obtain the total vertical force acting on each tower, the vertical force from the cable 5.2 Derivation of Bridge Quantities 173 system must be added to the vertical force from self-weight, ்ܸ = ௩ܰ + ௧ܹ(ݖ) (5.32) The self-weight of the tower varies along its height and, consequently, the self-weight is a function of the distance from the tower top, denoted by the letter, ݖ. The area required in each tower can be obtained as, ܣ௧ = ௩ܰ+ ௧ܹ(ݖ)ߚ௧ߪ௧ (5.33) where ߪ௧ is the design stress of the tower and ߚ௧ is the reduction coefficient to account for the effect of out-of-plane loading (ߚ௧ = 0.7). Notwithstanding, the self-weight of the tower is also a function of the tower area, ௧ܹ(ݖ) = ܣ௧ߛ௧ݖ (5.34) where ߛ௧ is the unit weight of the tower material. Substituting Equation (5.34) into Equation (5.33) results in, ܣ௧ = ௩ܰߚ௧ߪ௧ − ߛ௧ݖ (5.35) Accordingly, the total quantity of material required in both towers can be calculated as, ܳ௧ = 2ߩ௧ ௩ܰ න ൬1ߚ௧ߪ௧ − ߛ௧ݖ൰ାಳ݀ݖ (5.36)which then simplifies to, ܳ௧ =2 ௩ܰ݃ ݈݊ ൦11 − (ℎ் + ℎ)ߛ௧ߚ௧ߪ௧൪ (5.37) The only parameter left to assign is ℎ which represents the height of the towers below deck. This is again a difficult parameter to generalize. Typically, it is desirable that the tower height below deck be made as short as possible in order to minimize costs. Nonetheless, the majority of cable bridges are constructed over large waterways and a clearance envelope is often required at mid-span so that vessel navigation is not curtailed. Therefore, assuming the following: The base of the towers is at water level; The vertical profile of the roadway is parabolic; 5.2 Derivation of Bridge Quantities 174 The vertical clearance required at mid-span is 50 metres; and The vertical slope of the roadway cannot exceed 5%. Then, the minimum tower height below deck can be calculated as, ℎ = 50 ൬1 −ܮ4000൰ [݅݊ ݉݁ݐݎ݁ݏ] (5.38)5.2.6 Superstructure Quantity The superstructure quantity is defined as the quantity of material required to support the roadway. Using the assumed weight of the superstructure, ߱௦, an expression for the superstructure quantity can be readily obtained, ܳ௦ =߱௦ܮ݃ߙ௦ௗ (1 + 2ܮௌோ) (5.39) However, ߱௦ has thus far been used to represent the entire dead load acting on the superstructure which consists partly of superimposed dead load. The contribution from superimposed dead load must be removed and, therefore, an additional reduction coefficient, ߙ௦ௗ ≥ 1.0, has been included for this purpose. 5.2.7 Load Correction All of the expressions derived in previous sections depend on two main loading parameters: the dead load of the superstructure (߱௦) and the magnitude of the live load (߱). In contrast to the magnitude of the live load which can be considered constant, the superstructure dead load will be affected by the values of the other parameters. It is, therefore, necessary to develop an expression that accounts for the variation in the superstructure dead load. If ߱௦ is used to represent the superstructure dead load of a bridge with known parameters (i.e. a reference bridge), then the variation in the dead load of the superstructure for a bridge with different parameters can be expressed as, Δ߱௦ = ߛ௦[(ܣ௦ − ܣ௦) + (ܣ௦ − ܣ௦) + (ܣ௦ − ܣ௦)] (5.40) where ߛ௦ is the unit weight of the superstructure material. In addition, ܣ௦/ܣ௦ is the area required for miscellaneous transverse support members (i.e. floor beams, diaphragms, etc.), ܣ௦/ܣ௦ is the area required for longitudinal axial demands, and ܣ௦/ܣ௦ is the area required for longitudinal bending demands. Accordingly, when the longitudinal bridge proportions are varied (ܣ௦ − ܣ௦) 5.2 Derivation of Bridge Quantities 175 can be assumed equal to zero, upon which, Δ߱௦ = ߛ௦[(ܣ௦ − ܣ௦) + (ܣ௦ − ܣ௦)] (5.41) The area required for longitudinal axial demands can be derived by integrating the axial demands in the superstructure. The governing loading scenario and the corresponding axial forces are shown in Figure 5.14. Figure 5.14: Longitudinal Axial Demands in Superstructure A free body diagram of the pertinent forces has already been depicted in Figure 5.10. From Figure 5.10, it is clear that the superstructure axial force components ܨ௦ଵ and ܨଵ are, together, equal to the horizontal force in the suspension cable. Thus, ܨ௦ଵ + ܨଵ = ܪ (5.42) Substituting in the appropriate expression for ܪ then yields, ܨ௦ଵ + ܨଵ =18ℎ்ோ ൣܳ݃ + ൫߱௦ + ߱൯ܮோ(2 − ܮோ)൧ (5.43) The uniform axial force components in the side span (ܨ௦ଶ and ܨଶ) are a result of the anchorage force required to equilibrate the unbalanced portion of the dead and live load in the stayed regions of ߱߱௦ ܨ௦ଷܨଷܨଶ ܨଵ ܨ௦ଵ ܨ௦ଶ ܨ௦ଶ Dead Load Axial ForcesLive Load Axial Forces5.2 Derivation of Bridge Quantities 176 the bridge. These force components will, therefore, be equal to the horizontal component of the anchorage force or, written another way, ܨ௦ଶ + ܨଶ = ܸ௦തതതതܮௌோℎ்ோ (5.44) where ܸ௦തതതത is the vertical component of the anchorage force. ܸ௦തതതത can be obtained from Equation (5.20); however, it is important to exclude the horizontal component of the suspension cable force since its contribution has already been accounted for in Equation (5.43). Accordingly, ܨ௦ଶ + ܨଶ =124ℎ்ோ ൣ3൫߱௦ + ߱൯ܮ(1 − ܮோ)ଶ + 2ܳ௦௧݃(1 − ܮோ) − 4ܳ௦௧௦݃ܮௌோ− 12߱௦ܮܮௌோଶ൧ (5.45) The portion of the side span dead load which is balanced produces a variable axial force in the side span which is represented by ܨ௦ଶ. The magnitude of the axial force can be found by integrating the axial force in a small segment of the side span stay cable array, ܨ௦ଶ = න߱௦ݔℎ் ݀ݔ௫= 12߱௦ݔଶℎ் (5.46) where the origin for ݔ is situated at the anchor pier. Similarly, the axial force in the main span stayed regions of the bridge due to dead and live load can be expressed as, ܨ௦ଷ + ܨଷ = න൫߱௦ + ߱൯ݔℎ் ݀ݔ௫= 12൫߱௦ + ߱൯ݔଶℎ் (5.47) where in this case, the origin for ݔ is situated at the stay cable-hanger junction. Considering each of the axial force components in Figure 5.14, the volume of material required for the axial demands is, ܸܱܮ௦ =1ߪ௦ ൫ܨ௦ଵ + ܨଵ൯ܮ(1 + 2ܮௌோ) + 2൫ܨ௦ଶ + ܨଶ൯(ܮௌோܮ)+ 2න (ܨ௦ଶ)݀ݔೄೃ+ 2න ൫ܨ௦ଷ + ܨଷ൯݀ݔ(ଵିೃ)ଶ (5.48) which simplifies to, 5.2 Derivation of Bridge Quantities 177 ܸܱܮ௦ =ܮ24ℎ்ோߪ௦ ቂܮ݉൫߱ݏ + ߱൯൫1 − ܮܴ3 + 3ܮܴ + 6ܮܴܵ൯ − 16ܮ݉ܮܴܵ3߱ݏ+ 4ܳݏݐ݉݃ܮܴܵ(1 − ܮܴ) − 8ܳݏݐݏ݃(ܮܴܵ)2 + 3ܳܿ݉݃(2ܮܴܵ + 1)ቃ (5.49) Clearly, the cross sectional area required for the longitudinal axial demands varies along the length of the bridge; however, in this context it will be sufficiently accurate to use the average area required, which can be readily obtained from Equation (5.49), ܣ௦ =ܸܱܮ௦ܮ(1 + 2ܮௌோ) (5.50) In contrast to the longitudinal axial demands, the change in the longitudinal bending demands is difficult to estimate without employing sophisticated analyses. In addition, the area required for the longitudinal bending demands depends on the depth of the superstructure which is a difficult parameter to generalize. Therefore, for the time being it will be assumed that the change in longitudinal bending demands has zero effect on the superstructure quantity. The validity of this assumption will be revisited later in the chapter. Accordingly, Equation (5.41) becomes, Δ߱௦ = ߛݏ(ܣܽݏ − ܣݏܽ) (5.51) and the dead load of the superstructure can be written as, ߱௦ = ߱௦ + ߛݏ(ܣܽݏ − ܣݏܽ) (5.52) However, because ߱ ௦ depends on ܣ௦, which is also a function of ߱ ௦, an unavoidable consequence of this approach is that iteration is required to determine the various bridge component quantities. Also, because the superstructure quantity directly depends on ߱௦, it is important to make one more modification. Specifically, it is important that the magnitude of the superimposed dead load be made constant and independent of ߱ ௦. This can be achieved by linking the superimposed dead load to the superstructure dead load of the reference bridge. On that account, Equation (5.39) becomes, ܳ௦ =ܮ(1 + 2ܮௌோ)݃ [߱௦ − (ߙ௦ௗ − 1)߱௦] (5.53) 5.2 Derivation of Bridge Quantities 178 5.2.8 Summary of Equations ܮ = Main span length ߱௦ = Superstructure dead load ܮோ = Suspension ratio ߱௦ = Reference superstructure dead load ܮௌோ = Side-to-main span length ratio ߱ = Live load ℎ்ோ = Tower height-to-span ratio ߩ / ߛ = Density/Unit weight of material ‘m’ ߚ௧ = Factor for out-of-plane loading on towers ߪ = Design stress of material ‘m’ ߙ௦ௗ = Factor for superimposed dead load ݃ = Standard acceleration due to gravity Stay Cable Steel Quantity Main Span ܳ௦௧ =ߩ௦௧ߪ௦௧ ܮଶ൫߱௦ + ߱൯(1 − ܮோ)ℎ்ோ ቈ(1 − ܮோ)ଶ12ℎ்ோଶ+ 1 (5.54) Side Span ܳ௦௧௦ =ߩ௦௧ߪ௦௧ ܮଶ൫߱௦ + ߱൯ℎ்ோܮௌோ ቈ23 ൬ܮௌோℎ்ோ൰ଶ+ 2 (5.55) Suspension Cable Quantity ܳ =1݃ ܮ݉൫߱௦ + ߱൯ܮோ(2 − ܮோ)ߦߟ1 − ߦߟ ቈ(1 − ܮோ)ߟ + ܮோ(ߟଶ + 5)6 (5.56) where, ߟ = ඨ1 + 16 ൬ ℎܴܶ2 − ܮோ൰ଶ and ߦ =ߛ ܮ8 ߪℎ்ோ Anchor Cable Quantity ܳ௦ =112ߩ௦ߪ௦ ܮൣ3ܮ൫߱௦ + ߱൯ + 3ܳ݃ + 2ܳ௦௧݃(1 − ܮோ) − 4ܳ௦௧௦݃(ܮௌோ)− 12ܮܮௌோଶ߱௦൧ ൬ܮௌோℎ்ோ +ℎ்ோܮௌோ൰ (5.57) Hanger Cable Quantity ܳ =13ߩߪ ܮଶ൫߱௦ + ߱൯ℎ்ோܮோଶ ൬12 − ܮோ൰ (5.58)5.2 Derivation of Bridge Quantities 179 Tower Quantity ܳ௧ =2 ௩ܰ݃ ݈݊ ൦11 − (ℎ்ோܮ + ℎ)ߛ௧ߚ௧ߪ௧൪ (5.59) where, ௩ܰ =124ܮௌோ ൣ3ܮ൫߱௦ + ߱൯(2ܮௌோ + 1)ଶ + 2ܳ௦௧݃(6ܮௌோ − ܮோ + 1) + 8ܳ௦௧௦݃(ܮௌோ)+ 3ܳ݃(4ܮௌோ + 1) + 6ܳ௦݃(ܮௌோ)൧ and, ℎ = 50 ൬1 −ܮ4000൰ [݅݊ ݉݁ݐݎ݁ݏ] Superstructure Quantity ܳ௦ =ܮ(1 + 2ܮௌோ)݃ [߱௦ − (ߙ௦ௗ − 1)߱௦] (5.60) Load Correction Equation ߱௦ = ߱௦ + ߛ௦(ܣ௦ − ܣ௦) (5.61) where, ܣ௦ =124ℎ்ோߪ௦(2ܮௌோ + 1) ൣܮ൫߱௦ + ߱൯൫1 − ܮோଷ + 3ܮோ + 6ܮௌோ൯ − 16ܮܮௌோଷ߱௦+ 4ܳ௦௧݃ܮௌோ(1 − ܮோ) − 8ܳ௦௧௦݃(ܮௌோ)ଶ + 3ܳ݃(2ܮௌோ + 1)൧ and, ܣ௦ is computed using the same expression as ܣ௦ with the corresponding parameters for the reference bridge. 5.3 Span Proportions 180 5.3 Span Proportions 5.3.1 Tower Height-to-Span Ratio Based on the quantities given by Equations (5.54) to (5.61), the expected cost of a self-anchored discontinuous hybrid cable bridge, ܥு, can be expressed as, ܥு = ܿ௦௧(ܳ௦௧ + ܳ௦௧௦) + ܿ(ܳ + ܳ௦) + ܿܳ + ܿ௧ܳ௧ + ܿ௦ܳ௦ + ܿܳ (5.62)where ܿ represents the unit cost of component ‘m’ and ܳ represents the quantity of the bridge foundations. For simplicity, the anchor cables and the suspension cable are considered as a collective entity and it is assumed both cables share the same material/cost parameters. This is justified given that the anchor cables and suspension cable share a common load path. In regard to the foundation quantity, ܳ was not discussed in Section 5.2 because it is a parameter which cannot be generalized. Many different types of foundations exist and the type chosen will depend on a wide variety of local conditions. Nevertheless, an accurate estimate of the optimum tower height-to-span ratio can still be obtained if it is assumed that the tower height-to-span ratio has a negligible effect on the foundation quantity. This is because the optimum tower height-to-span ratio depends only on the rate of change in the quantities. This is reflected in the mathematical equation which gives the condition upon which the optimum ratio is found, ݀ܥு݀ℎ்ோ = 0 (5.63)Neglecting the change in the foundation quantity, Equation (5.63) may also be written as, ݀(ܳ௦௧ + ܳ௦௧௦)݀ℎ்ோ +ܿܿ௦௧݀(ܳ + ܳ௦)݀ℎ்ோ +ܿܿ௦௧݀ܳ݀ℎ்ோ +ܿ௧ܿ௦௧݀ܳ௧݀ℎ்ோ +ܿ௦ܿ௦௧݀ܳ௦݀ℎ்ோ = 0 (5.64)whereupon it also becomes clear that the optimal solution does not depend on the specific values assigned for the unit costs – only the ratios of the unit costs affect the solution. This is highly convenient given that specific unit costs may vary greatly from site-to-site whereas the ratios of the unit costs can be more or less generalized. Although less apparent, the optimal solution will also not depend on the specific values assigned for the live load (߱) and superstructure dead load (߱௦). Examining closely the equations for the quantities of the various components, it can be seen that the superstructure dead load can be entirely factored out of Equation (5.64) so that, from a loading perspective, the optimal solution depends only on the live load ratio of the reference bridge, ߱ோ, which is defined as, 5.3 Span Proportions 181 ߱ோ =߱߱ݏ (5.65)Notwithstanding, the iterative nature and complexity of Equations (5.54) to (5.61) make it necessary to evaluate Equation (5.64) numerically. However, values first need to be assigned to the input parameters. The assigned values for the material and cost input parameters are given in Table 5.1. The material input parameters are based on engineering experience and reflect current design standards. For simplicity, the material input parameters for the superstructure and towers are defined for all-steel or all-concrete scenarios. In addition, the material input parameters are assumed equal for each of the cable types. The cost input parameters are based on historical unit price information for conventional cable bridge projects. The information was obtained from a comprehensive structure study report compiled by multiple professional engineering firms (Parsons, 2008). The use of the unit cost parameters reflects the current method by which the cost of large infrastructure projects is assessed. Accordingly, the value of the parameters incorporates all costs related to the construction of a particular component. This mainly includes material, fabrication, transportation, erection, and testing costs. Table 5.1: Material and Cost Input Parameters Input Parameter Superstructure Towers Cables (Concrete) (Steel) (Concrete) (Steel) (Stays) (Suspension & Anchor) (Hangers)ߛ (kN/m3) 24 77 24 77 90 90 90 ߪ (MPa) 25 250 25 250 800 800 800 ߚ௧ n/a n/a 0.7 0.7 n/a n/a n/a ߙ௦ௗ 1.1 1.1 n/a n/a n/a n/a n/a ܿ ܿ௦௧⁄ 0.1 0.85 0.125 0.6 1.0 0.75 1.0 Also required is the assignment of the input parameters for the reference bridge. For familiarity, the reference bridge is designated as a standard cable-stayed bridge (ܮோ = 0). All of the input material and cost parameters for the reference bridge are assigned values consistent with those given in Table 5.1. The only unique input parameters which need to be assigned for the reference bridge are the tower height-to-span ratio (ℎ்ோ) and the live load ratio (߱ோ). The tower height-to-span ratio is set at a conventional value for cable-stayed bridges, ℎ்ோ = 0.25. The live load ratio depends on the superstructure material and, accordingly, for the all-steel and all-concrete scenarios the live load ratio is assigned values of ߱ோ = 0.6 and ߱ோ = 0.2, respectively. Using the assigned input values, Table 5.2 gives the calculated optimum tower height-to-span ratio for a self-anchored discontinuous hybrid cable bridge with a span length of 500 metres. Optimal values are presented considering multiple suspension ratios (ܮோ) and multiple side-to-main span 5.3 Span Proportions 182 length ratios (ܮௌோ). For consistency, the span lengths of the reference bridge were set to equal the span lengths defined for the hybrid cable bridge when computing the optimal values. Table 5.2: Optimal Tower Height-to-Span Ratio for Bridge with 500 metre Main Span Tower Material (Concrete) (Steel) 0.2 0.4 0.6 0.2 0.4 0.6 Superstructure Material (Concrete) 0.3 0.23 0.24 0.25 0.21 0.23 0.24 0.4 0.24 0.25 0.26 0.22 0.24 0.25 0.5 0.24 0.25 0.26 0.22 0.24 0.25 (Steel) 0.3 0.27 0.29 0.30 0.25 0.27 0.28 0.4 0.29 0.30 0.32 0.27 0.29 0.30 0.5 0.30 0.31 0.32 0.28 0.29 0.30 Parameters: ܮ = 500m, ܮ = ܮ, ܮௌோ = ܮௌோ, ℎ்ோ = 0.25, also refer to Table 5.1 Based on Table 5.2, the optimum tower height-to-span ratio, Decreases when steel is used in place of concrete for the towers; Increases when steel is used in place of concrete for the superstructure; Increases with increasing suspension ratio; and Increases when the side-to-main span length ratio is increased. However, in each case, the change to the optimum tower height-to-span ratio is minor. The cause for these trends can be explained by examining the cost function (Equation (5.62)) which is plotted in Figure 5.15 as a function of the tower height-to-span ratio, for a suspension ratio of 0.4 and a side-to-main span ratio of 0.3. For clarity, the cost function is broken down on a component-by-component basis. Moreover, the cost of each component is normalized with respect to the cost of the reference bridge. From Figure 5.15, it can be seen that when steel is used in place of concrete for the towers, the cost of the towers increases relative to the other components. Consequently, the overall optimum shifts towards the optimum for the tower cost. Similarly, when steel is used in place of concrete for the superstructure, the overall cost becomes largely controlled by the cost of the superstructure. Accordingly, the overall optimum shifts towards the optimum for the superstructure cost. In addition, although not apparent from Figure 5.15, increasing the suspension ratio leads to slight increases in the cost of the superstructure and suspension cable. As a result, the overall optimum increases since the cost of both these components diminishes when the tower height-to-span ratio is 5.3 Span Proportions 183 increased. Likewise, as the side-to-main span length increases, the total length of the bridge increases. This again has the effect of increasing the relative contribution of the superstructure cost. Tower Material (Concrete) (Steel) Superstructure Material (Concrete) (Steel) Optimum Value Total Stay Cables Hangers Suspension & Anchor Cables Superstructure Towers Figure 5.15: Cost Function Normalized with Respect to Cost of Reference Bridge Parameters: ܮ = 500m, ܮோ = 0.4, ܮௌோ = 0.3, ܮ = ܮ, ܮௌோ = ܮௌோ, ℎ்ோ = 0.25, also refer to Table 5.1 *Excludes foundation cost Another important observation from Figure 5.15 is that the hanger cost is inconsequential relative to the cost of the other components. This is also true when the value of the suspension ratio is increased. Thus, the cost of the hangers can be effectively negated in the calculation of the total cost. It is also of note that the optimum tower height-to-span ratio is fairly impervious to changes in the 0.1 0.2 0.3 0.4 0.500.20.40.60.811.21.40.1 0.2 0.3 0.4 0.500.20.40.60.811.21.40.1 0.2 0.3 0.4 0.500.20.40.60.811.21.40.1 0.2 0.3 0.4 0.500.20.40.60.811.21.4 5.3 Span Proportions 184 main span length. As confirmation, Figure 5.16 shows how the optimum tower height-to-span ratio varies with the main span length for a suspension ratio of 0.4 and a side-to-main span ratio of 0.3. The positive trend occurs because as the span length is increased the cost of the suspension/anchor cable steel and the superstructure become slightly more dominant. The only exception is for the case of a steel superstructure and concrete tower where it is the cost of the tower which becomes more dominant. Concrete Superstructure; Concrete Tower Steel Superstructure; Steel Tower Concrete Superstructure; Steel Tower Steel Superstructure; Concrete Tower Figure 5.16: Optimum Tower Height-to-Span Ratio versus Main Span Length Parameters: ܮோ = 0.4, ܮௌோ = 0.3, ܮ = ܮ, ܮௌோ = ܮௌோ, ℎ்ோ = 0.25, also refer to Table 5.1 Considering all of the data presented above, the optimum tower height-to-span ratio of self-anchored discontinuous hybrid cable bridges can be specified to be within the range of 0.2 to 0.3. However, it is important to revisit some of the initial assumptions made in the derivation of the optimal range. In regards to the assumed values for the input parameters (Table 5.1), the sensitivity of the optimum tower height-to-span ratio to changes in the assumed values was computed by varying each input parameter independently to within plus or minus twenty percent of its original assumed value. The results are plotted in Figure 5.17. In accordance with Figure 5.16, the results are only marginally dependent on the main span length. It is not surprising that, based on Figure 5.15, the optimum tower height-to-span ratio is most affected by the design stress and unit cost of the towers and superstructure. Nevertheless, a twenty percent change in the design stress or unit cost of the towers or superstructure returns less than a ten percent change in the optimum tower height-to-span ratio. Comparatively, the sensitivity with respect to all other input parameters is minor. This includes the input parameters for the reference bridge (߱ோ, ℎ்ோ), and the input factor which accounts for superimposed dead load (ߙ௦ௗ). The only exception is the design stress and unit cost of the suspension/anchor cable which have a notable 200 400 600 800 10000.200.220.240.260.280.30_ ()5.3 Span Proportions 185 influence when the superstructure is composed of concrete. Tower Material (Concrete) (Steel) Superstructure Material (Concrete) Percent Change in Optimum Tower Height-to-Span Ratio (Steel) Percent Change in Optimum Tower Height-to-Span Ratio Pecent Change in Input Parameters Pecent Change in Input Parameters Input Parameters: ܿ ܿ௦௧⁄ ܿ ܿ௦௧⁄ ܿ௦ ܿ௦௧⁄ ܿ௧ ܿ௦௧⁄ ߱ோ ߪ௦௧ ߪ ߪ ߪ௦ ߚ௧ߪ௧ ℎ்ோ ߩ௦௧ ߩ ߩ ߙ௦ௗ Figure 5.17: Sensitivity of Optimum Tower Height-to-Span Ratio Parameters: ܮ = 500m, ܮோ = 0.4, ܮௌோ = 0.3, ܮ = ܮ, ܮௌோ = ܮௌோ, ℎ்ோ = 0.25, also refer to Table 5.1 During the derivation of the optimal range it was also assumed that the rate of change in the superstructure bending moment envelope could be neglected. Although the rate of change in the bending moment envelope is too complex to compute algebraically, this assumption can be justified from the results presented in Chapter 4. Specifically, Figure 4.24 and Figure 4.26 show that the superstructure bending moment envelope is not highly sensitive to the tower height-to-span ratio when the tower height-to-span ratio is varied within the optimal range. 20− 10− 0 10 2010−5−051020− 10− 0 10 2010−5−051020− 10− 0 10 2010−5−051020− 10− 0 10 2010−5−05105.3 Span Proportions 186 The optimal range was also derived assuming that the tower foundation cost could be neglected. The tower foundation cost is, again, too complex to compute algebraically. However, the vertical force from dead and live load constitutes a large portion of the foundation demands. Therefore, the cost of the foundation can be gauged by examining the magnitude of the vertical force acting on the foundation from dead and live load. This force is given by, ܰ = ௩ܰ +12ܳ௧݃ (5.66)where ܰ ௩ is the vertical force at the top of the towers from the cable system (Equation (5.31)), and ܳ ௧ is the tower quantity. The relationship between ܰ and the tower height-to-span ratio is plotted in Figure 5.18. For ease of comparison, the ordinates are normalized with respect to the resulting force when the tower height-to-span ratio equals 0.25. Concrete Superstructure; Concrete Tower Steel Superstructure; Steel Tower Concrete Superstructure; Steel Tower Steel Superstructure; Concrete Tower Figure 5.18: Vertical Force at Tower Foundation versus Tower Height-to-Span Ratio Parameters: ܮ = 500m, ܮோ = 0.4, ܮௌோ = 0.3, ܮ = ܮ, ܮௌோ = ܮௌோ, ℎ்ோ = 0.25, also refer to Table 5.1 It can be seen from Figure 5.18 that when the tower height-to-span ratio is varied within the optimal range the vertical force on the foundation varies by less than 5%. Still, the tower foundation cost generally constitutes a large portion of the overall bridge cost. Therefore, for cases when the tower is composed of concrete, it would be prudent to slightly reduce the value of the optimum tower height-to-span ratio reported in Table 5.2. Moreover, there are other justifications for reducing the tower height-to-span ratio. Aesthetically, lofty towers can be overly striking on most landscapes and, structurally, nonlinear effects and tower bending moments will increase with increasing tower height. Ultimately, for the aforementioned reasons, a tower height-to-span ratio in the range of 0.20-0.25 is 0.20 0.25 0.300.951.001.05 5.3 Span Proportions 187 recommended. This is in disagreement with pre-established notions that the optimum tower height-to-span ratio should be computed by achieving consistency between the tower height-to-span ratio of the stayed region and the historically established optimum tower height-to-span ratio for cable-stayed bridges. Or, alternatively, that the optimum tower height-to-span ratio should be computed by achieving consistency between the sag ratio of the suspension cable and the historically established optimum tower height-to-span ratio for self-anchored suspension bridges. However, these notions are flawed in that they violate the basic principle of sub-optimization which states, ‘Optimizing each subsystem independently will not in general lead to a system optimum, or more strongly, improvement of a particular subsystem may actually worsen the overall system’ (Machol, 1965). In addition, it is logical that the optimal range for the tower height-to-span ratio should match so closely with the historically established optimal range for cable-stayed bridges. Based on Equations (4.23) and (5.31), the maximum axial force in the superstructure and towers does not depend significantly on the suspension ratio. This becomes clear when the contribution from the weight of the cable steel is neglected. Furthermore, the cost of the cable steel only changes marginally when the suspension ratio is varied. This is because the additional cost of the suspension cable and hanger steel is offset by the discounted cost of the stay cable steel. It could, therefore, be reasoned that the optimum tower height-to-span ratio should remain fairly constant regardless of the suspension ratio and this is what the calculations presented above reflect. 5.3.2 Side-to-Main Span Ratio The span lengths of a bridge are normally constrained by the site topography and, therefore, the side-to-main span ratio is a parameter which cannot be freely assigned. It is also a parameter which affects many design aspects. For these reasons, attempting to specify a single optimal value for the side-to-main span ratio would be misguided. However, the effects associated with the side-to-main span ratio are more or less independent of the suspension ratio. This will become clear later on in this section. Therefore, an optimal range for the side-to-main span ratio can be determined based on the established optimal range for cable-stayed bridges. Most authors agree that the optimal side-to-main span ratio for a cable-stayed bridge lies within the range 0.35 to 0.45 (Podolny & Scalzi, 1976; Leonhardt, 1991; Farquhar, 2008). The rationale behind this range is based on a mixture of qualitative and quantitative reasoning. Details discussed below will provide guidance for the selection of the appropriate side-to-main span ratio in discontinuous hybrid cable bridges. The lower limit for the optimal side-to-main span range is normally governed by the uplift force at the anchor pier and by the tower/tower foundation cost. An uplift force is generated at the anchor 5.3 Span Proportions 188 pier when there is an imbalance of loading between the main span and the side spans. Thus, the magnitude of the uplift force increases as the side-to-main span ratio decreases. In addition, the maximum uplift force occurs when only the main span is loaded with live load. Large uplift forces are undesirable because the presence of large tensile forces adversely affects the design of the anchor pier and anchor pier foundation. If the contribution from the self-weight of the cables is neglected then, from Equation (5.19), the maximum uplift force at the anchor pier can be approximated as, ܴ ≅߱௦ܮ8ܮௌோ ൣ1 + ߱ோ − 4ܮௌோଶ൧ (5.67) The relationship between the side-to-main span ratio (ܮௌோ) and the uplift force (ܴ) is plotted in Figure 5.19. The ordinates of the plot are normalized so that they are independent of the superstructure dead load (߱௦), and the main span length (ܮ). In doing so, it is assumed that the main span length is fixed and the effect the side-to-main span ratio has on the superstructure dead load is negligible. Figure 5.19: Uplift Force at Anchor Pier versus Side-to-Main Span Ratio As an example, it can be seen from Figure 5.19 that transitioning from a side-to-main span ratio of 0.4 to 0.2 amplifies the uplift force at the anchor pier by more than a factor of 3. From a design perspective, the consequences resulting from decreasing the side-to-main span ratio are perhaps made clearer by considering the magnitude of added dead load required in the side spans to balance the uplift force. Accordingly, denoting ߱ ௦௦ and ߱ ௦ as the superstructure dead load in the side and main spans, respectively, Equation (5.67) can be re-written as, 0.10 0.20 0.30 0.40 0.500.00.51.01.52.0࣓ ߱ோ = 0.2 ߱ோ = 0.4 ߱ோ = 0.6 ߱ோ = 0 5.3 Span Proportions 189 ܴ ≅߱௦ܮ8ܮௌோ 1 + ߱ோ − 4ܮௌோଶ ൬߱௦௦߱௦൰൨ (5.68) where ߱ ோ is the unfactored live load ratio for the main span only (߱ோ = ߱ ߱௦⁄ ). Solving for the superstructure dead load ratio (߱௦௦ ߱௦⁄ ) when ܴ equals zero then gives the balancing condition as, ߱௦௦߱௦ ≅1 + ߱ோ4ܮௌோଶ (5.69) Figure 5.20: Dead Load Ratio Required to Balance Uplift Force at Anchor Pier For convenience, the balancing condition is plotted in Figure 5.20. Clearly, for a side-to-main span ratio of 0.2 the dead load of the superstructure in the side span is required to be anywhere from 6 to 10 times greater than the dead load in the main span to prevent uplift under service loads. Designing for a variance this large is not practical. Normally, in regards to the superstructure, even if two different materials are judiciously employed to balance the uplift force, a conventional concrete section is only in the order of 4 times heavier than a conventional steel section. Therefore, if uplift forces are to be avoided under service loads without any additional ballast, a lower limit of roughly 0.35 must be imposed on the side-to-main span ratio. However, it should be recognized that this 0.10 0.20 0.30 0.40 0.501.02.03.04.05.06.07.08.09.010.0࣓࣓ ߱ோ = 0.2߱ோ = 0.4߱ோ = 0.6߱ோ = 0 ߱௦߱ ܮܮௌோܮ ܮௌோܮ0 0 ߱௦௦ ߱௦௦߱ோ = ߱ ߱௦⁄ 5.3 Span Proportions 190 value is conservative because the influence of the superstructure bending stiffness was ignored in its formation. The cost of the tower and tower foundation can again be gauged with respect to the total vertical force acting on the foundation (Equation (5.66)). However, for simplicity, it is appropriate here to neglect the contribution from the weight of the cable steel. Accordingly, the expression for the total vertical force at the tower foundation becomes, ܰ ≅ܮ8ܮௌோ ൫߱௦ + ߱൯(2ܮௌோ + 1)ଶΛ (5.70)where, Λ = 1 + ݈݊ ൦ 11 − (ℎ்ோܮ + ℎ)ߛ௧ߚ௧ߪ௧൪ Since all of the parameters can be assumed independent of ܮௌோ, it becomes apparent from Equation (5.70) that a simple common relationship exists between the total vertical force at the tower foundation and the side-to-main span ratio. When the relationship is plotted (Figure 5.21), it also becomes apparent that the total vertical force at the tower foundation starts to increase rapidly when the side-to-span ratio falls below 0.40. This is how the cost of the tower and tower foundation factor in to the lower limit of the side-to-main span ratio. Figure 5.21: Vertical Force at Tower Foundation versus Side-to-Main Span Ratio At the opposite end of the range, the effective stiffness and the stress range of the anchor cable normally set the upper limit for the side-to-main span ratio. The effective stiffness of the anchor cable controls the longitudinal deflection at the top of the towers. From Chapter 4 (Figure 4.23), it has already been observed that the effective stiffness of the anchor cable is greatest when the side-to-main 0.10 0.20 0.30 0.40 0.501.01.21.31.51.61.8൫࣓ + ࣓൯ࢫ 5.3 Span Proportions 191 span ratio is slightly less than the tower height-to-span ratio. Accordingly, based on this one aspect, the optimum side-to-main span ratio would be within, or slightly below, the range of 0.2 to 0.25. Since this is clearly below the lower limit specified above, the effectiveness of the anchor cable is simply an important design aspect to keep in mind when assigning the side-to-main span ratio. In that respect, the effective stiffness of the anchor cable is greater when the side-to-main span ratio is kept relatively small. The stress range of the anchor cable is important to consider because of its relation to fatigue. Fatigue can severely reduce the life span of a cable and, therefore, it is important that any concerns of fatigue are abated. This requires that the stress range in the anchor cable be kept within reasonable limits. Live load positioned in the main span increases the stress in the anchor cable and live load positioned in the side spans decreases the stress in the anchor cable. Therefore, the larger the side-to-main span ratio, the larger the stress range will be in the anchor cable. Historically, fatigue of the anchor cable has been evaluated using the two worst case loading scenarios – full main span lane loading alternating with full side span lane loading. However, the magnitude of the applied live load was reduced given that these loading scenarios are unlikely to occur regularly and fatigue is a phenomenon which in this case is associated with high frequency loading. Furthermore, to simplify the calculations involved the influence of the superstructure bending stiffness was neglected. On this basis, for vehicular bridges employing steel and concrete superstructures, the upper limit for the side-to-main span ratio was computed to be roughly, 0.35 and 0.4, respectively (Leonhardt & Zellner, 1980). Recent studies on the fatigue of anchor cables have revealed that approaches based on the worst case loading scenarios are overly conservative (Goodyear, 1987). Consequently, many design codes now specify that that the load from a single design truck be used to evaluate fatigue. This type of loading is more consistent with real fatigue loading conditions. When the stiffness of the superstructure is also taken into account, the upper limit for the side-to-main span ratio from a fatigue perspective is likely to be between 0.4 and 0.45 (Farquhar, 2008). 5.3.3 Suspension Ratio When the tower-to-height ratio is in the range of 0.2 to 0.25, costs will increase moderately with increasing suspension ratio primarily in response to increased superstructure and tower demands (Figure 4.24). As such, it is best to keep the suspension ratio to a minimum. Ultimately, aesthetics will dictate the upper limit of the suspension ratio. The importance of bridge aesthetics should not be undervalued. Bridges are designed and built to 5.3 Span Proportions 192 provide decades of service. An unsightly bridge, even if functional, can become a long-lasting scar on a city landscape resulting in property devaluation and public outcry. In contrast, an aesthetically pleasing bridge can more quickly gain the approval of client groups, approving authorities, and the public in general. Moreover, a well-balanced and pleasing design can often become a local or even national icon. Bridge aesthetics is thereby becoming increasingly more relevant during the bridge procurement process. There are many different theories regarding the best aesthetic practices. However, for long span bridges, there is one rule which is universal: the emphasis should be on the main span. The longer the main span is relative to the side spans, the longer the main span will appear and this lends to an overall slender appearance for the bridge. For the same reason, a discontinuous hybrid cable bridge will be more aesthetically appealing if the length of the stayed region in the main span is made greater than or equal to the length of the side span. This is demonstrated in Figure 5.22 which shows two discontinuous hybrid cable bridges with the same span lengths and tower heights. In the first case, the suspension ratio is relatively large so that the length of the stayed region in the main span is less than the length of the side span. In the second case, the suspension ratio is set so that the length of the stayed region in the main span is equal to the length of the side span. Figure 5.22: The Effect of the Suspension Ratio on Bridge Appearance Clearly, the second bridge in Figure 5.22 is more appealing. This also has to do with the fact that as the suspension ratio increases, a larger gap manifests between the suspension cable and the outermost stay cable. As a result, the suspension cable and the stay cables appear disjointed. Accordingly, from an aesthetics perspective, in order for the length of the stayed region in the main span to be greater than or equal to the length of the side span, the following condition must be satisfied, ܮௌோܮ ܮௌோܮ ܮ(1 − ܮோଵ) 2⁄ܮ(1 − ܮோଶ) 2⁄= > Bridge 1 Bridge 2 5.3 Span Proportions 193 ܮோ ≤ 1 − 2ܮௌோ (5.71) The optimal range for side-to-main span length ratio established in the previous section was 0.35 to 0.45. Substituting these values into Equation (5.71) yields the following range for the suspension ratio, 0.1 ≤ ܮோ ≤ 0.3 However, designing a hybrid bridge with a suspension ratio of 0.1 would be fruitless. A lower limit of 0.2 is more sensible; although as a consequence, to satisfy Equation (5.71) the upper limit of the side-to-main span ratio would need to be decreased to 0.4. Thereafter, the optimal ranges for the side-to-main span ratio and the suspension ratio become, 0.35 ≤ ܮௌோ_௧ ≤ 0.4 0.2 ≤ ܮோ_௧ ≤ 0.3 If necessary, the upper limit of these ranges could be extended by using a number of cross stays / cross hangers at the stay cable-hanger junction. When selecting the suspension ratio within the above range, it is also important to take into account the impact the suspension ratio has on construction demands. As previously mentioned, in order to erect a self-anchored discontinuous hybrid cable bridge the superstructure needs to be temporarily supported or, alternatively, the horizontal component of the suspension cable force needs to be temporarily restrained. For long span bridges, the latter option is undoubtedly more efficient. From Chapter 4 (Equation (4.9)), the horizontal component of the suspension cable force under dead load which must be restrained can be approximately equated to, (ܪ) ≅ܮ߱௦8ܵோ ߱߱௦ + ܮோ(2 − ܮோ)൨ (5.72) Equation (5.72) is plotted in Figure 5.23 as a function of the suspension ratio. The ordinates are normalized with respect to the case when the suspension ratio equals 0.2. As a result, the plot is virtually independent of the applied loading, the main span length, and the tower height-to-span ratio. 5.4 Economic Outcome 194 Figure 5.23: Horizontal Component of Suspension Cable Force under Dead Load versus Suspension Ratio From Figure 5.23, an increase in the suspension ratio from 0.2 to 0.3 results in roughly a 40% increase in the dead load horizontal cable force. Whether or not it is efficient to accommodate this increase depends on the starting value of the horizontal cable force in addition to the choice of the anchorage structure and the geological conditions at the bridge site, all of which are further discussed in Chapter 7. 5.4 Economic Outcome With the optimum portions established, it is now appropriate to study the expected costs associated with the construction of a self-anchored discontinuous hybrid cable bridge. Similar to the methodology used in Section 5.3.1 to evaluate the optimum sag ratio, it is convenient to examine the expected cost of the hybrid bridge system relative to a conventional cable-stayed bridge. This is because the two bridge systems possess many of the same features and, consequently, the results can be more or less generalized. Accordingly, using Equation (5.62), Figure 5.24 plots the expected cost of a self-anchored discontinuous hybrid cable bridge relative to a conventional cable-stayed bridge with the ordinates expressed in terms of percent change. Sub-plots are also presented for the individual bridge components. All input parameters are assumed equal between the two bridge systems and the comparison is made for the maximum recommended suspension ratio of 0.3. Because the parameters are assumed equal for the two bridge systems the plots are only slightly sensitive to the presumed input parameters (refer to Table 5.1). 0.20 0.22 0.24 0.26 0.28 0.301.001.101.201.301.401.50൫൯ 5.4 Economic Outcome 195 Tower Material (Concrete) (Steel) Superstructure Material (Concrete) (Steel) Total Cables Superstructure Towers Figure 5.24: Cost versus Span Length in Relation to a Cable-Stayed Bridge Parameters: ℎ்ோ = 0.25, ܮோ = 0.3, ܮௌோ = 0.35, ܮ = ܮ, ܮௌோ = ܮௌோ, ℎ்ோ = ℎ்ோ, also refer to Table 5.1 *Difference in foundation cost assumed negligible (refer to Equation (5.70)) Based on Equation (5.62), it can be observed from Figure 5.24 that the total cost of a self-anchored discontinuous hybrid cable bridge is slightly greater than the cost which would be incurred by a conventional cable-stayed bridge. Notwithstanding, considering that it is generally uneconomic to employ concrete superstructures for relatively long spans, the percent change in cost between the two systems can be expected to be less than 5% for any practical span length. This result should not be interpreted to underrate the potential economic advantages of self-anchored discontinuous hybrid cable bridges. There are many factors which are not accounted for in Equation (5.62). Primarily, in the hybrid system, Equation (5.62) treats the uniform compression force transferred to the superstructure by the suspension cable as a disadvantage because it assumes additional200 400 600 800 10000246810200 400 600 800 10000246810200 400 600 800 10000246810200 400 600 800 10000246810() () % % 5.5 End of Chapter Summary 196 superstructure material is be required to carry the force. In Figure 5.24, it can be seen that this is actually the major source of the cost discrepancy between the hybrid system and the conventional cable-stayed system. In reality, the compression force can theoretically be exploited as an advantage, ultimately saving costs. For steel superstructures, continual compression is beneficial from a durability standpoint because it reduces the likelihood of fatigue, thereby reducing costs associated with fabrication. For concrete and composite superstructures, continual compression reduces or eliminates the need for longitudinal post tensioning steel. As a result, the thickness, and more importantly the weight, of certain cross-sectional components can be reduced. This in turn generates cost savings as it reduces the load which must be supported by the other major bridge components. Equation (5.62) also neglects the relationship between cost and construction duration, which can have a significant impact on overall costs. This is discussed in more detail in Chapter 7. In consideration of the above, the cost of a self-anchored discontinuous hybrid cable bridge can be expected to be closely comparable to the cost of a conventional cable-stayed bridge. The economic span range of the two bridges should, therefore, also be comparable. 5.5 End of Chapter Summary The optimum stay cable and hanger arrangements were discussed at the beginning of the chapter. For reasons associated with cost, structural efficiency, and aesthetics, a fan or semi-fan arrangement can be considered optimum for an array of stay cables and a vertical arrangement can be considered optimum for an array of hangers. With respect to the aforementioned optimum cable arrangements, parametric equations were derived giving material estimates for principal components of a generalized self-anchored discontinuous hybrid cable bridge. These equations were then used to study optimum ranges for the span proportions. Based on convention, the optimum tower height-to-span ratio was studied primarily from a cost perspective. In that regard, the optimum tower height-to-span ratio was shown to be not greatly dependent on the choice of material for the superstructure and/or the towers. Furthermore, the optimum tower height-to-span ratio was shown to be fairly insensitive to changes in the assumed material and cost parameters of the principal components. After qualitatively including the influence of the foundation cost, a tower height-to-span ratio in the range of 0.2 to 0.25 was recommended. The optimum side-to-main span ratio can be considered independent of the suspension ratio. As such, the historically established optimum side-to-main span ratio for cable-stayed bridges is equally applicable to self-anchored discontinuous hybrid cable bridges. The rationale behind the historically established range of 0.35 to 0.45 was discussed in detail. Nevertheless, when examining the optimum 5.5 End of Chapter Summary 197 suspension ratio, due to the correlation between the suspension ratio and the side-to-main span ratio, aesthetics and function dictated that the optimum side-to-main span range should be slightly adjusted. Ultimately, a range of 0.35 to 0.4 was recommended for the side-to-main span ratio and a range of 0.2 to 0.3 was recommended for the suspension ratio. The upper limit of these ranges can be extended if cross stays / cross hangers are employed; however, construction of the suspended region becomes more challenging as the suspension ratio increases. For the established optimal proportions, the expected cost of a self-anchored discontinuous hybrid cable bridge is closely comparable to a conventional cable-stayed bridge. Additional economies unique to self-anchored discontinuous hybrid cable bridges may also be achieved by exploiting the continual compression force produced by the hybrid cable system. Based on these results it was deduced that the economic span range of a self-anchored discontinuous hybrid cable bridge is similar to that which has been established for conventional cable-stayed bridges. 𝐸𝐴𝑔𝐸𝐼𝑔𝐸𝐴𝑡𝐸𝐼𝑡 𝐹𝑖𝑥𝑒𝑑 𝑆𝑢𝑝𝑝𝑜𝑟𝑡 (TYP) 𝑅𝑜𝑙𝑙𝑒𝑟 𝑆𝑢𝑝𝑝𝑜𝑟𝑡 (TYP) 𝐿𝑚 50 (𝑇𝑌𝑃) 𝐸𝐴𝑔,𝐸𝐼𝑔 𝐿𝑚 500 𝜔𝑝 = 75𝑘𝑁/𝑚 𝑉𝑎𝑟𝑖𝑏𝑙𝑒 37.5𝑚 𝑉𝑎𝑟𝑖𝑏𝑙𝑒 0.36(𝐿𝑚) 0.36(𝐿𝑚) 0.28(𝐿𝑚) 0.225(𝐿𝑚) 𝜔𝑠 = 𝜔𝑝 0.6 𝐸𝐴𝑡,𝐸𝐼𝑡 𝐿𝑚 = 1000𝑚 0.36(𝐿𝑚) 0.36(𝐿𝑚) 𝑇𝑎𝑏𝑙𝑒 6.2 𝑥 𝑦 Hanger (TYP) Stay Cable (TYP) Main Span Suspension Cable Anchor Cable 𝜔𝑝 𝜔𝑠 Anchor and Stay Cables Modelled using Tangent Modulus Anchor Cable Modelled as Catenary; Stay Cables Modelled using Tangent Modulus Anchor and Stay Cables Modelled as Catenaries AS1 AS2 AS3 Deflections (m) Axial Forces (MN) Shear Forces (MN) Moments (MN·m) Cable Stress Range (MN) Absolute value of dead load axial force Cable Stress Range (MN) AS1 AS2 AS3 Main Span Side Span Deflections (m) Shear Forces (MN) Moments (MN·m) AS1 AS2 AS3 ƗAS2 Deflections (m) Axial Forces (MN) Shear Forces (MN) Moments (MN·m) Cable Stress Range (MN) Cable Stress Range Ɨ (MN) 𝐻𝑦𝑏𝑟𝑖𝑑 𝐵𝑟𝑖𝑑𝑔𝑒 𝐶𝑜𝑛𝑣𝑒𝑛𝑡𝑖𝑜𝑛𝑎𝑙 𝐶𝑎𝑏𝑙𝑒 𝑆𝑡𝑎𝑦𝑒𝑑 𝐵𝑟𝑖𝑑𝑔𝑒 Main Span Side Span Deflections (m) Shear Forces (MN) Moments (MN·m) AS1 AS2 AS3 𝐶𝑜𝑛𝑣𝑒𝑛𝑡𝑖𝑜𝑛𝑎𝑙 𝐶𝑎𝑏𝑙𝑒 𝑆𝑡𝑎𝑦𝑒𝑑 𝐵𝑟𝑖𝑑𝑔𝑒 𝐻𝑦𝑏𝑟𝑖𝑑 𝐵𝑟𝑖𝑑𝑔𝑒 𝐸𝐼𝑔 𝐸𝐼𝑡𝐸𝐴𝑐 Ɨ With Cable Clamp Without Cable Clamp Span Cable Stress Range (MN) Cable Stress Range Ɨ (MN) Cross Stay (TYP) Anchor Cable Stay Cable (TYP) Hanger (TYP) 𝑥𝑝𝑖𝐿𝑠𝑥𝑝𝑖 𝐿𝑠 𝑥𝑝𝑖 𝐿𝑠 𝑥𝑝𝑖 𝐿𝑠 𝑥𝑝𝑖 𝐿𝑠 𝑥𝑝𝑖 𝐿𝑠 𝑥𝑝𝑖 𝐿𝑠 𝑥𝑝𝑖 𝐿𝑠 ƗAS2 Deflections (m) Axial Forces (MN) Shear Forces (MN) Moments (MN·m) Cable Stress Range (MN) Cable Stress Range Ɨ (MN) 𝑥𝑝𝑖 𝐿𝑠 = 0 𝑥𝑝𝑖 𝐿𝑠 = 0.5 𝑥𝑝𝑖 𝐿𝑠 = 0.75 𝑥𝑝𝑖 𝐿𝑠 = 0.25 𝑳𝒔 𝒙𝒑𝒊 Main Span Side Span Deflections (m) Shear Forces (MN) Moments (MN·m) AS1 AS2 AS3 𝑥𝑝𝑖 𝐿𝑠 = 0.5 𝑥𝑝𝑖 𝐿𝑠 = 0.75 𝑥𝑝𝑖 𝐿𝑠 = 0.25 𝑥𝑝𝑖 𝐿𝑠 = 0 3 Equal Spaces 5 Equal Spaces 4 Equal Spaces 3 Equal Spaces 5 Equal Spaces 4 Equal Spaces ƗAS2 Deflections (m) Axial Forces (MN) Shear Forces (MN) Moments (MN·m) Cable Stress Range (MN) Cable Stress Range Ɨ (MN) 𝑁𝑜 𝐼𝑛𝑡.𝑃𝑖𝑒𝑟 3 𝐼𝑛𝑡.𝑃𝑖𝑒𝑟𝑠 4 𝐼𝑛𝑡.𝑃𝑖𝑒𝑟𝑠 2 𝐼𝑛𝑡.𝑃𝑖𝑒𝑟𝑠 Main Span Side Span Deflections (m) Shear Forces (MN) Moments (MN·m) AS1 AS2 AS3 𝑁𝑜 𝐼𝑛𝑡.𝑃𝑖𝑒𝑟 3 𝐼𝑛𝑡.𝑃𝑖𝑒𝑟𝑠 4 𝐼𝑛𝑡.𝑃𝑖𝑒𝑟𝑠 2 𝐼𝑛𝑡.𝑃𝑖𝑒𝑟𝑠 Guy Line (TYP) Falsework (TYP) Step 1 Step 2 Step 3 Step 4a Step 4b Step 5a Step 5b Step 6a Step 6b Step 7a Step 7b Step 8 Step 9 (𝐻𝑐𝑚)𝐷𝐿 ≅𝐿𝑚𝜔𝑠𝑚8𝑆𝑅[𝜔𝑐𝑚𝜔𝑠𝑚+ 𝐿𝑅(2 − 𝐿𝑅)]𝜔𝑠𝑚 𝜔𝑠𝜔𝑠𝑚 𝜔𝑠𝑠𝐿𝑚 𝑆𝑅 𝐿𝑅 𝜔𝑐𝐻𝑆𝑅 ≅ 𝜇𝑉𝑝𝑓𝜇 𝑉𝑝𝑓𝑉𝑝𝑓𝐿𝑅 1 − 2𝐿𝑆𝑅𝑉𝑝𝑓 ≅ (𝜔𝑠𝑠 − 𝜔𝑠𝑚)𝐿𝑆𝑅𝐿𝑚 − (𝐻𝑐𝑚)𝐷𝐿 (𝑆𝑅𝐿𝑆𝑅)𝐿𝑆𝑅𝐻𝑆𝑅(𝐻𝑐𝑚)𝐷𝐿= 𝜇2𝑆𝑅(1 − 𝐿𝑅)[𝜔𝑐𝑚𝜔𝑠𝑚+ (1 − 𝐿𝑅)2 (1 − 2𝜔𝑠𝑠𝜔𝑠𝑚) + 1𝜔𝑐𝑚𝜔𝑠𝑚− (1 − 𝐿𝑅)2 + 1]𝜔𝑐Vertical Component of Suspension Cable Force Surplus Dead Load from Side Span Superstructure 𝐿𝑚 𝐿𝑆𝑅 𝐿𝑅 𝜔𝑅 𝛾𝑐 𝜎𝑎𝑙𝑙𝑜𝑤0.20 0.22 0.24 0.26 0.28 0.300.01.02.03.04.00.20 0.22 0.24 0.26 0.28 0.300.01.02.03.04.00.20 0.22 0.24 0.26 0.28 0.300.01.02.03.04.0𝑆𝑅 = 0.25 𝑆𝑅 = 0.225 𝑆𝑅 = 0.20 𝑳𝑹 𝑳𝑹 𝑳𝑹 𝑯𝑺𝑹(𝑯𝒄𝒎)𝑫𝑳 𝑯𝑺𝑹(𝑯𝒄𝒎)𝑫𝑳 𝑯𝑺𝑹(𝑯𝒄𝒎)𝑫𝑳 𝝁 = 𝟎.𝟓 𝝁 = 𝟎.𝟑 𝝁 = 𝟎.𝟕 𝐿𝑚 𝐿𝑆𝑅 𝐿𝑅 𝜔𝑅 𝛾𝑐 𝜎𝑎𝑙𝑙𝑜𝑤0.20 0.22 0.24 0.26 0.28 0.301.02.03.04.05.00.20 0.22 0.24 0.26 0.28 0.301.02.03.04.05.00.20 0.22 0.24 0.26 0.28 0.301.02.03.04.05.0𝑆𝑅 = 0.25 𝑆𝑅 = 0.225 𝑆𝑅 = 0.20 𝑳𝑹 𝑳𝑹 𝑳𝑹 𝝎𝒔𝒔𝝎𝒔𝒎 𝝎𝒔𝒔𝝎𝒔𝒎 𝝎𝒔𝒔𝝎𝒔𝒎 𝝁 = 𝟎.𝟓 𝝁 = 𝟎.𝟑 𝝁 = 𝟎.𝟕 𝐿𝑅 𝐿𝑚 𝛾𝑐 𝜎𝑎𝑙𝑙𝑜𝑤0.20 0.22 0.24 0.26 0.28 0.3024578𝑆𝑅 = 0.25 𝑆𝑅 = 0.225 𝑆𝑅 = 0.20 𝑳𝑹 𝑨𝒄 (𝒔𝒖𝒔𝒑𝒆𝒏𝒔𝒊𝒐𝒏 𝒃𝒓𝒊𝒅𝒈𝒆)𝑨𝒄 (𝒉𝒚𝒃𝒓𝒊𝒅 𝒃𝒓𝒊𝒅𝒈𝒆) Locked-Coil Strand (TYP) Hanger Cable (TYP) 266 Appendix A PROCEDURE TO DETERMINE CABLE SHAPE IN THREE DIMENSIONS Appendix A: Procedure to Determine Cable Shape in Three Dimensions 267 Figure A1 displays a three dimensional view of a suspension cable whose ends, according to the prescribed global Cartesian coordinate system, are offset in the ܺ (longitudinal), ܻ (vertical), and ܼ (transverse) directions by the vectors Δ்ܺ, Δ்ܻ , and Δ்ܼ respectively. The cable is further represented as a series of cable segments, where each segment is bounded by nodes which have been placed at the ends of the cable and at hanger locations. A free body diagram for a given segment ݅, is depicted in Figure A2. Assuming that the cable is only subjected to gravity loads, then the only forces acting along the length of each cable segment are those due to the self-weight of the cable (hanger forces are considered to act at nodal locations). Hence, between nodes, the local curve of each cable segment takes the form of a catenary whose local coordinates lie in a two dimensional plane characterized by the ݔ∗ and ݕ axes. It should be noted that lowercase letters will be used throughout to represent the local axes of the individual cable segments in order to avoid confusion with the global axes which are denoted by uppercase letters. ଵܰ… Nodal Numbering Scheme ΔX Projected Length of Cable in ‘x’ Directionଵܵ… Segment Numbering Scheme ΔY Projected Length of Cable in ‘y’ Direction߱ ƗSelf-Weight of Suspension Cable ΔZ Projected Length of Cable in ‘z’ Directionܨℎ Hanger Force Acting at Node i ݊ Number of Cable Segments ƗNot Shown for Clarity Figure A1: 3D View of a Suspension Cable ܻܼܨℎଶଵܰ߂்ܺ ଵܵ߂்ܼ߂்ܻ ܨℎଷܨℎିଵܨℎܨℎାଵܨℎଷܰܰିଵܰܰାଵܰ ܵଶܵଷܵିଵܵܵାଵܵ ଶܰ ܺAppendix A: Procedure to Determine Cable Shape in Three Dimensions 268 Using the Equations already established in Section 3.1.1.1, the local ordinates of a given cable segment ݅ may thereby be expressed as (from Figure 3.1 and Equation 3.3), ݕ =ܪ߱ ܿݏℎ ൬߱ݔ∗ܪ + ܣ൰ + ܤ (A1)where, ܣ = ܽݏ݅݊ℎ ߱ℎ2ܪݏ݅݊ℎ ቀ߱ܽ2ܪ ቁ − ߱ܽ2ܪ and ܤ = −ܪ߱ ܿݏℎሺܣሻ ߱ ƗSelf-Weight of Suspension Cable ܪ ‘Horizontal’ Cable Force in Segment iሺܴݔ, ܴݕ, ܴݖሻ Cable Force Components at Beginning of Segment iሺ߂ݔ, ߂ݕ, ߂ݖሻ Projected Dimensions of Segment i ƗNot Shown for Clarity Figure A2: Free Body Diagram of Segment i ݕݖ߂ݔ ߂ݕ ݔݔ∗߂ݖܴݔܴݕ = −ܪ݀ݕ݀ݔ∗ฬ௫∗ୀ = −ܪݏ݅݊ℎሺܣሻ ܴݖܪܴݔܴݖ ܪ݀ݕ݀ݔ∗ฬ௫∗ୀ= ܪݏ݅݊ℎ ൬߱ܽܪ + ܣ൰ℎ Appendix A: Procedure to Determine Cable Shape in Three Dimensions 269 The direction of the ݔ∗ axis may vary from segment to segment depending upon the line of action of the hanger forces. As such, Equation (A1) needs to be transformed to a consistent local three dimensional coordinate system by making the following substitutions (with reference to Figure A2), ܪ = ඥሺܴݔሻଶ + ሺܴݖሻଶ (A2) ܽ = ඥሺΔݔሻଶ + ሺΔݖሻଶ (A3) ℎ = Δy (A4) ݔ∗ = ݔඨ1 + ൬ΔݖΔݔ൰ଶ (A5) where, the local ݖ and ݔ axes are related through the following relationship, ݖ = ݔ ൬ΔݖΔݔ൰ (A6)Equations (A1)-(A6) describe the three dimensional catenary curve of an individual cable segment with respect to a local Cartesian coordinate system positioned at the beginning of the segment. Considering all cable segments, assuming the self-weight of the suspension cable and the projected length of each cable segment in the ݔ direction are known parameters, there remain 5 × ݊ unknowns in the form of ܴݔଵ…, ܴݕଵ…, ܴݖଵ…, Δݕଵ…, and Δݖଵ…. However, given the support reactions at Node 1 ሺܴݔଵ, ܴݕଵ, ܴݖଵሻ, the transverse and vertical projected dimensions of Segment 1 can be determined from the geometrical conditions upon which, Δݖ = Δݔ ൬ܴݖܴݔ൰ (A7)and ܴݕ = −ܪݏ݅݊ℎሺܣሻ (A8)where, ܣ is a function of Δݕ. Thereafter, the parameters of all subsequent cable segments can be derived via the following equilibrium equations, 3D TRANSFORMATION EQUATIONS Appendix A: Procedure to Determine Cable Shape in Three Dimensions 270 ܴݔାଵ = ܴݔ + ܨℎ௫ାଵ (A9) ܴݕାଵ = ܨℎ௬ାଵ − ܪݏ݅݊ℎ ൬߱ܽܪ + ܣ൰ (A10) ܴݖାଵ = ܴݖ + ܨℎ௭ାଵ (A11) where, ܨℎ௫, ܨℎ௬, and ܨℎ௭ denote the respective longitudinal, vertical, and transverse components of the hanger force acting at node ݅. The line of action of each hanger force depends upon the shape of the cable. Consequently, the hanger force components cannot be determined independent to the cable coordinates. Nonetheless, using the geometric parameters obtained from Equations (A1)- (A11) and given the magnitude of the tensile force in each hanger |ܨℎ|, the hanger force components at a given node ݅ can be computed as (neglecting the sag effect of the hangers), ܨℎ௫ = |ܨℎ|ۉۇ ߜℎ௫ටߜℎ௫ଶ + ߜℎ௬ଶ + ߜℎ௭ଶیۊ(A12) ܨℎ௬ = |ܨℎ|ۉۇ ߜℎ௬ටߜℎ௫ଶ + ߜℎ௬ଶ + ߜℎ௭ଶیۊ(A13) ܨℎ௭ = |ܨℎ|ۉۇ ߜℎ௭ටߜℎ௫ଶ + ߜℎ௬ଶ + ߜℎ௭ଶیۊ(A14) where, ߜℎ௫ = ℎ௫ − Δݔିଵୀଵ ߜℎ௬ = ℎ௬ − Δyିଵୀଵ JOINT EQUILIBRIUM EQUATIONS Appendix A: Procedure to Determine Cable Shape in Three Dimensions 271 ߜℎ௭ = ℎ௭ − Δzିଵୀଵ As shown in Figure A3, ߜℎ௫, ߜℎ௬, and ߜℎ௭ represent the projected dimensions of the hanger in the ݔ, ݕ, and ݖ directions, respectively. And, ൫ℎ௫, ℎ௬, ℎ௭൯ denote a set of specified coordinates for the end node of hanger ݅ , opposite the cable, measured with respect to the global coordinate system assigned in Figure A1. If, on the other hand, the component of the hanger force in the direction of gravity is known, as opposed to the magnitude of the tensile force, then the other components of the force may be alternatively computed as, ܨℎ௫ = ܨℎ௬ ቆߜℎ௫ߜℎ௬ቇ (A15) ܨℎ௭ = ܨℎ௬ ቆߜℎ௭ߜℎ௬ቇ (A16)Equations (A1)-(A16) allow for the determination of the coordinates of a general three dimensional suspension cable with support reactions at one end equal to ܴ ݔଵ, ܴ ݕଵ, and ܴ ݖଵ. However, since these Figure A3: YZ Section at Node i *Nodes (2… i-1) not shown for clarityℎ௭Bridge Superstructure ܻܼଵܰܰℎ௬ߜℎ௭ߜℎ௬ SuspensionCableHangerAppendix A: Procedure to Determine Cable Shape in Three Dimensions 272 support reactions are typically unknown, iteration is required in order to obtain the correct values of the support reactions, for a given longitudinal span Δ்ܺ, which yield the specified end offsets of the cable (see Figure A1). Δ்ܻ = Δݕୀଵ= ݂ܵ݁ܿ݅݅݁݀ ܸ݈ܽݑ݁ (A17) Δ்ܼ = Δzୀଵ= ݂ܵ݁ܿ݅݅݁݀ ܸ݈ܽݑ݁ (A18)Still, there exist an infinite number of solutions which satisfy the support boundary conditions, and as such, an added parameter must be specified which dictates the sag of the cable curve. For this purpose, with respect to the global coordinate system in Figure A1, the vertical distance (distance in the ‘Y’ direction) from the origin to the cable at, ܺ = Δ்ܺ2 is chosen. This value, referred to hereon as the vertical cable sag, ௬݂, is computed for given set of support reactions as. Δݕୀଵ if there is a hanger at midspan ௬݂ = (A19) Δݕିଵୀଵ+ ݕ ൬ݔ =Δݔ2 ൰ if there is no hanger at midspan where, ݍ = ݊2 and =݊ + 12 Thus, the correct cable coordinates are obtained only when Equations (A17)-(A19) all converge to their desired target values. The entire iterative process is described in the following algorithm which uses a multi-dimensional form of Newton’s Method. Appendix A: Procedure to Determine Cable Shape in Three Dimensions 273 MULTI-DIMENISONAL CABLE SHAPE FINDING ALGORITHM Assumptions: 1. The cable has negligible bending stiffness. 2. The material of the cable obeys Hooke’s Law. 3. Infinitesimal strain theory applies. 4. The sag effect of the hangers is neglected. Initial Inputs: 1. The self-weight of the suspension cable, ߱ = ߛܣ. 2. An array containing Δݔଵ… 3. An array containing ܨℎଵ…, or alternatively ܨℎ௬ଵ… 4. Target values for ௬݂, Δ்ܻ , and ΔZ். Also, the tolerance accepted in achieving the target values, denoted as ܱܶܮ. 5. An initial guess for the support reactions at Node 1, denoted as ܴݔܽଵ, ܴݕܽଵ, and ܴݖܽଵ. Steps: 1. Set the support reactions at Node 1 equal to ሺܴݔܽଵ, ܴݕܽଵ, ܴݖܽଵሻ. 2. Compute Δz using Equation (A7). 3. Solve for Δݕ using Equation (A8). 4. Decompose ܨℎ using Equations (A12)-(A14). 5. Compute ܴݔାଵ, ܴݕାଵ, and ܴݖାଵ using the joint equilibrium equations, (A9)-(A11). 6. Repeat Steps 2 through 5 for ݅ = 1 … ݊. 7. Compute the vertical cable end offset (denoted as Δ்ܻ ), the transverse cable end offset (denoted as Δ்ܼ), and the vertical cable sag (denoted as ௬݂) corresponding to ሺܴݔܽଵ, ܴݕܽଵ, ܴݖܽଵሻ using Equations (A17)-(A19) combined with the geometric parameters obtained in Steps 2 through 6. 8. Determine the error in the target parameters, Δ்ܻ ா = Δ்ܻ − Δ்ܻ , ΔZ்ா = ΔZ் − ΔZ், and ௬݂ா = ௬݂ − ௬݂ 9. Check convergence a. If ቀหΔ்ܻ ாห ∧ หΔZ்ாห ∧ ቚ ௬݂ாቚቁ > ܱܶܮ advance to Step 10. b. If ቀหΔ்ܻ ாห ∧ หΔZ்ாห ∧ ቚ ௬݂ாቚቁ ≤ ܱܶܮ advance to Step 13. See Figure A1 Appendix A: Procedure to Determine Cable Shape in Three Dimensions 274 10. Numerically compute the Jacobian Matrix, ሾܬሿ =ۏێێێۍ ௗೌ್ௗோ௫ೌ್ௗೌௗோ௬ೌௗೌௗோ௭ೌୢଢ଼ೌ್ௗோ௫ೌ್ୢଢ଼ೌௗோ௬ೌୢଢ଼ೌௗோ௭ೌୢೌ್ௗோ௫ೌ್ୢೌௗோ௬ೌୢೌௗோ௭ೌ ےۑۑۑې a. Set, ܴݔܾଵ = ሺ1 − ܱܶܮሻܴݔܽଵ, ܴݕܿଵ = ሺ1 − ܱܶܮሻܴݕܽଵ, and ܴݖ݀ଵ =ሺ1 − ܱܶܮሻܴݖܽଵ. b. Repeat Steps 2 through 6 except with the support reactions at Node 1 equal to ሺܴݔܾଵ, ܴݕܽଵ, ܴݖܽଵሻ and label the cable parameters in Step 7 ൫Δ்ܻ , Δ்ܼ, ௬݂൯. c. Compute first column of the Jacobian Matrix, ௗೌ್ௗோ௫ೌ್ =ೌି್ோ௫భିோ௫భ ; ୢଢ଼ೌ್ௗோ௫ೌ್ =ଢ଼ೌିଢ଼್ோ௫భିோ௫భ ; ୢೌ್ௗோ௫ೌ್ =ೌି್ோ௫భିோ௫భ d. Repeat Steps 2 through 6 except with the support reactions at Node 1 equal to ሺܴݔܽଵ, ܴݕܿଵ, ܴݖܽଵሻ and label the cable parameters in Step 7 ൫Δ்ܻ , Δ்ܼ, ௬݂൯. e. Compute second column of the Jacobian Matrix, ௗೌௗோ௬ೌ =ೌିோ௬భିோ௬భ ; ୢଢ଼ೌௗோ௬ೌ =ଢ଼ೌିଢ଼ோ௬భିோ௬భ ; ୢೌௗோ௬ೌ =ೌିோ௬భିோ௬భ f. Repeat Steps 2 through 6 except with the support reactions at Node 1 equal to ሺܴݔܽଵ, ܴݕܽଵ, ܴݖ݀ଵሻ and label the cable parameters in Step 7 ൫Δ்ܻ ௗ, Δ்ܼௗ, ௬݂ௗ൯. g. Compute third column of Jacobian Matrix, ௗೌௗோ௭ೌ =ೌିோ௭భିோ௭ௗభ ; ୢଢ଼ೌௗோ௭ೌ =ଢ଼ೌିଢ଼ோ௭భିோ௭ௗభ ; ୢೌௗோ௭ೌ =ೌିோ௭భିோ௭ௗభ 11. Update the initial guess values for the support reactions at Node 1, ሾܴܽሿோௐ = ሾܴܽሿ +ሾΔܴܽሿ a. Set, ሾܴܽሿ = ܴݔܽଵܴݕܽଵܴݖܽଵ൩ b. Compute the requisite change in the support reactions, ሾΔܴܽሿ = ሾܬሿିଵ ௬݂ாΔ்ܻ ாΔZ்ா 12. Repeat Steps 1 through 11 until the convergence criterion in Step 9b is met. 13. With ሺܴݔܽଵ, ܴݕܽଵ, ܴݖܽଵሻ set as the end support reactions at Node 1, compute the local cable coordinates for each cable segment using Equations (A1)-(A16) combined with the geometric parameters obtained in Steps 2 through 6. 14. Convert the local coordinates of each cable segment to the global coordinate system shown in Figure A1. Appendix A: Procedure to Determine Cable Shape in Three Dimensions 275 ADDITIONAL NOTES All initial inputs should be entered as positive or negative values according to the coordinate systems specified in Figure A1 and Figure A2. As ܨℎଵ acts at a support node, its value should be set equal to zero. The convergence of Newton’s Method is sensitive to the initial guess values provided. For general bridge engineering applications, it is recommended that the parabolic approximation be used as a basis when determining the starting values for ܴݔܽଵ, ܴݕܽଵ, and ܴݖܽଵ. If the transverse force component of all hangers is zero ൫ܨℎ௭ଵ… = 0൯, then the third row and third column of the Jacobian Matrix must be omitted to prevent the matrix from becoming singular. To avoid possible convergence problems, the updated guess values for the support reactions at Node 1 should be prevented from changing signs. As such, it is recommended that the following limit, หሾΔܴܽሿห ≤ 0.5หሾܴܽሿห be placed on Step 11 for ݆ = 1 … ݎݓݏሺሾΔܴܽሿሻ. GENERAL COMMENTS Once the correct cable shape has been established using the algorithm presented, other geometric and force parameters can be determined. The magnitude of the tensile force at each end of the cable can be found using Equations (A20) & (A21). ଵܶ = ටܴݔଵଶ + ܴݕଵଶ + ܴݕଵଶ (A20) ܶାଵ = ටܴݔାଵଶ + ܴݕାଵଶ + ܴݕାଵଶ (A21)Also, the angles formed between the ends of the cable and the ݔ, ݕ, and ݖ axes are given by, ߠ௫ଵ = ܽܿݏ ቆ|ܴݔଵ|ଵܶቇ and ߠ௫ାଵ = ܽܿݏ ቆ|ܴݔାଵ|ܶାଵቇ (A22) ߠ௬ଵ = ܽܿݏ ቆ|ܴݔଵ|ଵܶቇ and ߠ௬ାଵ = ܽܿݏ ቆ|ܴݕାଵ|ܶାଵቇ (A23) ߠ௭ଵ = ܽܿݏ ቆ|ܴݖଵ|ଵܶቇ and ߠ௭ାଵ = ܽܿݏ ቆ|ܴݖାଵ|ܶାଵቇ (A24)Appendix A: Procedure to Determine Cable Shape in Three Dimensions 276 And, the unstressed length of the cable may be computed as (refer to Equations (3.10), (3.12) and (3.13)), ܷܵܮ = ܥ − Δୀଵ൩ (A25)where, ܥ =ܪ߱ ݏ݅݊ℎ ൬߱ܽܪ + ܣ൰ − ݏ݅݊ℎሺܣሻ൨ (A26)and, Δ =ܪܽܧܣ ቈ߱Δݕଶ2ܪܽ ܿݐℎ ൬߱ܽ2ܪ ൰ +12 +ܪ2߱ܽ ݏ݅݊ℎ ൬߱ܽܪ ൰ (A27) In some cases, prior to computing the cable shape, it may be desirable to use the unstressed length of the cable as a target parameter rather than the vertical sag in the cable. In those scenarios, the algorithm presented may be easily modified using Equations (A25) to (A27). If desired, the sag effect of the hangers may be factored into Equations (A12) to (A16) by utilizing the equations given in Section 3.1.1. In terms of calculating updated guess values for the support reactions, other multi-dimensional numerical techniques exist which may offer improved convergence and/or computational efficiency. Notwithstanding, for most practical cases, the aforementioned method was found to converge, within a tolerance of 1×10-10, in less than ten iterations.
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Structural and economic evaluation of self-anchored discontinuous hybrid cable bridges Sauer, Devin James 2017
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Title | Structural and economic evaluation of self-anchored discontinuous hybrid cable bridges |
Creator |
Sauer, Devin James |
Publisher | University of British Columbia |
Date Issued | 2017 |
Description | The self-anchored discontinuous hybrid cable bridge (SDHCB) is a novel type of bridge system which has the potential to overcome many of the deficiencies of conventional cable bridge structures while preserving their advantages. To date, research on the system is extremely limited. Accordingly, this thesis examines the structural and economic attributes of the system, as well its constructability. These areas of research are vital in evaluating the utility of the system and in advancing its development. The structural attributes of SDHCBs were studied in this thesis using a systematic approach. First, the behaviour of each of the two basic cable types found in hybrid cable bridges was studied under a wide range of parameters. Then, starting with a bare model of a SDHCB, a series of analyses was performed while progressively expanding the model so that the influence of various parameters and bridge components could be isolated, and accurately assessed. The model parameters were further refined through a cost analysis. Thereafter, upon reaching a complete and detailed model, the influence of various structural parameters was re-assessed, the structural benefits of employing various supplemental design components were appraised, and the constructability of the system was addressed. This work is significant in that it has provided a highly generalized and robust model of a SDHCB. Using this model, it is possible to ascertain how various design parameters such as geometric factors, material properties, and loading conditions affect the structural behaviour, cost, and constructability of the system. Many insights were also obtained from this research which led to the formation of a recommended, universal, design space for the system. In addition, the practicality and adaptability of the system were demonstrated through the development of several innovative construction schemes which aim to reduce construction duration and costs through the creation of multiple work fronts and the elimination of large temporary works. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2017-04-20 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0343995 |
URI | http://hdl.handle.net/2429/61293 |
Degree |
Doctor of Philosophy - PhD |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2017-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
Aggregated Source Repository | DSpace |
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