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Structural and economic evaluation of self-anchored discontinuous hybrid cable bridges Sauer, Devin James 2017

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                                                                                                                                                                                                                                                                                                                                                                                                𝑎𝑎𝑜ðīðī𝑠𝑎ðī𝑠𝑎𝑜ðī𝑐ðī𝑐𝑚ðī𝑔ðīℎðī𝑠𝑏ðī𝑠𝑏𝑜ðī𝑠𝑚ðī𝑠𝑚𝑜ðīð‘Ąð‘ðĩð‘ð‘ð‘ð‘šð‘ð‘“ð‘â„Žð‘ð‘ ð‘ð‘ ð‘Ąð‘ð‘Ąðķðķðŧðķðŧ𝑜ðķ𝑜, ðķ𝑓𝑑ðļðļ𝑐ðļ𝑒𝑓𝑓ðļ𝑠𝑒𝑐ðļð‘Ąð‘Žð‘›ð‘“ð‘“â„Žð‘“ð‘œð‘“ð‘ð‘“ð‘Ķðđ𝑐ðđℎðđℎð‘Ĩ, ðđℎð‘Ķ, ðđℎ𝑧ðđ𝑠ðđ𝑝𝑔ℎℎðĩℎ𝑜ℎ𝑇ℎ𝑇𝑅 ℎ𝑇: ðŋ𝑚ℎ𝑇𝑅𝑜ℎð‘Ĩ, ℎð‘Ķ, ℎ𝑧ðŧðŧ𝑐𝑚, ðŧ𝑐𝑠ðŧð‘†ð‘…ðžð‘”ðžð‘Ąðūðū𝑟ðŋ𝑐ðŋ𝑐𝑚 _1ðŋ𝑐𝑚_2ðŋ𝑐𝑜ðŋℎðŋ𝑚ðŋ𝑚𝑜ðŋ𝑝ðŋ𝑅 ðŋℎ: ðŋ𝑚ðŋ𝑠ðŋ𝑆𝑅 ðŋ𝑠: ðŋ𝑚ðŋ𝑆𝑅𝑜𝑀𝑀𝑆𝑂𝑛𝑁ð‘Ģ𝑁𝑓𝑝, 𝑞𝑄𝑄𝑐𝑚, 𝑄𝑐𝑠𝑄𝑓𝑄ðđ𝑎𝑛, 𝑄ðŧð‘Žð‘Ÿð‘ð‘„â„Žð‘„ð‘ ð‘„ð‘ ð‘Ąð‘„ð‘ ð‘Ąð‘š , ð‘„ð‘ ð‘Ąð‘ ð‘„ð‘Ąð‘…ð‘ð‘…ð‘Ĩ, 𝑅ð‘Ķ, 𝑅𝑧𝑆𝑅 𝑓: ðŋ𝑚𝑇𝑇𝑐𝑠𝑇ðī, 𝑇ðĩ𝑈𝑆ðŋ𝑈𝑆ðŋ𝑐 , 𝑈𝑆ðŋ𝑝𝑈𝑆ðŋ𝑜, 𝑈𝑆ðŋ𝑓𝑉ðī, 𝑉ðĩ𝑉𝑐𝑚,𝑉𝑐𝑠𝑉𝑝𝑓𝑉𝑠𝑉𝑇𝑉𝑂ðŋð‘ ð‘Žð‘Šð‘Ąð‘Ĩ𝑝ð‘Ĩ𝑝𝑖ð‘Ķðĩð‘Ķ𝑐 , ð‘Ķð‘ð‘§ð›žð‘ð›žð‘ ð‘‘ð‘™ð›―ð‘Ąð›ū𝑐ð›ū𝑐𝑚ð›ū𝑠ð›ūð‘ ð‘Ąð›ūð‘Ąð›ŋð›ŋ𝑆𝑂(ð›ŋ)ðđðŋ, (ð›ŋ)𝑃ðŋð›ŋ𝑐ð›ŋ𝑒ð›ŋℎ, ð›ŋð‘Ģð›ŋ𝑟ð›ŋ𝑠𝑓ð›ŋ𝑎ð›ŋðķð›ŋ𝑓ð›ŋℎð›ŋℎð‘Ĩ, ð›ŋℎð‘Ķ, ð›ŋℎ𝑧ð›ŋðŋ𝑐ð›ŋ𝜃𝑐ð›ŋ𝜎𝑐ð›Ĩ𝑒ð›Ĩ𝑒𝑜 , ð›Ĩ𝑒𝑓ð›Ĩð‘Ĩ, ð›Ĩð‘Ķ, ð›Ĩ𝑧ð›Ĩ𝑋𝑇 , ð›Ĩ𝑌𝑇 , ð›Ĩ𝑍𝑇ð›Ĩ𝜔𝑠𝜖𝑐𝜂𝜃ðī, 𝜃ðĩ𝜃𝑐𝜃ð‘Ĩ, 𝜃ð‘Ķ , 𝜃𝑧𝜆ΛΞ𝜉𝜌𝑐𝑚, ðœŒð‘ð‘ ðœŒâ„ŽðœŒð‘ ð‘ĄðœŒð‘ĄðœŽð‘Žð‘™ð‘™ð‘œð‘Ī𝜎𝑎,𝐷ðŋ, 𝜎𝑎,ðŋðŋ𝜎𝑏,ðŋðŋ𝜎𝑐 ðđ𝑐 ðī𝑐⁄𝜎𝑐𝑜, 𝜎𝑐𝑓𝜎𝑐𝑚, ðœŽð‘ð‘ ðœŽâ„ŽðœŽð‘ ð‘ĄðœŽð‘ ðœŽð‘Ąð›īðŧð‘ ð‘Ąð‘ , ð›īð‘‰ð‘ ð‘Ąð‘ ð›đ𝑐 𝜔𝑐𝑎/ðŧð›đ𝑐𝑝 𝜔𝑐𝑝𝑎/ðŧ?Ė…?𝑝 (𝜔𝑐 + 𝜔𝑠)ðŋ𝑚/ðŧ𝜔𝑐𝜔𝑐𝑚𝜔𝑐𝑝𝜔𝑝𝜔𝑠𝜔𝑠𝑚, 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 𝑓ðŧ𝑑2ð‘Ķ𝑑ð‘Ĩ2=𝜔𝑐ðŧ√1+ (𝑑ð‘Ķ𝑑ð‘Ĩ)2𝜃ðĩ  𝜃ðī 𝑓 𝒚 𝒙 𝑎2⁄  𝑎2⁄  ℎ 𝑉ðī 𝑉ðĩ 𝑇ðĩ 𝑇ðī ðŧ ðŧ ðđ𝑐 ðđ𝑐 ÎĻ𝑐 =𝜔𝑐𝑎ðŧÎĐ =ℎ𝑎𝑑ð‘Ķ𝑑ð‘Ĩ= 𝑠𝑖𝑛ℎ (ÎĻ𝑐ð‘Ĩ𝑎+ ðī)  ð‘Ķ =𝑎ÎĻ𝑐𝑐𝑜𝑠ℎ (ÎĻ𝑐ð‘Ĩ𝑎+ ðī) + ðĩð‘Ķ(ð‘Ĩ = 0) = 0 ð‘Ķ(ð‘Ĩ = 𝑎) = ℎ  ðī = 𝑎𝑠𝑖𝑛ℎ [ÎĻ𝑐ÎĐ2𝑠𝑖𝑛ℎ (ÎĻ𝑐2 )]−ÎĻ𝑐2ðĩ = −𝑎ÎĻ𝑐𝑐𝑜𝑠ℎ (ðī)𝑉ðī = ðŧ𝑑ð‘Ķ𝑑ð‘Ĩ|ð‘Ĩ=0=𝜔𝑐𝑎ÎĻ𝑐𝑠𝑖𝑛ℎ (ðī)𝑉ðĩ = ðŧ𝑑ð‘Ķ𝑑ð‘Ĩ|ð‘Ĩ=𝑎=𝜔𝑐𝑎ÎĻ𝑐𝑠𝑖𝑛ℎ(ÎĻ𝑐 + ðī)ℎ 𝑇ðī 𝑇ðĩℎ𝑇ðī = √ðŧ2 + 𝑉ðī2 =𝜔𝑐𝑎ÎĻ𝑐𝑐𝑜𝑠ℎ(ðī)𝑇ðĩ = √ðŧ2 + 𝑉ðĩ2 =𝜔𝑐𝑎ÎĻ𝑐𝑐𝑜𝑠ℎ(ÎĻ𝑐 + ðī)𝜃ðī = ð‘Žð‘Ąð‘Žð‘›ð‘‘ð‘Ķ𝑑ð‘Ĩ|ð‘Ĩ=0= ð‘Žð‘Ąð‘Žð‘›[𝑠𝑖𝑛ℎ(ðī)]𝜃ðĩ = ð‘Žð‘Ąð‘Žð‘›ð‘‘ð‘Ķ𝑑ð‘Ĩ|ð‘Ĩ=𝑎= ð‘Žð‘Ąð‘Žð‘›[𝑠𝑖𝑛ℎ(ÎĻ𝑐 +ðī)]ðķðķ = âˆŦ √1 + (𝑑ð‘Ķ𝑑ð‘Ĩ)2𝑑ð‘Ĩ𝑎0=𝑎ÎĻ𝑐[𝑠𝑖𝑛ℎ(ÎĻ𝑐 + ðī) − 𝑠𝑖𝑛ℎ (ðī)]𝑈𝑆ðŋ Δ𝑒ðķ = 𝑈𝑆ðŋ + Δ𝑒∆𝑒  =ðŧðļ𝑐ðī𝑐âˆŦ [1 + (𝑑ð‘Ķ𝑑ð‘Ĩ)2] 𝑑ð‘Ĩ𝑎0=ð›ū𝑐𝑎2ÎĻ𝑐ðļ𝑐[ÎĻ𝑐ÎĐ22ð‘ð‘œð‘Ąâ„Ž (ÎĻ𝑐2) +12+12ÎĻ𝑐𝑠𝑖𝑛ℎ(ÎĻ𝑐)]ðī𝑐 ðļ𝑐 ð›ū𝑐ðķ Δ𝑒𝑈𝑆ðŋ = ðķ − Δ𝑒ÎĻ𝑐ðđ𝑐ðđ𝑐 = ðŧðŋ𝑐𝑎= ðŧ√1 + ÎĐ2ÎĻ𝑐ÎĻ𝑐 =𝜔𝑐𝑎ðđ𝑐√1 + ÎĐ2𝜎𝑐ÎĻ𝑐 =ð›ū𝑐𝑎𝜎𝑐√1 + ÎĐ2𝜔𝑐𝑝 𝜔𝑐𝑝 = 𝜔𝑐√1+ ÎĐ2ÎĐ𝑑2ð‘Ķ𝑑ð‘Ĩ2=𝜔𝑐𝑝ðŧ𝑑ð‘Ķ𝑑ð‘Ĩ=ÎĻ𝑝2𝑎(2ð‘Ĩ − 𝑎) + ÎĐð‘Ķ =ÎĻ𝑝ð‘Ĩ2𝑎(ð‘Ĩ − 𝑎) + ÎĐð‘Ĩ𝜔𝑐  𝜔𝑐  𝜔𝑐𝑝  ÎĻ𝜔𝑐𝑝  ÎĻ𝑝 =𝜔𝑐𝑝𝑎ðŧ𝑉ðī = ðŧ𝑑ð‘Ķ𝑑ð‘Ĩ|ð‘Ĩ=0=𝜔𝑐𝑝𝑎ÎĻ𝑝(ÎР−ÎĻ𝑝2)𝑉ðĩ = ðŧ𝑑ð‘Ķ𝑑ð‘Ĩ|ð‘Ĩ=𝑎=𝜔𝑐𝑝𝑎ÎĻ𝑝(ÎĐ +ÎĻ𝑝2)𝑇ðī = √ðŧ2 + 𝑉ðī2 =𝜔𝑐𝑝𝑎ÎĻ𝑝√1 + (ÎР−ÎĻ𝑝2)2𝑇ðĩ = √ðŧ2 + 𝑉ðĩ2 =𝜔𝑐𝑝𝑎ÎĻ𝑝√1 + (ÎĐ +ÎĻ𝑝2)2𝜃ðī = ð‘Žð‘Ąð‘Žð‘›ð‘‘ð‘Ķ𝑑ð‘Ĩ|ð‘Ĩ=0= ð‘Žð‘Ąð‘Žð‘›(ÎР−ÎĻ𝑝2)𝜃ðĩ = ð‘Žð‘Ąð‘Žð‘›ð‘‘ð‘Ķ𝑑ð‘Ĩ|ð‘Ĩ=𝑎= ð‘Žð‘Ąð‘Žð‘›(ÎĐ +ÎĻ𝑝2)ðķ = âˆŦ √[1 + (𝑑ð‘Ķ𝑑ð‘Ĩ)2] 𝑑ð‘Ĩ𝑎0ðķ =𝑎2ÎĻ𝑝[(ÎĐ +ÎĻ𝑝2)√1 + ÎĐ2 + ÎĐÎĻ𝑝 +ÎĻ𝑝24− (ÎР−ÎĻ𝑝2)√1 + ÎĐ2 − ÎĐÎĻ𝑝 +ÎĻ𝑝24+ 𝑎𝑠𝑖𝑛ℎ (ÎĐ +ÎĻ𝑝2)− 𝑎𝑠𝑖𝑛ℎ (ÎР−ÎĻ𝑝2)]√1 + 𝑘 𝑘√1 + 𝑘   = ∑(−1)𝑚(2𝑚)!(1 − 2𝑚)(𝑚!)2(4𝑚)𝑘𝑚∞𝑚=0𝑘 = 0 |𝑘| â‰Ī  1ÎĐÎĐ = 0 ðķ𝑎𝑝𝑝𝑟𝑜ð‘Ĩ|ÎĐ=0 = âˆŦ [1 +12(𝑑ð‘Ķ𝑑ð‘Ĩ)2]𝑑ð‘Ĩ = ðŋ𝑐 [1 +124(𝜔𝑐ðŋ𝑐ðŧℎ𝑜𝑟𝑖𝑧)2]ðŋ𝑐0ðŧℎ𝑜𝑟𝑖𝑧 ðŋ𝑐ÎĐðķ𝑠𝑖𝑚𝑝𝑙𝑖𝑓𝑖𝑒𝑑 = 𝑎√1 + ÎĐ2 [1 +124(ÎĻ𝑝1 + ÎĐ2)2]∆𝑒  =ðŧðļ𝑐ðī𝑐âˆŦ [1 + (𝑑ð‘Ķ𝑑ð‘Ĩ)2] 𝑑ð‘Ĩ𝑎0=ð›ū𝑐𝑎2√1 + ÎĐ2ÎĻ𝑝ðļ𝑐[1 + ÎĐ2 +112ÎĻ𝑝2]ÎĻ𝑝ÎĻ𝑝 =𝜔𝑐𝑎ðđ𝑐(1 + ÎĐ2)ÎĻ𝑝 =ð›ū𝑐𝑎𝜎𝑐(1 + ÎĐ2) ðŋ𝑐  𝑓ℎ𝑜𝑟𝑖𝑧 𝑎2⁄  𝑎2⁄  ðŧℎ𝑜𝑟𝑖𝑧 =𝜔𝑐ðŋ𝑐28𝑓ℎ𝑜𝑟𝑖𝑧 ðŧℎ𝑜𝑟𝑖𝑧 ðŧ ðŧ =𝜔𝑐𝑎28𝑓ℎ𝑜𝑟𝑖𝑧= ðŧℎ𝑜𝑟𝑖𝑧 (𝑎ðŋ𝑐)2 ÎĻ𝑝𝑓𝑝ÎĻ𝑝 =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 ï€Ū1042 ï€Ū1043 ï€Ū1044 ï€Ū1045 ï€Ū104ðœļ𝒄𝒂𝝈𝒄      ð‘ķð‘đ      𝟖𝒇𝒑𝒂(𝟏 + ðœī𝟐) |(𝒚𝒄 − 𝒚𝒑)𝒎𝒂𝒙|𝒂 𝜎𝑐 𝑎 𝑓𝑝 𝜎𝑐 𝑎 < ðķð›ū𝑐 𝑎 𝛚 𝜎𝑐𝝈𝒄 𝒂 𝝈𝒄 𝝈𝒄 𝒂 𝝈𝒄 (𝒚𝒄 − 𝒚𝒑)𝒎𝒂𝒙 (𝒚𝒄 − 𝒚𝒑)𝒎𝒂𝒙 ÎĐ𝑈𝑆ðŋ𝑝𝑈𝑆ðŋ𝑐ðļ𝑐 ðŋ𝑚 ð›ū𝑐 𝜎𝑐ðŋ𝑚 1.0 0.5 0.0   ðŋ𝑚4 1.0 0.5 0.0   𝛀 = 𝟎𝛀 = 𝟎. 𝟓𝛀 = 𝟏𝛀 = 𝟏. 𝟓𝛀 = 𝟐ðļ𝑐0 0.02 0.04 0.06 0.08 0.1 0.12 0.1402 ï€Ū1064 ï€Ū1066 ï€Ū1068 ï€Ū1060 0.02 0.04 0.06 0.08 0.1 0.12 0.1402 ï€Ū1064 ï€Ū1066 ï€Ū1068 ï€Ū106|𝑞𝑚ð‘ģ𝒄 − 𝑞𝑚ð‘ģ𝒑|𝒂 ðœļ𝒄𝒂𝝈𝒄      ð‘ķð‘đ      𝟖𝒇𝒑𝒂(𝟏 + ðœī𝟐) 𝜎𝑐 𝑎 𝜎𝑐 𝑓𝑝 𝜎𝑐𝑜 𝜎𝑐𝑓ðķ𝑜 = ðŋ𝑐𝑜 [1 +124(1 + ÎĐ2)(ð›ū𝑐ðŋ𝑐𝑜𝜎𝑐𝑜)2]Δ𝑒𝑜 =ðŋ𝑐𝑜𝜎𝑐𝑜ðļ𝑐+112ðļ𝑐(ð›ū𝑐2ðŋ𝑐𝑜3𝜎𝑐𝑜)ðķ𝑓 = ðŋ𝑐𝑜 + Îīðŋ𝑐 +124(1 + ÎĐ2)(ð›ū𝑐𝜎𝑐𝑓)2(ðŋ𝑐𝑜 + ð›ŋðŋ𝑐)3Δ𝑒𝑓 =(ðŋ𝑐𝑜 + Îīðŋ𝑐)𝜎𝑐𝑓ðļ𝑐++112ðļ𝑐ð›ū𝑐2𝜎𝑐𝑓(ðŋ𝑐𝑜 + ð›ŋðŋ𝑐)3 𝑓𝑜 𝑓𝑜 − ð›ŋ𝑓 𝑎𝑜  𝜎𝑐𝑓 = 𝜎𝑐𝑜 + ð›ŋ𝜎𝑐  + 𝜎𝑐𝑜  𝜎𝑐𝑜  𝜎𝑐𝑓  ℎ𝑜 ℎ𝑜 + ð›ŋℎ 𝑎𝑜 + ð›ŋ𝑎 𝑈𝑆ðŋ𝑜 = 𝑈𝑆ðŋ𝑓Îīðŋ𝑐ðŋ𝑐𝑜−Îīðŋ𝑐ðŋ𝑐𝑜𝜎𝑐𝑓ðļ𝑐=(𝜎𝑐𝑓 − 𝜎𝑐𝑜)ðļ𝑐−ð›ū𝑐2𝑎224[1𝜎𝑐𝑓2(ðŋ𝑐𝑜 + Îīðŋ𝑐)3ðŋ𝑐𝑜3 −1𝜎𝑐𝑜2]+ð›ū𝑐2ðŋ𝑐𝑜212ðļ𝑐𝜎𝑐𝑓[(ðŋ𝑐𝑜 + Îīðŋ𝑐)3ðŋ𝑐𝑜3 −𝜎𝑐𝑓𝜎𝑐𝑜](ð›ŋðŋ𝑐 ≩ ðŋ𝑐𝑜)Îīðŋ𝑐ðŋ𝑐𝑜=(𝜎𝑐𝑓 − 𝜎𝑐𝑜)ðļ𝑐+ð›ū𝑐2𝑎224(1𝜎𝑐𝑜2−1𝜎𝑐𝑓2)Linear Term Nonlinear Term ~1~0~0ðļ𝑐ðļ𝑠𝑒𝑐ðļ𝑠𝑒𝑐 = (𝜎𝑐𝑓 − 𝜎𝑐𝑜)ðŋ𝑐𝑜Îīðŋ𝑐ðļ𝑠𝑒𝑐 =11ðļ𝑐+ð›ū𝑐2𝑎2(𝜎𝑐𝑓 + 𝜎𝑐𝑜)24𝜎𝑐𝑓2𝜎𝑐𝑜2 𝝈𝒄𝒇 (0,0) ðŋ𝑐𝑜 ðœđð‘ģ𝒄ð‘ģ𝒄𝒐 𝜎𝑐𝑜 𝑎𝒄 𝟏 ð›ŋ𝜎𝑐 ð›ŋ𝜎𝑐ð›ŋ𝜎𝑐 ð›ŋ𝜎𝑐ð›ŋ𝜎𝑐ð›ŋ𝜎𝑐 ≩ 𝜎𝑐𝑜𝜎𝑐𝑓 = 𝜎𝑐𝑜 𝜎𝑐𝑓 𝜎𝑐𝑜ðļð‘Ąð‘Žð‘› =11ðļ𝑐+ð›ū𝑐2𝑎212𝜎𝑐𝑜3 ð›ŋðŋ𝑐ðŋ𝑐𝑜 𝜎𝑐𝑜 𝑎𝒔𝒆𝒄 𝟏 𝜎𝑐𝑓 ð›ŋ𝜎𝑐 𝜎𝑐𝑜ðļ𝑐𝛔𝒄𝒐 = 𝟏𝟎𝟎ð‘ī𝑷𝒂𝛔𝒄𝒐 = 𝟐𝟎𝟎ð‘ī𝑷𝒂𝛔𝒄𝒐 = 𝟑𝟎𝟎ð‘ī𝑷𝒂𝛔𝒄𝒐 = 𝟒𝟎𝟎ð‘ī𝑷𝒂𝛔𝒄𝒐 = 𝟓𝟎𝟎ð‘ī𝑷𝒂𝛔𝒄𝒐 = 𝟔𝟎𝟎ð‘ī𝑷𝒂𝛔𝒄𝒐 = 𝟕𝟎𝟎ð‘ī𝑷𝒂𝛔𝒄𝒐 = 𝟖𝟎𝟎ð‘ī𝑷𝒂𝛔𝒄𝒐 = 𝟗𝟎𝟎ð‘ī𝑷𝒂𝛔𝒄𝒐 = 𝟏𝟎𝟎𝟎ð‘ī𝑷𝒂ðļ𝑐 ð›ū𝑐𝒂 (𝒎) 𝑎𝒕𝒂𝒏𝑎𝒄 𝜎𝑐𝑜 𝜎𝑐𝑜 𝑎 𝑃ð›ŋ𝜃𝑐 ≩ 𝜃𝑐Îī𝑐 = 𝜖𝑐ðŋ𝑐 =𝜎𝑐ðļ𝑒𝑓𝑓𝑎𝑐𝑜𝑠𝜃𝑐ðļ𝑒𝑓𝑓 ðļ𝑠𝑒𝑐 ðļð‘Ąð‘Žð‘›ð›ŋð‘Ģ =Îī𝑐𝑠𝑖𝑛𝜃𝑐=𝜎𝑐ðļ𝑒𝑓𝑓(1𝑠𝑖𝑛𝜃𝑐1𝑐𝑜𝑠𝜃𝑐)𝑎𝑎𝜃𝑐 = 45°𝜃𝑐 = 45°21.5° â‰Ī 𝜃𝑐 â‰Ī 26.5°𝜃𝑐 = 26.5°‘𝑎’   0 7.5 15 22.5 30 37.5 45 52.5 60 67.5 75 82.5 9000.511.522.533.544.55ð›ŋð‘Ģ 𝑎 ℎ 𝑃 Conventional Inclination of Longest Stay in a Cable-Stayed Bridge  ðœ―ð’„ (𝒅𝒆𝒈𝒓𝒆𝒆𝒔) ðœđ?Ė‚? ð›ŋ𝜃𝑐 𝜃𝑐   𝑖  ð‘Ķ𝑖 =ðŧ𝜔𝑐𝑐𝑜𝑠ℎ (𝜔𝑐ð‘Ĩ𝑖ðŧ+ ðī𝑖) + ðĩ𝑖 𝑁1â€Ķ𝑛 𝑇𝑚𝑎ð‘Ĩ𝑆1â€Ķ𝑛 ðŧðđℎ𝑖 {ð‘Ĩ𝑖 , ð‘Ķ𝑖}𝜔𝑐 Δð‘Ĩ𝑖𝑓 Δy𝑖ðŋ𝑚𝑓 ðŋ𝑚2 ð›Ĩð‘Ĩ𝑖  ð›Ĩð‘Ķ𝑖  ð‘Ķ𝑖  ðđℎ𝑖 ð‘Ĩ𝑖  ðŧ 𝑇𝑚𝑎ð‘Ĩ 𝜔𝑐  𝜔𝑐  ðŧ𝑑ð‘Ķ𝑖𝑑ð‘Ĩ𝑖|ð‘Ĩ𝑖=ð›Ĩð‘Ĩ𝑖 ðŧ ðŧ𝑑ð‘Ķ𝑖𝑑ð‘Ĩ𝑖|ð‘Ĩ𝑖=0 ðŧ ðđℎ1 ðđℎ𝑖−1 ðđℎ𝑖+1 ðđℎ𝑛 𝑆1 𝑆𝑖−1 𝑆𝑖 𝑆𝑖+1 𝑆𝑛 𝑁1 𝑁𝑖−1 𝑁𝑖  𝑁𝑖+1 𝑁𝑛 Free Body Diagram of Segment i ðī𝑖 = 𝑎𝑠𝑖𝑛ℎ [𝜔𝑐Δð‘Ķ𝑖2ðŧ𝑠𝑖𝑛ℎ (𝜔𝑐Δð‘Ĩ𝑖2ðŧ )]−𝜔𝑐Δð‘Ĩ𝑖2ðŧðĩ𝑖 = −ðŧ𝜔𝑐𝑐𝑜𝑠ℎ (ðī𝑖) 𝑛 + 1 Δð‘Ķ1â€Ķ𝑛 ðŧðŧ Δð‘Ķ1â€Ķ𝑛𝑖 = 12ðŧ𝑑ð‘Ķ1𝑑ð‘Ĩ1|ð‘Ĩ1=0= ðđℎ1𝑟𝑒𝑑ð‘Ē𝑐𝑒𝑠 ð‘Ąð‘œâ†’        ðŧ𝑠𝑖𝑛ℎ(ðī1) =ðđℎ12𝑖 = 2â€Ķ𝑛 ðŧ𝑑ð‘Ķ𝑖𝑑ð‘Ĩ𝑖|ð‘Ĩ𝑖=0= ðŧ𝑑ð‘Ķ𝑖−1𝑑ð‘Ĩ𝑖−1|ð‘Ĩ𝑖−1=Δð‘Ĩ𝑖−1+ ðđℎ𝑖𝑟𝑒𝑑ð‘Ē𝑐𝑒𝑠 ð‘Ąð‘œâ†’        ðŧ [𝑠𝑖𝑛ℎ(ðī𝑖) − 𝑠𝑖𝑛ℎ (𝜔𝑐Δð‘Ĩ𝑖−1ðŧ+ ðī𝑖−1)] = ðđℎ𝑖Δð‘Ķ1â€Ķ𝑛 𝑓ðŧ  𝑓 =∑Δy𝑖𝑛𝑖=1ðŧ      𝜔𝑐 = ð›ū𝑐ðī𝑐 Δð‘Ĩ1â€Ķ𝑛 ðđℎ1â€Ķ𝑛 𝑓𝑇𝑇𝑂ðŋ ðŧðī Δð‘Ķ1â€Ķ𝑛 ðŧðī  𝑓ðī ðŧðīΔð‘Ķ1â€Ķ𝑛 𝑓ðļ = 𝑓𝑇 − 𝑓ðī  𝑓ðļ > 𝑇𝑂ðŋ 𝑓ðļ â‰Ī 𝑇𝑂ðŋ 𝑑ðŧðīðĩ𝑑𝑓ðīðĩ= (ðŧðī−ðŧðĩ𝑓ðī−𝑓ðĩ) ðŧðĩ = (1 − 𝑇𝑂ðŋ)ðŧðī Δð‘Ķ1â€Ķ𝑛 ðŧðĩ  𝑓ðĩ ðŧðĩΔð‘Ķ1â€Ķ𝑛 ðŧðī 𝑁ðļ𝑊 = ðŧðī 𝑂ðŋ𝐷 + 𝑓ðļ (𝑑ðŧðīðĩ𝑑𝑓ðīðĩ)  ðŧðīΔð‘Ķ1â€Ķ𝑛   Δð‘Ĩ1â€Ķ𝑛 ðđℎ1â€Ķ𝑛ðđℎ1 Δð‘Ĩ1 ðŧðŧðī ðŧðī 𝑁ðļ𝑊ðŧðī 𝑁ðļ𝑊 â‰Ĩ 0.5ðŧðī 𝑂ðŋ𝐷   𝑈𝑆ðŋ = 2 [∑ðķ𝑖 − Δ𝑖𝑛𝑖=1]  ðķ𝑖 =ðŧ𝜔𝑐[𝑠𝑖𝑛ℎ (𝜔𝑐Δð‘Ĩ𝑖ðŧ+ ðī𝑖) − 𝑠𝑖𝑛ℎ (ðī𝑖)]  Δ𝑒𝑖 =ðŧΔð‘Ĩ𝑖ðļ𝑐ðī𝑐[𝜔𝑐Δð‘Ķ𝑖22ðŧΔð‘Ĩð‘–ð‘ð‘œð‘Ąâ„Ž (𝜔𝑐Δð‘Ĩ𝑖2ðŧ) +12+ðŧ2𝜔𝑐Δð‘Ĩ𝑖𝑠𝑖𝑛ℎ (𝜔𝑐Δð‘Ĩ𝑖ðŧ)] 𝑆𝑚,ð‘Ķ(ð‘Ĩ)|𝑆𝑚 =1ðŧ[∑ ðđℎ𝑗 (ð‘Ĩ −∑𝜆𝑖𝑗𝑖=1)+𝜔𝑐ð‘Ĩ22− 𝑉𝑠ð‘Ĩ𝑚−1𝑗=1]ð‘Ĩ ð‘Ķ𝑉𝑠 =12∑ ðđℎ𝑖 +𝜔𝑐ðŋ𝑚2𝑁ℎ𝑎𝑛𝑔𝑒𝑟𝑠𝑖=1ðŧ =1𝑓[𝑉𝑠ðŋ𝑚2− 𝜔𝑐ðŋ𝑚28−∑ðđℎ𝑗 (ðŋ𝑚2−∑𝜆𝑖𝑗𝑖=1)𝑁∗𝑗=1]𝑁ℎ𝑎𝑛𝑔𝑒𝑟𝑠−12, 𝑁∗ =𝑁ℎ𝑎𝑛𝑔𝑒𝑟𝑠2    𝜔𝑐𝜆 𝜔𝑠 𝑁1â€Ķ ðŋ𝑚𝑆1â€Ķ ðŧ𝑁ℎ𝑎𝑛𝑔𝑒𝑟𝑠 𝑉𝑠𝜔𝑐 Îŧ𝑚𝑓 ðđℎ𝑚ð‘Ķ ðđℎ𝑚+1 ð‘Ĩ ðŧ ðđℎ1 ðđℎ𝑚 ðđℎ𝑚−2 𝑉𝑠 ðđℎ𝑚−1 𝜆1 𝜆𝑚−2 𝜆𝑚−1 𝜆𝑚 𝜆𝑚+1 𝑁1 𝑁𝑚−2 𝑁𝑚−1 𝑁𝑚 𝑁𝑚+1 ðŧ ðŋ𝑚2 ðđℎ2 𝜆2 𝑓 𝑁2 𝑆1 𝑆2 𝑆𝑚−2 𝑆𝑚−1 𝑆𝑚 𝑆𝑚+1 (𝜔𝑐+𝜔𝑠)ðŋ𝑚28𝑓, ðŧ =(𝜔𝑐+𝜔𝑠)ðŋ𝑚28𝑓−𝜔𝑠𝜆28𝑓,ð‘Ķ(ð‘Ĩ) =(𝜔𝑐 +𝜔𝑠)ð‘Ĩ2ðŧ[ð‘Ĩ − ðŋ𝑚]ð‘Ĩðķ𝑚 = 𝜆𝑚√1+ ÎĐ𝑚2 [1 +124ÎĻ𝑐𝑚2(1 + ÎĐ𝑚2)]∆𝑒𝑚 =ðŧ𝜆𝑚ðļ𝑐ðī𝑐[1 + ÎĐ𝑚2 +112ÎĻ𝑐𝑚2(1 + ÎĐ𝑚2)]ÎĻ𝑐𝑚 =𝜔𝑐𝜆𝑚ðŧÎĐ𝑚 =ℎ𝑚𝜆𝑚 𝜆𝑚𝜆 ðŧℎ𝑚ℎ𝑚 = ð‘Ķ𝑚 − ð‘Ķ𝑚−1ð‘Ķ𝑚 𝑚ð‘Ķ𝑚 ðŧ𝑈𝑆ðŋð‘‡ð‘œð‘Ąð‘Žð‘™ = ∑ (𝑈𝑆ðŋ𝑚)𝑁ℎ𝑎𝑛𝑔𝑒𝑟𝑠+1𝑚=1 = ∑ (ðķ𝑚 − Δ𝑒𝑚)𝑁ℎ𝑎𝑛𝑔𝑒𝑟𝑠+1𝑚=1ð‘Ĩð‘Ķ𝑐 ð‘Ķ𝑝Îŧ ðŋ𝑚 𝜔𝑐 𝜔𝑠𝜔𝑝𝜎𝑎𝑙𝑙𝑜ð‘Ī  0 0.1 0.2 0.3 0.4 0.505 ï€Ū1051 ï€Ū1041.5 ï€Ū1042 ï€Ū104𝝎𝒄𝝎𝒔⁄        (𝒚𝒑 − 𝒚𝒄)𝒎𝒂𝒙ð‘ģ𝒎 𝑓 =ðŋ𝑚10𝑈𝑆ðŋ𝑐𝑈𝑆ðŋ𝑝 ð›ū𝑐 𝜎𝑎𝑙𝑙𝑜ð‘Ī 𝜔𝑝 𝜔𝑠250 500 750 1000 1250 1500 1750 200005 ï€Ū1051 ï€Ū1041.5 ï€Ū1042 ï€Ū104(𝒚𝒑 − 𝒚𝒄)𝒎𝒂𝒙ð‘ģ𝒎 ð‘ģ𝒎(𝒎) 𝑓 =ðŋ𝑚10𝜔𝑝 𝜔𝑠ðķ = âˆŦ √[1 + (𝑑ð‘Ķ𝑑ð‘Ĩ)2] 𝑑ð‘Ĩðŋ𝑚0=ðŋ𝑚2ÎĻĖ…𝑝[ÎĻĖ…𝑝√1+ÎĻĖ…𝑝24+ 2𝑠𝑖𝑛ℎ (ÎĻĖ…𝑝2)]ð›ū𝑐 𝜎𝑎𝑙𝑙𝑜ð‘Ī 𝜔𝑝 𝜔𝑠 ðļ𝑐ð‘ģ𝒎 (𝒎) (𝑞𝑚ð‘ģ𝒄 − 𝑞𝑚ð‘ģ𝒑)ð‘ģ𝒎 𝑓 =ðŋ𝑚10Δ𝑒 =ðŧðļ𝑐ðī𝑐âˆŦ [1 + (𝑑ð‘Ķ𝑑ð‘Ĩ)2] 𝑑ð‘Ĩðŋ𝑚0=ðŧðŋ𝑚ðļ𝑐ðī𝑐[1 +112ÎĻĖ…𝑝2]ÎĻĖ…𝑝 =(𝜔𝑐 +𝜔𝑠)ðŋ𝑚ðŧ= 8(𝑓ðŋ𝑚) 𝜔𝑐 , 𝜔𝑠  𝜔𝑝 𝜎𝑎𝑙𝑙𝑜ð‘Īω𝑐 =(ω𝑠 +ω𝑝) 𝜉 √1 + 16 𝑆𝑅21 − 𝜉 √1 + 16 𝑆𝑅2𝑆𝑅 𝑓: ðŋ𝑚𝜉𝜉 =ð›ū𝑐 ðŋ𝑚8 𝜎𝑎𝑙𝑙𝑜ð‘Ī 𝑆𝑅𝜔𝑠 𝜔𝑝𝜔𝑝:𝜔𝑠        ð›ū𝑐 𝜎𝑎𝑙𝑙𝑜ð‘Ī0 250 500 750 1000 1250 1500 1750 200000.050.10.150.20.250.30.350.40.450.5ð‘ģ𝒎 (𝒎) 𝝎𝒄𝝎𝒔 +𝝎𝒑 𝑆𝑅 = 0.20 𝑆𝑅 = 0.15       𝑆𝑅 = 0.10       𝜔𝑝:𝜔𝑠 = 0.6ð‘Ĩ𝑝 ðŋ𝑝ðŋ𝑝2 â‰Ī   ð‘Ĩ𝑝  â‰Ī  ðŋ𝑚 −ðŋ𝑝2     ðŋ𝑚 ð›ū𝑐 ðļ𝑐 𝜎𝑎𝑙𝑙𝑜ð‘Ī 𝜔𝑝 𝜔𝑠 𝑆𝑅𝜔𝑠  𝜔𝑝 ð‘Ĩ𝑝 ðŋ𝑝 ð›ŋ𝑚𝑎ð‘Ĩ ðŋ𝑚 ð‘ģ𝒑ð‘ģ𝒎⁄  𝒙𝒑ð‘ģ𝒎⁄  ðœđ𝒎𝒂𝒙ð‘ģ𝒎 ð‘Ĩ𝑝 ðŋ𝑚⁄ = 0.21 & 0.79ðŋ𝑝 ðŋ𝑚⁄ = 0.4ð‘Ĩ𝑝 ðŋ𝑚⁄ = 0.5ðŋ𝑝 ðŋ𝑚⁄ = 0.38                  ðŋ𝑚 ð›ū𝑐 ðļ𝑐 𝜎𝑎𝑙𝑙𝑜ð‘Ī 𝜔𝑝 𝜔𝑠 𝑆𝑅ð‘ģ𝒑ð‘ģ𝒎⁄  𝒙𝒑ð‘ģ𝒎⁄  ð‘Ĩ𝑝 ðŋ𝑝 ð›ŋ𝑚𝑎ð‘Ĩ  ðŋ𝑚 𝜔𝑠 𝜔𝑝 ðœđ𝒎𝒂𝒙ð‘ģ𝒎 Îī𝑠𝑓    ðŋ𝑚 ð›ū𝑐 ðļ𝑐 𝜎𝑎𝑙𝑙𝑜ð‘Ī 𝜔𝑝 𝜔𝑠 𝑆𝑅𝜔𝑠 𝜔𝑝 ð‘Ĩ𝑝 ðŋ𝑝 ð›ŋ𝑚𝑎ð‘Ĩ ðŋ𝑚 ð‘ģ𝒑ð‘ģ𝒎⁄  𝒙𝒑ð‘ģ𝒎⁄  ðœđ𝒔𝒇ðœđ𝒎𝒂𝒙 (%) ð›ŋ𝑠𝑓 ðļ𝑐ðī𝑐 ðļ𝑐ðī𝑐 = ∞ 𝜔𝑝 𝜔𝑠      ðŋ𝑚 ð›ū𝑐 ðļ𝑐 𝜎𝑎𝑙𝑙𝑜ð‘Ī 𝜔𝑝 𝜔𝑠 𝑆𝑅𝝎𝒔 𝜔𝑝 ð‘Ĩ𝑝 ðŋ𝑝 ð›ŋ𝑚𝑎ð‘Ĩ ðŋ𝑚 ð‘ģ𝒑ð‘ģ𝒎⁄  𝒙𝒑ð‘ģ𝒎⁄  ðœđ𝒎𝒂𝒙ð‘ģ𝒎 𝟐𝝎𝒔  𝜔𝑝 ð‘Ĩ𝑝 ðŋ𝑝 ð›ŋ𝑚𝑎ð‘Ĩ ðŋ𝑚     Suspension Cables Superstructure Hangers Transitory Load  0 100 200 300 400 500 6001020304050607080𝑚𝒑𝒂𝒏− 𝒕𝒐 − ð‘Ŧ𝒆𝒑𝒕𝒉 ð‘đ𝒂𝒕𝒊𝒐 𝑚𝒑𝒂𝒏−𝒕𝒐−ð‘ū𝒊𝒅𝒕𝒉 ð‘đ𝒂𝒕𝒊𝒐 ïķ ïķ ïķ    𝒙𝒑 ð‘ģ𝒎⁄ = 𝟎. 𝟖,  ð‘ģ𝒑 ð‘ģ𝒎⁄ = 𝟎. 𝟒  𝒙𝒑 ð‘ģ𝒎⁄ = 𝟎. 𝟓,  ð‘ģ𝒑 ð‘ģ𝒎⁄ = 𝟎. 𝟒  𝒙𝒑 ð‘ģ𝒎⁄ = 𝟎. 𝟓,  ð‘ģ𝒑 ð‘ģ𝒎⁄ = 𝟏. 𝟎 ð›ū𝑐 ðļ𝑐 𝜎𝑎𝑙𝑙𝑜ð‘Ī 𝜔𝑝 𝜔𝑠 𝑆𝑅250 500 750 100000.0010.0020.0030.0040.0050.0060.007ðœđ𝒎𝒂𝒙ð‘ģ𝒎 ð‘ģ𝒎 (𝒎) LS3ð›ŋ𝑚𝑎ð‘Ĩ 𝜔𝑝 𝜔𝑠 ð›ŋ𝑚𝑎ð‘Ĩ 𝜔𝑝 𝜔𝑠 ð›ŋ𝑚𝑎ð‘Ĩ 𝜔𝑝 𝜔𝑠 LS2LS1𝛞𝑝 𝛞𝑝   𝒙𝒑 ð‘ģ𝒎⁄ = 𝟎. 𝟖,  ð‘ģ𝒑 ð‘ģ𝒎⁄ = 𝟎. 𝟒  𝒙𝒑 ð‘ģ𝒎⁄ = 𝟎. 𝟓,  ð‘ģ𝒑 ð‘ģ𝒎⁄ = 𝟎. 𝟒  𝒙𝒑 ð‘ģ𝒎⁄ = 𝟎. 𝟓,  ð‘ģ𝒑 ð‘ģ𝒎⁄ = 𝟏. 𝟎 ðŋ𝑚 ð›ū𝑐 ðļ𝑐 𝜎𝑎𝑙𝑙𝑜ð‘Ī 𝑆𝑅 𝜔𝑝 𝜔𝑠𝛞𝑝 = 10 0.002 0.004 0.006 0.008 0.01 0.012 0.01400.250.50.7511.251.51.752ðœķ𝒑  ðœđ𝒎𝒂𝒙ð‘ģ𝒎 ð›ŋ𝑚𝑎ð‘Ĩ 𝛞𝑝𝜔𝑝 𝜔𝑠 ð›ŋ𝑚𝑎ð‘Ĩ 𝛞𝑝𝜔𝑝 𝜔𝑠 ð›ŋ𝑚𝑎ð‘Ĩ 𝛞𝑝𝜔𝑝 𝜔𝑠 LS1 LS2 LS3ð›ŋ𝑚𝑎ð‘Ĩ SO    (ðŋ𝑚 = 250𝑚) (ðŋ𝑚 = 500𝑚)   (ðŋ𝑚 = 750𝑚) (ðŋ𝑚 = 1000𝑚)ð›ū𝑐 ðļ𝑐 𝜎𝑎𝑙𝑙𝑜ð‘Ī 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 1.75 211.11.21.31.41.51.6ðœđ𝒎𝒂𝒙 𝑚ð‘ķðœđ𝒎𝒂𝒙 ðœđ𝒎𝒂𝒙 𝑚ð‘ķðœđ𝒎𝒂𝒙 ðœķ𝒑  ðœķ𝒑  ðœķ𝒑  ðœķ𝒑  LS1LS2LS3LS1LS2LS3LS1LS2LS3LS1LS2LS3𝜔𝑝: 𝜔𝑐𝜔𝑝: (𝜔𝑐 +𝜔𝑠 +𝜔𝑝)𝜔𝑐 = 𝜔𝑠 (1 + 𝜔𝑅) · ð›Ŧω𝑅 𝜔𝑝: 𝜔𝑠Κð›Ŧ = 𝜉 √1 + 16 𝑆𝑅21 − 𝜉 √1 + 16 𝑆𝑅2𝜉 =ð›ū𝑐 ðŋ𝑚8 𝜎𝑎𝑙𝑙𝑜ð‘Ī 𝑆𝑅𝜔𝑝𝜔𝑝𝜔𝑐=𝜔𝑅  (1 + 𝜔𝑅) · ð›Ŧð›ŋ𝑒2ð›ŋ𝑒1≅ (𝜔𝑅2  1 + 𝜔𝑅2) (1 + 𝜔𝑅1  𝜔𝑅1)ð›ŋ𝑒1 ð›ŋ𝑒2𝜔𝑝𝜔𝑐 +𝜔𝑠 +𝜔𝑝=𝜔𝑅  (1 + 𝜔𝑅)(ð›Ŧ + 1)ð›ŋ𝑠𝑓2ð›ŋ𝑠𝑓1≅ (𝜔𝑅2  1 + 𝜔𝑅2) (1 + 𝜔𝑅1  𝜔𝑅1)ð›ŋ𝑚𝑎ð‘Ĩ2ð›ŋ𝑚𝑎ð‘Ĩ1=ð›ŋ𝑠𝑓2 + ð›ŋ𝑒2ð›ŋ𝑠𝑓1 + ð›ŋ𝑒1ð›ŋ𝑚𝑎ð‘Ĩ2ð›ŋ𝑚𝑎ð‘Ĩ1≅ð›ŋ𝑠𝑓2ð›ŋ𝑠𝑓1≅ð›ŋ𝑒2ð›ŋ𝑒1≅ (𝜔𝑅2  1 + 𝜔𝑅2) (1 + 𝜔𝑅1  𝜔𝑅1)ð›ŋ𝑚𝑎ð‘Ĩ  ð›ŋ𝑠𝑓ð‘Ĩ𝑝ðŋ𝑝ð‘Ĩ𝑝 ðŋ𝑚 ðŋ𝑝 ðŋ𝑚𝛞𝑝 = 1𝛚ð‘đ𝟐 = 𝟎.𝟐 𝛚ð‘đ𝟐 = 𝟎.𝟑 𝛚ð‘đ𝟐 = 𝟎.𝟒 𝛚ð‘đ𝟐 = 𝟎. 𝟓 𝛚ð‘đ𝟐 = 𝟎. 𝟔   ðŋ𝑚 ð›ū𝑐 ðļ𝑐 𝜎𝑎𝑙𝑙𝑜ð‘Ī 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 0.600.250.50.7511.251.51.7522.252.5𝝎ð‘đ𝟏 𝝎ð‘đ𝟏 ðœđ𝒆𝟐ðœđ𝒆𝟏 ðœđ𝒔𝒇𝟐ðœđ𝒔𝒇𝟏     (ðŋ𝑚 = 250𝑚) (ðŋ𝑚 = 1000𝑚)ð›ū𝑐 ðļ𝑐 𝜎𝑎𝑙𝑙𝑜ð‘Ī 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 0.611.051.11.151.21.251.3ðœđ𝒎𝒂𝒙 𝑚ð‘ķðœđ𝒎𝒂𝒙 𝝎ð‘đ  𝝎ð‘đ  LS1LS2LS3LS1LS2LS3   𝒙𝒑 ð‘ģ𝒎⁄ = 𝟎. 𝟖,  ð‘ģ𝒑 ð‘ģ𝒎⁄ = 𝟎. 𝟒  𝒙𝒑 ð‘ģ𝒎⁄ = 𝟎. 𝟓,  ð‘ģ𝒑 ð‘ģ𝒎⁄ = 𝟎. 𝟒  𝒙𝒑 ð‘ģ𝒎⁄ = 𝟎. 𝟓,  ð‘ģ𝒑 ð‘ģ𝒎⁄ = 𝟏. 𝟎 ðŋ𝑚 ð›ū𝑐 ðļ𝑐 𝜎𝑎𝑙𝑙𝑜ð‘Ī 𝜔𝑝 𝜔𝑠ðķ𝑎𝑝𝑝𝑟𝑜ð‘Ĩ = ðŋ𝑚 [1 +83(𝑓ðŋ𝑚)2−325(𝑓ðŋ𝑚)4+â‹Ŋ ]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 ðœđ𝒎𝒂𝒙ð‘ģ𝒎 𝑚ð‘đ ð›ŋ𝑚𝑎ð‘Ĩ 𝜔𝑝 𝜔𝑠 𝜔𝑠 ð›ŋ𝑚𝑎ð‘Ĩ 𝜔𝑝 ð›ŋ𝑚𝑎ð‘Ĩ 𝜔𝑝 𝜔𝑠 LS1LS3LS2ð›ŋ𝑓 = [1516 𝑆𝑅 (5 − 24𝑆𝑅2)]Îīðķ𝑎𝑝𝑝𝑟𝑜ð‘Ĩð›ŋðķð›ŋ𝑓  Δ𝑒1 = Δ𝑒2 = 0.0325 · ðŋ𝑚1  (ð‘ð‘œð‘Ąð‘’: 𝑇ℎ𝑒 𝑎𝑚𝑜ð‘Ēð‘›ð‘Ą 𝑜𝑓 ð‘’ð‘™ð‘Žð‘ ð‘Ąð‘–ð‘ ð‘’ð‘™ð‘œð‘›ð‘”ð‘Žð‘Ąð‘–ð‘œð‘› 𝑖𝑠 ð‘Ąâ„Žð‘’ 𝑠𝑎𝑚𝑒 𝑖𝑛 ð‘ð‘œð‘Ąâ„Ž 𝑐𝑎𝑏𝑙𝑒𝑠)ð›ŋ𝑚𝑎ð‘Ĩ2 > ð›ŋ𝑚𝑎ð‘Ĩ1 ð›ŋ𝑚𝑎ð‘Ĩ1 𝑚ð‘đ𝟏 = 𝟎.𝟏 𝑚ð‘đ𝟐 = 𝟎. 𝟎𝟓 𝐷𝑒𝑓𝑜𝑟𝑚𝑒𝑑 ð‘ƒð‘œð‘ ð‘–ð‘Ąð‘–ð‘œð‘› ðŋ𝑚1 ðŋ𝑚2 = ðŋ𝑚1   ðŋ𝑚 ðļðī𝑔 ðļ𝐞𝑔 𝑓 𝜔𝑝 𝜔𝑠 𝜔𝑝𝑓 ðļðī𝑔, ðļ𝐞𝑔  ðŋ𝑚 ð‘Ĩ𝑝 ðŋ𝑝 𝑅𝑜𝑙𝑙𝑒𝑟 𝑆ð‘Ēð‘ð‘ð‘œð‘Ÿð‘Ą (TYP) 𝜔𝑝 𝜔𝑠 ðŋ𝑚 50⁄  (𝑇𝑌𝑃) 𝑅𝑖𝑔𝑖𝑑  𝑇𝑟ð‘Ē𝑠𝑠 ðļð‘™ð‘’ð‘šð‘’ð‘›ð‘Ąð‘  ðŋ𝑚 ðļðī𝑔,ðļ𝐞𝑔  𝑓 𝑅𝑖𝑔𝑖𝑑  𝑇𝑟ð‘Ē𝑠𝑠 ðļð‘™ð‘’ð‘šð‘’ð‘›ð‘Ąð‘  ðŋ𝑝 ð‘Ĩ𝑝 𝑃𝑖𝑛 𝑆ð‘Ēð‘ð‘ð‘œð‘Ÿð‘Ą 𝜔𝑝 𝜔𝑠 ðŋ𝑝 ðŋ𝑝ðŋ𝑚ðļð‘Žð‘Ÿð‘Ąâ„Ž − ðī𝑛𝑐ℎ𝑜𝑟𝑒𝑑 𝑆ð‘Ķð‘ ð‘Ąð‘’ð‘š 𝑆𝑒𝑙𝑓 − ðī𝑛𝑐ℎ𝑜𝑟𝑒𝑑 𝑆ð‘Ķð‘ ð‘Ąð‘’ð‘š 0 2 4 6012340 2 4 601234trace 1111trace 2trace 30 2 4 602550751000 2 4 60255075100trace 1111trace 2trace 3𝑎𝑰𝒈 (ð‘īð‘ĩ𝒎𝟐  ×  𝟏𝟎𝟓) ðœđ𝒑𝒆𝒂𝒌 (𝒎) 𝑚ð‘đ = 𝟎. 𝟏𝟎 𝑚ð‘đ = 𝟎. 𝟏𝟓 𝑚ð‘đ = 𝟎. 𝟐𝟎  𝑎𝑰𝒈 (ð‘īð‘ĩ𝒎𝟐  × 𝟏𝟎𝟓) ð‘ī𝒑𝒆𝒂𝒌 (ð‘īð‘ĩ𝒎) 𝑎𝑰𝒈 (ð‘īð‘ĩ𝒎𝟐  ×  𝟏𝟎𝟓) 𝑎𝑰𝒈 (ð‘īð‘ĩ𝒎𝟐  ×  𝟏𝟎𝟓) ðŋ𝑝ðŋ𝑚ð‘Ĩ𝑝ðŋ𝑚𝑚ð‘đ = 𝟎. 𝟏𝟎 𝑚ð‘đ = 𝟎. 𝟏𝟓 𝑚ð‘đ = 𝟎. 𝟐𝟎  ðŋ𝑝ðŋ𝑚ð‘Ĩ𝑝ðŋ𝑚ðļð‘Žð‘Ÿð‘Ąâ„Ž − ðī𝑛𝑐ℎ𝑜𝑟𝑒𝑑 𝑆ð‘Ķð‘ ð‘Ąð‘’ð‘š 𝑆𝑒𝑙𝑓 − ðī𝑛𝑐ℎ𝑜𝑟𝑒𝑑 𝑆ð‘Ķð‘ ð‘Ąð‘’ð‘š 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 3𝑎𝑰𝒈 (ð‘īð‘ĩ𝒎𝟐  ×  𝟏𝟎𝟓) ðœđ𝒑𝒆𝒂𝒌 𝑚ð‘ķðœđ𝒑𝒆𝒂𝒌 𝑚ð‘đ = 𝟎. 𝟏𝟎 𝑚ð‘đ = 𝟎. 𝟏𝟓 𝑚ð‘đ = 𝟎. 𝟐𝟎  𝑎𝑰𝒈 (ð‘īð‘ĩ𝒎𝟐  × 𝟏𝟎𝟓) ð‘ī𝒑𝒆𝒂𝒌 𝑚ð‘ķð‘ī𝒑𝒆𝒂𝒌  𝑎𝑰𝒈 (ð‘īð‘ĩ𝒎𝟐  ×  𝟏𝟎𝟓) 𝑎𝑰𝒈 (ð‘īð‘ĩ𝒎𝟐  ×  𝟏𝟎𝟓) 𝑚ð‘đ = 𝟎. 𝟏𝟎 𝑚ð‘đ = 𝟎. 𝟏𝟓 𝑚ð‘đ = 𝟎. 𝟐𝟎    áī‡áī‡         ðŋ𝑚 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 ðŋ𝑚 𝑓 ðŋ𝑚 𝑚ð‘đ = 𝒇 ð‘ģ𝒎⁄  ðœđ𝒑𝒆𝒂𝒌 (𝒎) ðļ𝐞𝑔 = 25000𝑀𝑁𝑚2 ðļðī𝑔 = 200000𝑀𝑁     ð‘đ𝒐𝒍𝒍𝒆𝒓 𝑚𝒖𝒑𝒑𝒐𝒓𝒕 (TYP) ðļðī𝑔, ðļ𝐞𝑔  𝑷𝒊𝒏 𝑚𝒖𝒑𝒑𝒐𝒓𝒕 ðļðī𝑔, ðļ𝐞𝑔  𝑎ð‘Ļ𝒈 (ð‘īð‘ĩ × 𝟏𝟎𝟓) ðļ𝐞𝑔 = 25000𝑀𝑁𝑚2 𝑆𝑅 = 0.2     𝑚ð‘đ = 𝒇 ð‘ģ𝒎⁄  ð‘ī𝒑𝒆𝒂𝒌 (ð‘īð‘ĩ𝒎) ðļ𝐞𝑔 = 25000𝑀𝑁𝑚2 ðļðī𝑔 = 200000𝑀𝑁         ðŋ𝑚 𝜔𝑝 𝜔𝑠 𝜔𝑝 ðļðī𝑔0.1 0.15 0.20.911.11.21.31.41.50.1 0.15 0.20.911.11.21.31.41.5ðœđ𝒑𝒆𝒂𝒌(ðœđ𝒑𝒆𝒂𝒌)ð‘đ ð‘ī𝒑𝒆𝒂𝒌(ð‘ī𝒑𝒆𝒂𝒌)ð‘đ 𝑚ð‘đ = 𝒇 ð‘ģ𝒎⁄  𝑚ð‘đ = 𝒇 ð‘ģ𝒎⁄  Flexible Superstructure (ðļ𝐞𝑔 = 25000𝑀𝑁𝑚2) Stiff Superstructure (ðļ𝐞𝑔 = 600000𝑀𝑁𝑚2)                                     Sag Ratio = 0.1 Sag Ratio = 0.2 ðŋ𝑚 ðļ𝐞𝑔 áī‡ 𝜔𝑝 𝜔𝑠 𝜔𝑝4.531.501.534.54.531.501.534.5300200100010020030030020010001002003006040200204060604020020406030020010001002003003002001000100200300Deflection Envelope (m) Bending Moment Envelope (MN·m) Percent Change (%) Percent Change (%) System 2 System 1  ðŋℎ 𝑓ℎðŋ𝑚 𝑓𝑓 𝑓ℎ𝑓ℎ ðŋℎ  ðŋ𝑚 𝑓ðŋℎ 𝑓ℎðŋ𝑚 2⁄  ðŋℎ 𝑓ℎ  𝑓 Suspended Region Stayed Region Stayed Region ðŋ𝑚 2⁄   𝑁1â€Ķ𝑛 ðŋℎ𝑆1â€Ķ𝑛 𝑇𝑚𝑎ð‘Ĩðđℎ𝑖 ðŧ𝜔𝑐 {ð‘Ĩ𝑖 , ð‘Ķ𝑖}𝑓 Δð‘Ĩ𝑖ðŋ𝑚 Δy𝑖Ɨ𝑓 ðŋ𝑚 2⁄  ð›Ĩð‘Ĩ𝑖  ð›Ĩð‘Ķ𝑖  ð‘Ķ𝑖  ðđℎ𝑖 ð‘Ĩ𝑖  ðŧ 𝑇𝑚𝑎ð‘Ĩ 𝜔𝑐  𝜔𝑐  ðŧ𝑑ð‘Ķ𝑖𝑑ð‘Ĩ𝑖|ð‘Ĩ𝑖=ð›Ĩð‘Ĩ𝑖 ðŧ ðŧ𝑑ð‘Ķ𝑖𝑑ð‘Ĩ𝑖|ð‘Ĩ𝑖=0 ðŧ ðđℎ1 ðđℎ𝑖−1 ðđℎ𝑛 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 ð›Ĩð‘Ĩ𝑛 =  ðŋ𝑚 − ðŋℎ 2 ⁄ ðŋℎ 2⁄  Stayed Region  ðŋ𝑚 − ðŋℎ 2⁄  𝑓𝜔𝑐𝑝𝜔𝑐𝑝 = 𝜔𝑐√1+ ÎĐ2 ÎĐÎĐ =2 yðĩðŋ𝑚 − ðŋℎ  ðŋ𝑚 − ðŋℎ 2⁄  ðŋℎ 𝑓ℎ  𝑓 ðī ðĩ ðķ 𝐷 𝜔𝑐𝑝  𝜔𝑐𝑝  𝜔𝑐  ð‘Ķðĩ  ðŋ𝑚 − ðŋℎ 2⁄  𝑭𝒐𝒓  𝟎 â‰Ī 𝒙 <  ð‘ģ𝒎 − ð‘ģ𝒉 𝟐⁄ð‘Ķ ð‘Ĩ =1ðŧ[𝜔𝑐𝑝ð‘Ĩ22− 𝑉𝑠ð‘Ĩ]𝑭𝒐𝒓  ð‘ģ𝒎 − ð‘ģ𝒉 𝟐⁄ â‰Ī 𝒙 â‰Ī  ð‘ģ𝒎 + ð‘ģ𝒉 𝟐⁄ð‘Ķ ð‘Ĩ |𝑆𝑚 =1ðŧ[∑ ðđℎ𝑗 (ð‘Ĩ − ∑𝜆𝑖𝑗𝑖=1) +𝜔𝑐𝑝𝜆12 2ð‘Ĩ − 𝜆1 +𝜔𝑐2 ð‘Ĩ − 𝜆1 2 − 𝑉𝑠ð‘Ĩ𝑚−1𝑗=1]𝑭𝒐𝒓  ð‘ģ𝒎 + ð‘ģ𝒉 𝟐⁄ â‰Ī 𝒙 < ð‘ģ𝒎ð‘Ķ ð‘Ĩ =1ðŧ[𝜔𝑐𝑝2 2𝜆1 + ðŋℎ − ð‘Ĩ 2 − 𝑉𝑠ð‘Ĩ]ð‘Ĩ ð‘Ķ𝑉𝑠 =12∑ ðđℎ𝑖 + 𝜔𝑐𝑝𝜆1 +𝜔𝑐ðŋℎ2𝑁ℎ𝑎𝑛𝑔𝑒𝑟𝑠𝑖=1ðŧ =1𝑓[𝑉𝑠ðŋ𝑚2−𝜔𝑐𝑝𝜆12 ðŋ𝑚 − 𝜆1 −𝜔𝑐ðŋℎ28− ∑ðđℎ𝑗 (ðŋ𝑚2− ∑𝜆𝑖𝑗𝑖=1)𝑁∗𝑗=1]ðŧ =1𝑓ℎ[(𝑉𝑠 − 𝜔𝑐𝑝𝜆1)ðŋℎ2−𝜔𝑐ðŋℎ28− ∑ðđℎ𝑗 (ðŋ𝑚2− ∑𝜆𝑖𝑗𝑖=1)𝑁∗𝑗=1]𝑁ℎ𝑎𝑛𝑔𝑒𝑟𝑠−12, 𝑁∗ =𝑁ℎ𝑎𝑛𝑔𝑒𝑟𝑠2    𝜔𝑐  𝑁1â€Ķ ðŋ𝑚𝑆1â€Ķ ðŋℎ𝑁ℎ𝑎𝑛𝑔𝑒𝑟𝑠 Ɨ ðŧ𝜔𝑐 Ɨ 𝑉𝑠𝑓 Îŧ𝑚𝑓ℎ Ɨ ðđℎ𝑚Ɨð‘Ķ ðđℎ𝑚+1 ð‘Ĩ ðŧ ðđℎ1 ðđℎ𝑚 𝑉𝑠 ðđℎ𝑚−1 𝜆1 =  ðŋ𝑚 − ðŋℎ 2⁄   𝜆𝑚−1 𝜆𝑚 𝜆𝑚+1 𝑁1 𝑁𝑚−1 𝑁𝑚 𝑁𝑚+1 ðŧ ðŋ𝑚 2⁄   ðđℎ2 𝜆2 𝑓 𝑁2 𝑆1 𝑆2 𝑆𝑚−1 𝑆𝑚 𝑆𝑚+1 ðŋℎ 2⁄   𝑑0𝑑0 = 0𝜆 𝜔𝑠𝜔𝑐𝑝 ðŋ𝑚−ðŋℎ 2+ðŋℎ 2ðŋ𝑚−ðŋℎ  𝜔𝑐+𝜔𝑠 8𝑓,ðŧ =𝜔𝑐𝑝 ðŋ𝑚−ðŋℎ 2+ðŋℎ 2ðŋ𝑚−ðŋℎ  𝜔𝑐+𝜔𝑠 −𝜔𝑠𝜆28𝑓, 𝜔𝑐+𝜔𝑠 ðŋℎ28𝑓ℎ,ðŧ = 𝜔𝑐+𝜔𝑠 ðŋℎ2−𝜔𝑠𝜆28𝑓ℎ, ðŋ𝑚 2⁄   ðŋ𝑚 − ðŋℎ 2⁄  𝑑0 ðŋℎ 2⁄  Stay Cables ð‘Ĩ[𝜔𝑐𝑝 ð‘Ĩ−ðŋ𝑚 −ðŋℎ(𝜔𝑐+𝜔𝑠−𝜔𝑐𝑝)]2ðŧ, 𝑓𝑜𝑟  0 â‰Ī ð‘Ĩ <  ðŋ𝑚 − ðŋℎ 2⁄ð‘Ķ ð‘Ĩ =[ ðŋ𝑚−ðŋℎ 2(𝜔𝑐+𝜔𝑠−𝜔𝑐𝑝)+4ð‘Ĩ 𝜔𝑐+𝜔𝑠  ð‘Ĩ−ðŋ𝑚 ]8ðŧ, 𝑓𝑜𝑟  ðŋ𝑚 − ðŋℎ 2⁄ â‰Ī ð‘Ĩ â‰Ī  ðŋ𝑚 + ðŋℎ 2⁄ ð‘Ĩ−ðŋ𝑚 [𝜔𝑐𝑝 ð‘Ĩ +ðŋℎ(𝜔𝑐+𝜔𝑠−𝜔𝑐𝑝)]2ðŧ, 𝑓𝑜𝑟  ðŋ𝑚 + ðŋℎ 2⁄ â‰Ī ð‘Ĩ < ðŋ𝑚 ðŋ𝑚 − ðŋℎ 2⁄ â‰Ī ð‘Ĩ â‰Ī  ðŋ𝑚 + ðŋℎ 2⁄ ð‘Ĩ𝑑0 < 0ðķ𝑚 = 𝜆𝑚√1+ 𝛚𝑚2 [1 +124ð›đ𝑐𝑚2(1 + 𝛚𝑚2)]∆𝑒𝑚 =ðŧ𝜆𝑚ðļ𝑐ðī𝑐[1 + 𝛚𝑚2 +112ð›đ𝑐𝑚2(1 + 𝛚𝑚2)]ð›đ𝑐𝑚 =𝜔𝑐𝜆𝑚ðŧ𝛚𝑚 =ℎ𝑚𝜆𝑚ℎ𝑚ℎ𝑚 = ð‘Ķ𝑚 − ð‘Ķ𝑚−1ð‘Ķ𝑚 𝑚ðŧ𝑈𝑆ðŋð‘‡ð‘œð‘Ąð‘Žð‘™ = ∑  𝑈𝑆ðŋ𝑚 𝑁ℎ𝑎𝑛𝑔𝑒𝑟𝑠+1𝑚=1 = ∑ (ðķ𝑚 − ð›Ĩ𝑒𝑚)𝑁ℎ𝑎𝑛𝑔𝑒𝑟𝑠+1𝑚=1𝜔𝑐𝑝 = 𝜔𝑐√1 + (2 ð‘Ķðĩðŋ𝑚 − ðŋℎ)2ð‘Ķðĩ = ð‘Ķ (ð‘Ĩ =ðŋ𝑚 − ðŋℎ2)𝜔𝑐𝑝 ð‘Ķðĩ𝜔𝑐𝑝𝜔𝑐 𝜔𝑐𝑝 ð‘Ķðĩ𝜔𝑐𝑝𝜔𝑐𝑝 ≅ 𝜔𝑐√1+ 4𝑓2 [𝜔𝑐 ðŋ𝑚 + ðŋℎ + 2𝜔𝑠ðŋℎ𝜔𝑐ðŋ𝑚2 + 𝜔𝑠ðŋℎ 2ðŋ𝑚 − ðŋℎ ]2𝑓ℎ = 𝑓 − ð‘Ķðĩð‘Ķðĩ𝑓ℎðŋℎ≅ ðŋ𝑅 (𝑓ðŋ𝑚) [𝜔𝑐 + 𝜔𝑠𝜔𝑐𝑝 1 − ðŋ𝑅 2 + ðŋ𝑅 𝜔𝑐 + 𝜔𝑠  2 − ðŋ𝑅 ]ðŋ𝑅ðŋ𝑅 =ðŋℎðŋ𝑚  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 0.30.040.060.080.10.120.140.160.180.2ð‘ģ𝒉 = 𝟎. 𝟐 ð‘ģ𝒎 ð‘ģ𝒉 = 𝟎. 𝟑 ð‘ģ𝒎 ð‘ģ𝒉 = 𝟎. 𝟒 ð‘ģ𝒎 ð‘ģ𝒉 = 𝟎. 𝟓 ð‘ģ𝒎 𝒇 ð‘ģ𝒎⁄  𝒇𝒉 ð‘ģ𝒉⁄  𝒇 ð‘ģ𝒎⁄  𝒇 ð‘ģ𝒎⁄  𝒇 ð‘ģ𝒎⁄  𝒇𝒉 ð‘ģ𝒉⁄  𝒇𝒉 ð‘ģ𝒉⁄  𝒇𝒉 ð‘ģ𝒉⁄  ð‘ģ𝒉 = 𝟎. 𝟔 ð‘ģ𝒎 𝒇 ð‘ģ𝒎⁄  𝒇𝒉 ð‘ģ𝒉⁄  𝒇 𝒇𝒉 ð‘ģ𝒉 ð‘ģ𝒎 ð‘Ķ𝑐 ð‘Ķ𝑝ð‘Ĩð‘Ĩ𝜔𝑐: 𝜔𝑠  ð‘ģ𝒉 = 𝟎. 𝟐 ð‘ģ𝒎ð‘ģ𝒉 = 𝟎. 𝟒 ð‘ģ𝒎 𝐋𝐄𝐆𝐄𝐍𝐃ð‘ģ𝒉 = 𝟎. 𝟔 ð‘ģ𝒎ð‘ģ𝒉 = 𝟏. 𝟎 ð‘ģ𝒎𝜔𝑐: 𝜔𝑠0 0.05 0.1 0.15 0.201.25 ï€Ū1052.5 ï€Ū1053.75 ï€Ū1055 ï€Ū1050 0.05 0.1 0.15 0.205 ï€Ū1051 ï€Ū1041.5 ï€Ū1042 ï€Ū1040 0.05 0.1 0.15 0.202 ï€Ū1044 ï€Ū1046 ï€Ū1048 ï€Ū104𝝎𝒄 = ð‘Ē𝑛𝑖𝑓𝑜𝑟𝑚 ð‘Īð‘’ð‘–ð‘”â„Žð‘Ą 𝑜𝑓 𝑠ð‘Ē𝑠𝑝𝑒𝑛𝑠𝑖𝑜𝑛 𝑐𝑎𝑏𝑙𝑒 𝝎𝒔 = ð‘Ē𝑛𝑖𝑓𝑜𝑟𝑚 ð‘Īð‘’ð‘–ð‘”â„Žð‘Ą 𝑜𝑓 𝑠ð‘Ēð‘ð‘’ð‘Ÿð‘ ð‘Ąð‘Ÿð‘Ēð‘ð‘Ąð‘Ē𝑟𝑒  𝝎𝒄 𝝎𝒔⁄  𝝎𝒄 𝝎𝒔⁄  𝒇 = 𝟎. 𝟑 ð‘ģ𝒎 𝝎𝒄 𝝎𝒔⁄   |(𝒚𝒑 − 𝒚𝒄)𝒎𝒂𝒙|ð‘ģ𝒎 𝒇 𝒇𝒉 ð‘ģ𝒉 ð‘ģ𝒎 𝒇 = 𝟎. 𝟏 ð‘ģ𝒎 𝒇 = 𝟎. 𝟐 ð‘ģ𝒎 |(𝒚𝒑 − 𝒚𝒄)𝒎𝒂𝒙|ð‘ģ𝒎   ð‘ģ𝒉 = 𝟎. 𝟐 ð‘ģ𝒎ð‘ģ𝒉 = 𝟎. 𝟒 ð‘ģ𝒎 𝐋𝐄𝐆𝐄𝐍𝐃ð‘ģ𝒉 = 𝟎. 𝟔 ð‘ģ𝒎ð‘ģ𝒉 = 𝟏. 𝟎 ð‘ģ𝒎ð›ū𝑐 𝜎𝑎𝑙𝑙𝑜ð‘Ī 𝜔𝑝 𝜔𝑠200 400 600 800 100001.25 ï€Ū1052.5 ï€Ū1053.75 ï€Ū1055 ï€Ū105200 400 600 800 100005 ï€Ū1051 ï€Ū1041.5 ï€Ū1042 ï€Ū104200 400 600 800 100002 ï€Ū1044 ï€Ū1046 ï€Ū1048 ï€Ū104ð‘ģ𝒎  𝒎  ð‘ģ𝒎  𝒎   𝒇 = 𝟎. 𝟑 ð‘ģ𝒎 ð‘ģ𝒎  𝒎    |(𝒚𝒑 − 𝒚𝒄)𝒎𝒂𝒙|ð‘ģ𝒎 𝒇 𝒇𝒉 ð‘ģ𝒉 ð‘ģ𝒎 𝒇 = 𝟎. 𝟏 ð‘ģ𝒎 𝒇 = 𝟎. 𝟐 ð‘ģ𝒎 |(𝒚𝒑 − 𝒚𝒄)𝒎𝒂𝒙|ð‘ģ𝒎 𝜔𝑝: 𝜔𝑠  ð‘ģ𝒉 = 𝟎. 𝟐 ð‘ģ𝒎ð‘ģ𝒉 = 𝟎. 𝟒 ð‘ģ𝒎 𝐋𝐄𝐆𝐄𝐍𝐃ð‘ģ𝒉 = 𝟎. 𝟔 ð‘ģ𝒎ð‘ģ𝒉 = 𝟏. 𝟎 ð‘ģ𝒎ð›ū𝑐 𝜎𝑎𝑙𝑙𝑜ð‘Ī 𝜔𝑝 𝜔𝑠 ðļ𝑐 𝜆 ·ðŋ𝑚200 400 600 800 100002 ï€Ū1064 ï€Ū1066 ï€Ū1068 ï€Ū1061 ï€Ū105200 400 600 800 100001 ï€Ū1052 ï€Ū1053 ï€Ū1054 ï€Ū1055 ï€Ū105200 400 600 800 100005 ï€Ū1051 ï€Ū1041.5 ï€Ū1042 ï€Ū1042.5 ï€Ū104ð‘ģ𝒎  𝒎  ð‘ģ𝒎  𝒎   𝒇 = 𝟎. 𝟑 ð‘ģ𝒎 ð‘ģ𝒎  𝒎    |(𝑞𝑚ð‘ģ𝒄 − 𝑞𝑚ð‘ģ𝒑)|ð‘ģ𝒎 𝒇 𝒇𝒉 ð‘ģ𝒉 ð‘ģ𝒎 𝒇 = 𝟎. 𝟏 ð‘ģ𝒎 𝒇 = 𝟎. 𝟐 ð‘ģ𝒎 |(𝑞𝑚ð‘ģ𝒄 − 𝑞𝑚ð‘ģ𝒑)|ð‘ģ𝒎 𝒇ð‘ģ𝒎ð‘ģ𝒉ð‘ģ𝒎𝝎𝒑𝝎𝒔ð‘ģ𝒎  𝒎 |(ð‘Ķ𝑝 − ð‘Ķ𝑐)𝑚𝑎ð‘Ĩ| ðŋ𝑚⁄ð›ū𝑐 𝜎𝑎𝑙𝑙𝑜ð‘Ī𝒇ð‘ģ𝒎ð‘ģ𝒉ð‘ģ𝒎𝝎𝒑𝝎𝒔ð‘ģ𝒎  𝒎 |(𝑈𝑆ðŋ𝑐 − 𝑈𝑆ðŋ𝑝)𝑚𝑎ð‘Ĩ| ðŋ𝑚⁄  ð›ū𝑐 𝜎𝑎𝑙𝑙𝑜ð‘Ī ðļ𝑐 𝜆 ·ðŋ𝑚 𝜔𝑐 , 𝜔𝑠  𝜔𝑝     𝑇𝑚𝑎ð‘Ĩ = ðŧ√1 + (𝑑ð‘Ķ𝑑ð‘Ĩ|ð‘Ĩ=0)2ðī𝑐_𝑟𝑒𝑞 =𝑇𝑚𝑎ð‘Ĩ𝜎𝑎𝑙𝑙𝑜ð‘Ī=ðŧ𝜎𝑎𝑙𝑙𝑜ð‘Ī√1 + (𝑑ð‘Ķ𝑑ð‘Ĩ|ð‘Ĩ=0)2ðī𝑐_𝑟𝑒𝑞𝜔𝑐 =ð›ū𝑐ðŧ𝜎𝑎𝑙𝑙𝑜ð‘Ī√1+ (𝑑ð‘Ķ𝑑ð‘Ĩ|ð‘Ĩ=0)2𝜔𝑐 =ð›ū𝑐ðŋ𝑚[𝜔𝑐 + 𝜔𝑠ðŋ𝑅 2 − ðŋ𝑅 ]8 𝑆𝑅𝜎𝑎𝑙𝑙𝑜ð‘Ī√1 + 16𝑆𝑅2 (𝜔𝑐 + 𝜔𝑠ðŋ𝑅𝜔𝑐 − 𝜔𝑠ðŋ𝑅2 + 2𝜔𝑠ðŋ𝑅)2𝑆𝑅𝑓 ðŋ𝑚 ðŋ𝑅 ðŋℎ ðŋ𝑚𝜔𝑐ðŋ𝑅𝜔𝑐𝜔𝑐𝜔𝑐 =ð›ū𝑐ðŋ𝑚[𝜔𝑐 + 𝜔𝑠ðŋ𝑅 2 − ðŋ𝑅 ]8 𝑆𝑅𝜎𝑎𝑙𝑙𝑜ð‘Ī√1 + 16(𝑆𝑅2 − ðŋ𝑅)2𝜔𝑠 (𝜔𝑠 + 𝜔𝑝)𝜔𝑐𝜔𝑐 =(𝜔𝑠 + 𝜔𝑝) 𝜉 ðŋ𝑅  2 − ðŋ𝑅 √1 + 16(𝑆𝑅2 − ðŋ𝑅)21 − 𝜉 √1 + 16 (𝑆𝑅2 − ðŋ𝑅)2Îū𝜉 =ð›ū𝑐 ðŋ𝑚8 𝜎𝑎𝑙𝑙𝑜ð‘Ī 𝑆𝑅ðŋ𝑅ðŋ𝑅𝑆𝑅 ðŋ𝑚Suspension Ratio = 0.2    Suspension Ratio = 0.4  Suspension Ratio = 0.6  ð›ū𝑐 𝜎𝑎𝑙𝑙𝑜ð‘Ī0 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 100000.030.060.090.120.15ð‘ģ𝒎  𝒎  ð‘ģ𝒎  𝒎  ð‘ģ𝒎  𝒎  𝝎𝒄𝝎𝒔 +𝝎𝒑 𝝎𝒄𝝎𝒔 +𝝎𝒑 𝝎𝒄𝝎𝒔 +𝝎𝒑 𝑆𝑅 = 0.3 𝑆𝑅 = 0.2       𝑆𝑅 = 0.1       𝑓ℎ ðŋℎ ð›ŋ𝑚𝑎ð‘Ĩ ðđðŋ  ð›ŋ𝑚𝑎ð‘Ĩ 𝑃ðŋ(ð›ŋ𝑝𝑒𝑎𝑘)ðđðŋ𝑘ðļ𝑘𝐚     ðŋ𝑚 ð›ū𝑐 ðļ𝑐 𝜎𝑎𝑙𝑙𝑜ð‘Ī 𝜔𝑝 𝜔𝑠ð‘Ĩ𝑝 ðŋ𝑝 ðŋℎ = 0.2 · ðŋ𝑚 0.1 · ðŋℎ ð‘Ĩ𝑝 ðŋ𝑝 𝜔𝑠 𝜔𝑝 ðŋ𝑝 ð‘Ĩ𝑝 𝜔𝑝 𝜔𝑠 𝜔𝑠  𝜔𝑝 ðœđ𝒎𝒂𝒙ð‘ģ𝒎 ðœđ𝒎𝒂𝒙ð‘ģ𝒎 ðœđ𝒎𝒂𝒙ð‘ģ𝒎 ð‘ģ𝒑 ð‘ģ𝒉⁄  ð‘ģ𝒑 ð‘ģ𝒉⁄  ð‘ģ𝒑 ð‘ģ𝒉⁄  𝒙𝒑ð‘ģ𝒉⁄  𝒙𝒑ð‘ģ𝒉⁄  𝒙𝒑ð‘ģ𝒉⁄  0.1 · ðŋℎ ðŋℎ = 0.6 · ðŋ𝑚 0.1 · ðŋℎ ðŋℎ = 1.0 · ðŋ𝑚 Suspension Ratio = 0.2 Suspension Ratio = 0.6 Suspension Ratio = 1.0     ðŋ𝑚 ð›ū𝑐 ðļ𝑐 𝜎𝑎𝑙𝑙𝑜ð‘Ī 𝜔𝑝 𝜔𝑠 ðœđ𝒎𝒂𝒙 𝑷ð‘ģ −  ðœđ𝒎𝒂𝒙 𝑭ð‘ģ(ðœđ𝒑𝒆𝒂𝒌)𝑭ð‘ģ  ðœđ𝒎𝒂𝒙 𝑷ð‘ģ −  ðœđ𝒎𝒂𝒙 𝑭ð‘ģ(ðœđ𝒑𝒆𝒂𝒌)𝑭ð‘ģ 0.1 · ðŋℎ 0.1 · ðŋℎ 0.1 · ðŋℎ 0.1 · ðŋℎ ðŋℎ = 0.6 · ðŋ𝑚 ðŋ𝑚 ðŋℎ ð‘ģ𝒑 ð‘ģ𝒉⁄  ð‘ģ𝒑 ð‘ģ𝒉⁄  𝒙𝒑ð‘ģ𝒉⁄  𝒙𝒑ð‘ģ𝒉⁄  ðŋℎ = 0.2 · ðŋ𝑚 ðŋ𝑚 ðŋℎ ð‘Ĩ𝑝 ðŋ𝑝 𝜔𝑠 𝜔𝑝 ðŋ𝑝 ð‘Ĩ𝑝 𝜔𝑝 𝜔𝑠 Suspension Ratio = 0.6 Suspension Ratio = 0.2 ðŋℎ 𝑓ℎ  𝑘𝑠𝑝𝑟𝑖𝑛𝑔 = 𝑘ðļ + 𝑘𝐚         ðŋ𝑚 ð›ū𝑐 ðļ𝑐 𝜎𝑎𝑙𝑙𝑜ð‘Ī 𝜔𝑝 𝜔𝑠ðŋℎ       ðŋ𝑚 ð›ū𝑐 ðļ𝑐 𝜎𝑎𝑙𝑙𝑜ð‘Ī 𝜔𝑝 𝜔𝑠0.2 0.4 0.6 0.8 100.0050.010.0150.020.2 0.4 0.6 0.8 100.010.020.030.04ð‘ģð‘đ = ð‘ģ𝒉 ð‘ģ𝒎⁄  𝑆𝑅 = 0.3 𝑆𝑅 = 0.2 𝑆𝑅 = 0.1 ð‘ģð‘đ = ð‘ģ𝒉 ð‘ģ𝒎⁄  ðœđ𝒑𝒆𝒂𝒌ð‘ģ𝒎 ðœđ𝒑𝒆𝒂𝒌ð‘ģ𝒉 𝑆𝑅 = 0.3 𝑆𝑅 = 0.2 𝑆𝑅 = 0.1 ðŋ𝑚 500⁄ ðŋ𝑚 50⁄   𝑇𝑌𝑃  ðŋ𝑚 ðļðī𝑔, ðļ𝐞𝑔  𝑓 𝑅𝑖𝑔𝑖𝑑  𝑇𝑟ð‘Ē𝑠𝑠 ðļð‘™ð‘’ð‘šð‘’ð‘›ð‘Ąð‘  𝑅𝑖𝑔𝑖𝑑  𝑇𝑟ð‘Ē𝑠𝑠 ðļð‘™ð‘’ð‘šð‘’ð‘›ð‘Ąð‘  𝑃𝑖𝑛 𝑆ð‘Ēð‘ð‘ð‘œð‘Ÿð‘Ą 𝑅𝑜𝑙𝑙𝑒𝑟 𝑆ð‘Ēð‘ð‘ð‘œð‘Ÿð‘Ą (TYP) ðŋℎ  ðŋ𝑚 − ðŋℎ 2⁄   ðŋ𝑚 − ðŋℎ 2⁄   ðŋ𝑚 500⁄  𝜔𝑝 𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝜔𝑠 𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒 ðŋ𝑚 100⁄áī‡ð‘†ð‘… 𝑓 ðŋ𝑚⁄ ðŋ𝑚 ðļ𝐞𝑔 áī‡ ðļðī𝑔 áī‡ 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                           ðŋ𝑚 ðļ𝐞𝑔 áī‡6 ðļðī𝑔 áī‡ 𝜔𝑝 𝜔𝑠 𝜔𝑝0.2 0.3 0.4 0.5 0.69633690.1 0.15 0.2 0.25 0.39633690.2 0.3 0.4 0.5 0.65253501751753505250.1 0.15 0.2 0.25 0.35253501751753505250.2 0.3 0.4 0.5 0.63002001001002003000.1 0.15 0.2 0.25 0.3300200100100200300          𝑆𝑅 = 0.1 𝑆𝑅 = 0.2 𝑆𝑅 = 0.3          ðŋ𝑅 = 0.2 ðŋ𝑅 = 0.4 ðŋ𝑅 = 0.6ðŋ𝑅 𝑆𝑅Peak Deflections (m)  Peak Bending Moments (MN·m) Peak Axial Force (MN)    ðđ𝑝𝑒𝑎𝑘 ≅ ∑ðŧð‘ ð‘Ąð‘  + ðŧ𝑐𝑠 =(𝜔𝑠 + 𝜔𝑝)[ðŋ𝑚 1 − ðŋ𝑅 ]28 ℎ𝑇+ ðŧ𝑐𝑚ðŧ𝑐𝑚𝜔𝑐𝑝 ≈ 𝜔𝑐ℎ𝑇 ≈ 𝑓ðđ𝑝𝑒𝑎𝑘 ≅(ð›ū𝑐𝑚ðī𝑐𝑚 + 𝜔𝑠 + 𝜔𝑝)ðŋ𝑚8 𝑆𝑅ð›ū𝑐𝑚 ðī𝑐𝑚ðŧ𝑐𝑚/ 𝑉𝑐𝑚 = Horizontal/Vertical reaction from suspension cable    ðŧ𝑐𝑠/ 𝑉𝑐𝑠 = Horizontal/Vertical reaction from anchor cable   ∑ðŧð‘ ð‘Ąð‘ / âˆ‘ð‘‰ð‘ ð‘Ąð‘ = Cumulative horizontal/vertical reaction from stay cables in side span  ℎ𝑇  = Tower height  𝑉𝑇𝑏𝑎𝑠𝑒= Reaction at tower base  𝜔𝑠/ 𝜔𝑝 = Dead/Live loading  ðŋ𝑅 = Suspension ratio  ðŋ𝑚 = Main span length   ðŋ𝑚 1 − ðŋ𝑅 2⁄  ℎ𝑇 ðŧ𝑐𝑚 (𝜔𝑠 + 𝜔𝑝) ðŧ𝑐𝑚 ∑ðŧð‘ ð‘Ąð‘  + ðŧ𝑐𝑠  âˆ‘ð‘‰ð‘ ð‘Ąð‘  +𝑉𝑐𝑠 + 𝑉𝑐𝑚  𝑉𝑇𝑏𝑎𝑠𝑒 Stay Cables ∑ðŧð‘ ð‘Ąð‘  + ðŧ𝑐𝑠  Main Span Side Span C  Tower L  Suspension Ratio = 0.2 Suspension Ratio = 0.4 Suspension Ratio = 0.6             ðŋ𝑚 ðļðī𝑔 áī‡5 𝜔𝑝 𝜔𝑠 𝜔𝑝0 5 10 15 20 25 3096303690 5 10 15 20 25 3096303690 5 10 15 20 25 3096303690 5 10 15 20 25 30180012006000600120018000 5 10 15 20 25 30180012006000600120018000 5 10 15 20 25 3018001200600060012001800          𝑆𝑅 = 0.1 𝑆𝑅 = 0.2 𝑆𝑅 = 0.3Peak Bending Moments (MN·m) Peak Deflections (m)  ðļ𝐞𝑔  𝑀𝑁𝑚 ð‘Ĩ 106 ðļ𝐞𝑔  𝑀𝑁𝑚 ð‘Ĩ 106 ðļ𝐞𝑔  𝑀𝑁𝑚 ð‘Ĩ 106 Suspension Ratio = 0.2 Suspension Ratio = 0.4 Suspension Ratio = 0.6       ðŋ𝑚 ðļðī𝑔 áī‡5 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 = 0.3ðœđ𝒑𝒆𝒂𝒌𝑚ð‘ķðœđ𝒑𝒆𝒂𝒌−− ð‘ī𝒑𝒆𝒂𝒌𝑚ð‘ķð‘ī𝒑𝒆𝒂𝒌++ ð‘ī𝒑𝒆𝒂𝒌𝑚ð‘ķð‘ī𝒑𝒆𝒂𝒌−− ðļ𝐞𝑔  𝑀𝑁𝑚 ð‘Ĩ 106 ðļ𝐞𝑔  𝑀𝑁𝑚 ð‘Ĩ 106 ðļ𝐞𝑔  𝑀𝑁𝑚 ð‘Ĩ 106 ð›ŋð‘Ģ1 =𝜎𝑐−𝑓ðļ𝑒𝑓𝑓−𝑓(1𝑠𝑖𝑛𝜃𝑐−𝑓1𝑐𝑜𝑠𝜃𝑐−𝑓)ðŋ𝑚2ð›ŋℎ =𝜎𝑐−𝑎ðļ𝑒𝑓𝑓−𝑎(1𝑐𝑜𝑠𝜃𝑐−𝑎1𝑠𝑖𝑛𝜃𝑐−𝑎) ℎ𝑇ð›ŋð‘Ģ2 =𝜎𝑐−𝑎ðļ𝑒𝑓𝑓−𝑎(1𝑐𝑜𝑠𝜃𝑐−𝑎1𝑠𝑖𝑛𝜃𝑐−𝑎)ðŋ𝑚2𝜎𝑐−𝑓 ðļ𝑒𝑓𝑓−𝑓 𝜎𝑐−𝑎 ðļ𝑒𝑓𝑓−𝑎𝜎𝑐−𝑓 = (𝜔𝑅1 + 𝜔𝑅) 𝜎𝑎𝑙𝑙𝑜ð‘Ī𝜎𝑐−𝑎 = (𝜔𝑅1 + 𝜔𝑅 − 4ðŋ𝑆𝑅2)𝜎𝑎𝑙𝑙𝑜ð‘Ī𝜔𝑅 𝜔𝑝: 𝜔𝑠 ðŋ𝑆𝑅ðŋ𝑠: ðŋ𝑚 ð›ŋℎ ðŋ𝑠 ðŋ𝑚 2⁄  ℎ𝑇 𝜃𝑐−𝑓 𝜃𝑐−𝑎  𝜔𝑠 𝜔𝑝 ð›ŋð‘Ģ1 ð›ŋð‘Ģ2 (From Elongation of Forestay) (From Elongation of Anchor Cable) Forestay Anchor Cable ð›ŋð‘Ģ1 + ð›ŋð‘Ģ2ð›ŋð‘Ģ1= 1 + (1 + 𝜔𝑅1 + 𝜔𝑅 − 4ðŋ𝑆𝑅2) (𝑠𝑖𝑛𝜃𝑐−𝑓𝑠𝑖𝑛𝜃𝑐−𝑎𝑐𝑜𝑠𝜃𝑐−𝑓𝑐𝑜𝑠𝜃𝑐−𝑎)𝜃𝑐−𝑓 𝜃𝑐−𝑎ð›ŋð‘Ģ1 + ð›ŋð‘Ģ2ð›ŋð‘Ģ1= 1 + (1 + 𝜔𝑅1 + 𝜔𝑅 − 4ðŋ𝑆𝑅2) (2ðŋ𝑆𝑅) (ℎ𝑇𝑅 + ðŋ𝑆𝑅24ℎ𝑇𝑅2 + 1)ℎ𝑇𝑅 ℎ𝑇: ðŋ𝑚                           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          ðŋ𝑆𝑅 = 0.35 ðŋ𝑆𝑅 = 0.40 ðŋ𝑆𝑅 = 0.45ðŋ𝑆𝑅 ℎ𝑇𝑅ð›ŋð‘Ģ1 + ð›ŋð‘Ģ2ð›ŋð‘Ģ1Typical Range |ð›ŋ𝑒|𝑝𝑒𝑎𝑘|ð›ŋ𝑟|𝑝𝑒𝑎𝑘 ðŋ𝑚 ðŋ𝑆𝑅 ðļ𝐞𝑔 áī‡ ðļðī𝑔 áī‡ 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                            ðŋ𝑚  ðŋ𝑆𝑅 ðļ𝐞𝑔 áī‡6 ðļðī𝑔 áī‡ 𝜔𝑝 𝜔𝑠 𝜔𝑝                               ðŋ𝑚  ðŋ𝑆𝑅 ðļ𝐞𝑔 áī‡6 ðļðī𝑔 áī‡ 𝜔𝑝 𝜔𝑠 𝜔𝑝0.1 0.15 0.2 0.25 0.312340.2 0.3 0.4 0.5 0.69633690.1 0.15 0.2 0.25 0.39633690.2 0.3 0.4 0.5 0.65253501751753505250.1 0.15 0.2 0.25 0.3525350175175350525𝑆𝑅|ð›ŋ𝑒|𝑝𝑒𝑎𝑘|ð›ŋ𝑟|𝑝𝑒𝑎𝑘ðŋ𝑅 = 0.6 ðŋ𝑅 = 0.4 ðŋ𝑅 = 0.2           𝑆𝑅 = 0.1 𝑆𝑅 = 0.2 𝑆𝑅 = 0.3          ðŋ𝑅 = 0.2 ðŋ𝑅 = 0.4 ðŋ𝑅 = 0.6ðŋ𝑅 𝑆𝑅Peak Deflections (m)  Peak Bending Moments (MN·m) Suspension Ratio = 0.2 Suspension Ratio = 0.4 Suspension Ratio = 0.6                   ðŋ𝑚  ðŋ𝑆𝑅 ðļðī𝑔 áī‡5 𝜔𝑝 𝜔𝑠 𝜔𝑝0 5 10 15 20 25 3096303690 5 10 15 20 25 3096303690 5 10 15 20 25 3096303690 5 10 15 20 25 30180012006000600120018000 5 10 15 20 25 30180012006000600120018000 5 10 15 20 25 3018001200600060012001800          𝑆𝑅 = 0.1 𝑆𝑅 = 0.2 𝑆𝑅 = 0.3Peak Bending Moments (MN·m) Peak Deflections (m)  ðļ𝐞𝑔  𝑀𝑁𝑚 ð‘Ĩ 106 ðļ𝐞𝑔  𝑀𝑁𝑚 ð‘Ĩ 106 ðļ𝐞𝑔  𝑀𝑁𝑚 ð‘Ĩ 106 𝑑 ∝ ðŋ𝑚ð›ŋ𝑝𝑒𝑎𝑘 =5𝜔𝑝ðŋ𝑚4384ðļ𝐞𝑔𝐞𝑔 ∝ ðŋ𝑚3Suspension Ratio = 0.2 Suspension Ratio = 0.4 Suspension Ratio = 0.6           ðļ𝐞𝑔 ðŋ𝑚3 ðļðī𝑔 áī‡5 𝜔𝑝 𝜔𝑠 𝜔𝑝250 500 750 1000963369250 500 750 1000963369250 500 750 1000963369250 500 750 10000.60.40.20.20.40.6250 500 750 10000.60.40.20.20.40.6250 500 750 10000.60.40.20.20.40.6          𝑆𝑅 = 0.1 𝑆𝑅 = 0.2 𝑆𝑅 = 0.3ðŋ𝑚  𝑚 ðŋ𝑚  𝑚 ðŋ𝑚  𝑚 ðœđ𝒑𝒆𝒂𝒌ð‘ģ𝒎 ð‘ī𝒑𝒆𝒂𝒌ð‘ģ𝒎𝟐 [ð‘īð‘ĩ𝒎] (𝜎𝑏,ðŋðŋ)𝑝𝑒𝑎𝑘 =𝑀𝑝𝑒𝑎𝑘𝑐𝐞𝑔𝑐𝑐 𝑑𝑐 = 𝑘𝑑𝑘 𝑑ðŋ𝑚 𝐞𝑔 ðŋ𝑚3(𝜎𝑏,ðŋðŋ)𝑝𝑒𝑎𝑘 ∝𝑀𝑝𝑒𝑎𝑘ðŋ𝑚2𝑑 ∝ ðŋ𝑚 𝐞𝑔 ∝ ðŋ𝑚3(𝜎𝑎,ðŋðŋ)𝑝𝑒𝑎𝑘 ≅𝜔𝑝ðŋ𝑚8 𝑆𝑅ðī𝑔ðī𝑔𝐞𝑔 𝑑𝑑𝐞𝑔 ≅ðī𝑔𝑑24𝑑 ∝ ðŋ𝑚 𝐞𝑔 ∝ ðŋ𝑚3ðī𝑔 ∝ ðŋ𝑚𝑑 ∝ ðŋ𝑚𝐞𝑔 ∝ ðŋ𝑚3(𝜎𝑎,𝐷ðŋ)𝑝𝑒𝑎𝑘 ≅ ð›ū𝑐𝑚ðī𝑐𝑚 + 𝜔𝑠 ðŋ𝑚8 𝑆𝑅ðī𝑔(𝜎𝑎,𝐷ðŋ)𝑝𝑒𝑎𝑘 ≅(ð›ū𝑠ðī𝑔𝛞𝑠𝑑𝑙)ðŋ𝑚8 𝑆𝑅ðī𝑔 𝑑 ðī𝑔 2⁄  ðī𝑔 2⁄  ðī𝑔 Transverse Stiffener Longitudinal Stiffener (TYP) Top Plate Bottom Plate Idealized Top Plate Idealized Bottom Plate 𝛞𝑠𝑑𝑙ðī𝑔𝜔𝑅 = 𝜔𝑝 𝜔𝑠⁄ðļ𝐞𝑔 áī‡    ðŋ𝑚  ðŋ𝑆𝑅 𝑆𝑅 ðŋ𝑅 ðļ𝐞𝑔 áī‡ ðļðī𝑔 𝜔𝑝32101237550250255075Deflection Envelope (m) Bending Moment Envelope (MN·m) 𝜔𝑅 = 0.6𝜔𝑅 = 0.4𝜔𝑅 = 0.2𝑆𝑅 = 0.2ðŋ𝑅 = 0.4ð›ŋ𝑝𝑒𝑎𝑘2ð›ŋ𝑝𝑒𝑎𝑘1≅(𝜔𝑅21 + 𝜔𝑅2)(𝜔𝑅11 + 𝜔𝑅1)ð›ŋ𝑝𝑒𝑎𝑘2ð›ŋ𝑝𝑒𝑎𝑘1≅(𝜔𝑅21 + 𝜔𝑅2) + (𝜔𝑅21 + 𝜔𝑅2 − 4ðŋ𝑆𝑅2)ðū𝑟(𝜔𝑅11 + 𝜔𝑅1) + (𝜔𝑅11 + 𝜔𝑅1 − 4ðŋ𝑆𝑅2)ðū𝑟ðū𝑟ðū𝑟ðū𝑟 = (2ðŋ𝑆𝑅)(ℎ𝑇𝑅 + ðŋ𝑆𝑅24ℎ𝑇𝑅2 + 1)                  ðŋ𝑚  ðŋ𝑆𝑅 𝑆𝑅 ðŋ𝑅 ðļ𝐞𝑔 áī‡  ðļðī𝑔 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   ÜģāŪŋāŊ”āŊĄ ≅ßĐāŊĶāŊ§ßŠāŊĶāŊ§ āĩŦßąāŊĶ + ßąāŊĢāĩŊÜŪāŊ āŽķ āĩĪ2 āĩŽâ„ŽāŊÜŪāŊ āĩ° +16 āĩŽÜŪāŊ â„ŽāŊāĩ°āĩĻ (5.1)  ÜģāŊāŊ”āŊĨāŊĢ ≅ßĐāŊĶāŊ§ßŠāŊĶāŊ§ āĩŦßąāŊĶ + ßąāŊĢāĩŊÜŪāŊ āŽķ āĩĪāĩŽâ„ŽāŊÜŪāŊ āĩ° +14 āĩŽÜŪāŊ â„ŽāŊāĩ°āĩĻ (5.2) ßĐāŊĶāŊ§ and  ߊāŊĶāŊ§ are the density and design stress of the stay cable material; ßą āŊĶ and ßą āŊĢ are the magnitude of the uniformly distributed dead and live load; ÜŪāŊ  is the main span length; and ℎāŊ 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 (ℎāŊ: ÜŪāŊ ).   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 0.500.01.02.03.04.0ÜŠÜ―ÝŽÝŒÜĻÜ―ÝŠāĒŽāĒ€ āĄļāĒ“⁄āĄ―āĢ‹āĒ™āĒšāĢŒāĒ™āĒš āĩŦāĢ“āĒ™ + āĢ“āĒ–āĩŊāĄļāĒ“āŦ› 	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 (SchÞller, 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, etcâ€Ķ) 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.  ℎāŊ is the height of the towers above deck; Üū is the height above deck of the first tower connection; and Ü― is the length of the array.        Figure 5.6: Idealized Array of Stay Cables Ü―â„ŽāŊÜū	āĩŦßąāŊĶ + ßąāŊĢāĩŊݔC		TowerL	݀ݔ(ℎāŊ − Üū)Ü― 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 (ßąāŊĶ + ßąāŊĢ) acting over a tributary length of ݀ݔ, is given by the following expression:  ݀Üķ = āĩŦßąāŊĶ + ßąāŊĢāĩŊāķ§á‰‚ቀℎāŊ − ÜūÜ― ቁ ݔ + ÜūቃāŽķ+ ݔāŽķቂቀℎāŊ − ÜūÜ― ቁ ݔ + Üūቃ݀ݔ + 12݀ÜķߊāŊĶāŊ§ ߛāŊĶāŊ§āķĻāĩĪāĩŽâ„ŽāŊ − ÜūÜ― āĩ°Ý” + ÜūāĩĻāŽķ+ ݔāŽķ (5.3) where ߊāŊĶāŊ§ and ߛāŊĶāŊ§ 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 ݀Üķ is isolated, the following equation is obtained,   ݀Üķ =āĩŦßąāŊĶ + ßąāŊĢāĩŊāķ§á‰‚ቀℎāŊ − ÜūÜ― ቁ ݔ + ÜūቃāŽķ+ ݔāŽķቂቀℎāŊ − ÜūÜ― ቁ ݔ + Üūቃ ቈ1 −12ߛāŊĶāŊ§ßŠāŊĶāŊ§ āķ§á‰‚ቀℎāŊ − ÜūÜ― ቁ ݔ + ÜūቃāŽķ+ ݔāŽķ቉݀ݔ (5.4) The quantity of cable steel in the infinitesimal segment, ݀Üģ, will then be given by the density of the stay cable material, ßĐāŊĶāŊ§, multiplied by the cable area required, multiplied by the length of the segment:  ݀Üģ = ßĐāŊĶāŊ§Ý€ÜķߊāŊĶāŊ§āķĻāĩĪāĩŽâ„ŽāŊ − ÜūÜ― āĩ°Ý” + ÜūāĩĻāŽķ+ ݔāŽķ (5.5) When Equation (5.4) is substituted in for ݀Üķ, the quantity becomes,  ݀Üģ = āĩŦßąāŊĶ + ßąāŊĢāĩŊßĐāŊĶāŊ§ßŠāŊĶāŊ§á‰‚ቀℎāŊ − ÜūÜ― ቁ ݔ + ÜūቃāŽķ+ ݔāŽķቂቀℎāŊ − ÜūÜ― ቁ ݔ + Üūቃ ቈ1 −12ߛāŊĶāŊ§ßŠāŊĶāŊ§ āķ§á‰‚ቀℎāŊ − ÜūÜ― ቁ ݔ + ÜūቃāŽķ+ ݔāŽķ቉݀ݔ (5.6) Thereupon, the total quantity of cable steel in the array can be obtained by integrating ݀Üģ over the length of the array,  Üģ = āĩŦßąāŊĶ + ßąāŊĢāĩŊßĐāŊĶāŊ§ßŠāŊĶāŊ§ āķąá‰‚ቀℎāŊ − ÜūÜ― ቁ ݔ + ÜūቃāŽķ+ ݔāŽķቂቀℎāŊ − ÜūÜ― ቁ ݔ + Üūቃ ቈ1 −12ߛāŊĶāŊ§ßŠāŊĶāŊ§ āķ§á‰‚ቀℎāŊ − ÜūÜ― ቁ ݔ + ÜūቃāŽķ+ ݔāŽķ቉āŊ”āŽī݀ݔ (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 (Üū = ℎāŊ).  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 (Üū = 3/4ℎāŊ).  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: Ü― = 500m, ߊāŊ”āŊŸāŊŸāŊĒāŊŠ = 800MPa, ߛāŊĶāŊ§āŊ”āŊŽ = 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 ߛāŊĶāŊ§āŊ”āŊŽ equal to zero, at which point, the expression becomes,   ÜģāŊ”āŊĢāŊĢāŊĨāŊĒāŊŦ =ßĐāŊĶāŊ§ßŠāŊĶāŊ§ Ü―āŽķāĩŦßąāŊĶ + ßąāŊĢāĩŊ āĩŽÜ―3ℎāŊāĩ° ቈ1 + 3 āĩŽâ„ŽāŊÜ― āĩ°āŽķ቉ (5.8) Note that if Ü―  is replaced by ÜŪāŊ /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 Üĩ݁݉݅ − ÜĻÜ―ÝŠÜĻÜ―ÝŠāĒŽāĒ€ (āŦ›āĒ‡)⁄āĄ―āĢ‹āĒ™āĒšāĢŒāĒ™āĒš āĩŦāĢ“āĒ™ + āĢ“āĒ–āĩŊāĒ‡āŦ›		5.2  Derivation of Bridge Quantities   164    Figure 5.8: The Influence of Self-Weight on Cable Steel Quantity in a Fan Arrangement  Parameters: ߊāŊĶāŊ§ = 800MPa, ߛāŊĶāŊ§ = 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 ÜŪāŊ‹ as the suspension ratio, ℎāŊāŊ‹ as the tower height-to-main span ratio, ÜŪāŊŒāŊ‹ as the side-to-main span length ratio, and ÜŪāŊ  as the main span length, the final expression may be written as,    ÜģāŊĶāŊ§ =ßĐāŊĶāŊ§ßŠāŊĶāŊ§ ÜŪāŊ āŽķāĩŦßąāŊĶ + ßąāŊĢāĩŊ ቈ(1 − ÜŪāŊ‹)āŽ·12ℎāŊāŊ‹ + (1 − ÜŪāŊ‹)ℎāŊāŊ‹ +23ÜŪāŊŒāŊ‹āŽ·â„ŽāŊāŊ‹ + 2ℎāŊāŊ‹ÜŪāŊŒāŊ‹á‰‰ (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,   ÜĢāŊ–āŊ  =1ߛāŊ–āŊ āĩŦßąāŊĶ + ßąāŊĢāĩŊ ßĶ ÜŪāŊ‹ (2 − ÜŪāŊ‹)āķĻ1 + 16 ቀ ℎāŊāŊ‹2 − ÜŪāŊ‹á‰āŽķ1 − ßĶ āķĻ1 + 16 ቀ ℎāŊāŊ‹2 − ÜŪāŊ‹á‰āŽķ (5.10) where as before,   ßĶ = ߛāŊ–āŊ  ÜŪāŊ 8 ߊāŊ–āŊ â„ŽāŊāŊ‹ Using Equation (5.10), the quantity of cable steel can then be expressed as,  0.00 0.10 0.20 0.30 0.40 0.500.960.970.980.991.00āĒŽāĒ€ (āŦ›āĒ‡)⁄āĄ―āĒ‡āĒ–āĒ–āĒ˜āĒ•āĒžāĄ― 		Ü― = 500݉	Ü― = 400݉	Ü― = 300݉	Ü― = 200݉	Ü― = 100݉	5.2  Derivation of Bridge Quantities   165  Üģ = ßĐāŊ–āŊ ÜĢāŊ–āŊ āĩĢ2ÜŪāŊ–āŊ _āŽĩ + ÜŪāŊ–āŊ _āŽķāĩ§ (5.11) where ßĐāŊ–āŊ  and ÜĢāŊ–āŊ  are the density and area of the cable material.  In addition, ÜŪāŊ–āŊ _āŽĩ is the length of the cable in the stayed region (the value is multiplied by two because there are two stayed regions) and ÜŪāŊ–āŊ _āŽķ 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 ÜŪāŊ–āŊ _āŽĩ and ÜŪāŊ–āŊ _āŽķ.  The expression for the cable quantity then becomes,  Üģ = ßĐāŊ–āŊ ÜĢāŊ–āŊ  ቎2āķą āķĻ1 + āĩŽÝ€Ý•(ݔ)݀ݔ āĩ°āŽķ݀ݔ + āķą āķĻ1 + āĩŽÝ€Ý•(ݔ)݀ݔ āĩ°āŽķ݀ݔāŊ…āģ˜(āŽĩāŽūāŊ…āģƒ)āŽķāŊ…āģ˜(āŽĩāŽŋāŊ…āģƒ)āŽķāŊ…āģ˜(āŽĩāŽŋāŊ…āģƒ)āŽķāŽī቏ (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,   ÜŪāŊ–āŊ _āŽĩ_āŊ”āŊĢāŊĢāŊĨāŊĒāŊŦ = āķĻቆÜŪāŊ (1 − ÜŪāŊ‹)2 ቇāŽķ+ ݕāŪŧāŽķ (5.13) where ݕāŪŧ 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 ݕāŪŧ, then after substitution of Equations (4.9) and (4.11), Equation (5.13) reduces to,   ÜŪāŊ–āŊ _āŽĩ_āŊ”āŊĢāŊĢāŊĨāŊĒāŊŦ =ÜŪāŊ (1 − ÜŪāŊ‹)2 āķĻ1 + ℎāŊāŊ‹āŽķ āĩŽ 42 − ÜŪāŊ‹āĩ°āŽķ (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,  ÜŪāŊ–āŊ _āŽķ_āŊ”āŊĢāŊĢāŊĨāŊĒāŊŦ = āķą ቈ1 +12 āĩŽÝ€Ý•(ݔ)݀ݔ āĩ°āŽķ቉ ݀ݔāŊ…āģ˜(āŽĩāŽūāŊ…āģƒ)āŽķāŊ…āģ˜(āŽĩāŽŋāŊ…āģƒ)āŽķ (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  ÜŪāŊ–āŊ _āŽķ_āŊ”āŊĢāŊĢāŊĨāŊĒāŊŦ = ÜŪāŊ ÜŪāŊ‹ ቈ1 +83 ℎāŊāŊ‹āŽķ āĩŽ 12 − ÜŪāŊ‹āĩ°āŽķ቉ (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),   ÜģāŊ”āŊĢāŊĢāŊĨāŊĒāŊŦ = ßĐāŊ–āŊ ÜĢāŊ–āŊ ÜŪāŊ  āĩĶ(1 − ÜŪāŊ‹)āķĻ1 + āĩŽ4ℎāŊāŊ‹2 − ÜŪāŊ‹āĩ°āŽķ+ ÜŪāŊ‹ ቈ1 +83 āĩŽâ„ŽāŊāŊ‹2 − ÜŪāŊ‹āĩ°āŽķ቉āĩŠ (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: ÜŪāŊ  = 1000m, ߊāŊ–āŊ  = 800MPa, ߛāŊ–āŊ  = 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,   ÜģāŊ–āŊ  =1݃ ÜŪāŊ āĩŦßąāŊĶ + ßąāŊĢāĩŊÜŪāŊ‹(2 − ÜŪāŊ‹)ßĶߟ1 − ßĶߟ ቈ(1 − ÜŪāŊ‹)ߟ + ÜŪāŊ‹(ߟāŽķ + 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 0.500.980.991.001.011.02āĒŽāĒ€āĄūāĄ―āĒ‡āĒ–āĒ–āĒ˜āĒ•āĒžāĄ― 		 ÜŪāŊ‹ = 0.2 ÜŪāŊ‹ = 0.4 ÜŪāŊ‹ = 0.65.2  Derivation of Bridge Quantities   167  ߟ = āķĻ1 + 16 āĩŽ ℎāŊāŊ‹2 − ÜŪāŊ‹āĩ°āŽķ 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 ÜŠāŊ–āŊ / āŊ–ÜļāŊ  =	Horizontal/Vertical	reaction	from	suspension	cable	in	main	span		ÜģāŊ–āŊ 	=	Suspension	cable	quantity		ÜģāŊ–āŊĶ	=	Anchor	cable	quantity		ÜģāŊĶāŊ§āŊ 	=	Main	span	stay	cable	quantity		ÜģāŊĶāŊ§āŊĶ	=	Side	span	stay	cable	quantity		݃=	Standard	acceleration	due	to	gravity		ℎāŊ	=	Tower	height		ÜīāŊĢ=	Reaction	at	anchor	pier		ßąāŊĶ/	ßąāŊĢ	=	Dead/Live	loading		ÜŪāŊ‹	=	Suspension	ratio		ÜŪāŊ 	=	Main	span	length		ÜŪāŊĶ =	Side	span	lengthÜŪāŊ (1 − ÜŪāŊ‹) 6⁄ÜŪāŊ (1 − ÜŪāŊ‹) 2⁄ÜŪāŊĶ	ÜŪāŊĶ 3⁄ 	ℎāŊÜŠāŊ–āŊ 	ÜģāŊ–āŊĶ݃4 + āŊ–ÜļāŊ 	ßąāŊĶßąāŊĢÜīāŊĢ	ÜģāŊĶāŊ§āŊ Ýƒ 2⁄ÜģāŊĶāŊ§āŊĶ݃ 2⁄ 	ÜģāŊ–āŊĶ݃4 	Main	SpanSide	Span	C		Tower	L		ÜŠāŊ–āŊ 5.2  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,   ÜīāŊĢ =āĩŦßąāŊĶ + ßąāŊĢāĩŊÜŪāŊ (1 − ÜŪāŊ‹)āŽķ8ÜŪāŊŒāŊ‹ + ÜŠāŊ–āŊ â„ŽāŊāŊ‹ÜŪāŊŒāŊ‹ +ÜģāŊĶāŊ§āŊ Ýƒ12ÜŪāŊŒāŊ‹ (1 − ÜŪāŊ‹) −ÜģāŊĶāŊ§āŊĶ݃6− 12ßąāŊĶÜŪāŊ ÜŪāŊŒāŊ‹ −ÜģāŊ–āŊĶ݃4  (5.19) Vertical equilibrium at the anchor pier then gives the vertical component of the anchor cable chord tension,  āŊ–ÜļāŊĶ =āĩŦßąāŊĶ + ßąāŊĢāĩŊÜŪāŊ (1 − ÜŪāŊ‹)āŽķ8ÜŪāŊŒāŊ‹ + ÜŠāŊ–āŊ â„ŽāŊāŊ‹ÜŪāŊŒāŊ‹ +ÜģāŊĶāŊ§āŊ Ýƒ12ÜŪāŊŒāŊ‹ (1 − ÜŪāŊ‹) −ÜģāŊĶāŊ§āŊĶ݃6 −12ßąāŊĶÜŪāŊ ÜŪāŊŒāŊ‹ (5.20) Equation (4.9) can be substituted in for ÜŠāŊ–āŊ  (during the substitution ßąāŊĶ must be replaced by ßąāŊĶ + ßąāŊĢ 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,  āŊ–ÜķāŊĶ =124ÜŪāŊŒāŊ‹ āĩĢ3ÜŪāŊ āĩŦßąāŊĶ + ßąāŊĢāĩŊ + 3ÜģāŊ–āŊ Ýƒ + 2ÜģāŊĶāŊ§āŊ Ýƒ(1 − ÜŪāŊ‹) − 4ÜģāŊĶāŊ§āŊĶ݃(ÜŪāŊŒāŊ‹)− 12ÜŪāŊ ÜŪāŊŒāŊ‹āŽķßąāŊĶāĩ§āķ§ÜŪāŊŒāŊ‹āŽķ + ℎāŊāŊ‹āŽķℎāŊāŊ‹  (5.21) It then follows that the total quantity of cable steel in both anchor cables is, ÜģāŊ–āŊĶ = 2ßĐāŊ–āŊĶ āŊ–ÜķāŊĶߊāŊ–āŊĶ ÜŪāŊ āķ§ÜŪāŊŒāŊ‹āŽķ + ℎāŊāŊ‹āŽķ (5.22) where ßĐāŊ–āŊĶ and ߊāŊ–āŊĶ are the density and design stress of the anchor cable material.  When expanded, Equation (5.22) becomes,  ÜģāŊ–āŊĶ =112ßĐāŊ–āŊĶߊāŊ–āŊĶ ÜŪāŊ āĩĢ3ÜŪāŊ āĩŦßąāŊĶ + ßąāŊĢāĩŊ + 3ÜģāŊ–āŊ Ýƒ + 2ÜģāŊĶāŊ§āŊ Ýƒ(1 − ÜŪāŊ‹) − 4ÜģāŊĶāŊ§āŊĶ݃(ÜŪāŊŒāŊ‹)− 12ÜŪāŊ ÜŪāŊŒāŊ‹āŽķßąāŊĶāĩ§ āĩŽÜŪāŊŒāŊ‹â„ŽāŊāŊ‹ +ℎāŊāŊ‹ÜŪāŊŒāŊ‹āĩ° (5.23)5.2  Derivation of Bridge Quantities   169 where from Equation (5.9),   ÜģāŊĶāŊ§āŊ  =ßĐāŊĶāŊ§ßŠāŊĶāŊ§ ÜŪāŊ āŽķāĩŦßąāŊĶ + ßąāŊĢāĩŊ(1 − ÜŪāŊ‹)ℎāŊāŊ‹ ቈ(1 − ÜŪāŊ‹)āŽķ12ℎāŊāŊ‹āŽķ+ 1቉ (5.24)and,  ÜģāŊĶāŊ§āŊĶ =ßĐāŊĶāŊ§ßŠāŊĶāŊ§ ÜŪāŊ āŽķāĩŦßąāŊĶ + ßąāŊĢāĩŊℎāŊāŊ‹ÜŪāŊŒāŊ‹ ቈ23 āĩŽÜŪāŊŒāŊ‹â„ŽāŊāŊ‹āĩ°āŽķ+ 2቉ (5.25) Note from Equation (5.23) that the contribution from Üģ āŊĶāŊ§āŊ  and Üģ āŊĶāŊ§āŊĶ amount to zero when the stay cables are symmetrically arranged about the centre of the towers (i.e. when ÜŪāŊŒāŊ‹ = (1 − ÜŪāŊ‹)/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  āĩŦßąāŊĶ + ßąāŊĢāĩŊݕāŪŧ ݂ÜŪāŊ (1 − ÜŪāŊ‹) 2⁄ÜŪāŊ (1 − ÜŪāŊ‹) 2⁄ ÜŪāŊ›ÜŪāŊ Ý•(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,  Üģ = ßĐāŊ›āĩŦÜĢāŊ›_āŊ§āŊĒāŊ§āŊ”āŊŸāĩŊāĩŦÜŪāŊ›_āŊ”āŊĐāŊšāĩŊ (5.26) When expanded, the expression becomes,   Üģ = ßĐāŊ›āĩŦßąāŊĶ + ßąāŊĢāĩŊÜŪāŊ›ßŠāŊ› āĩŽâ„ŽāŊ − ݕāŪŧ3 āĩ° (5.27) where ߊāŊ› is the design stress of the hangers.  Substituting in Equations (4.9) and (4.11) then gives,   Üģ = 13ßĐāŊ›ßŠāŊ› ÜŪāŊ āŽķāĩŦßąāŊĶ + ßąāŊĢāĩŊℎāŊāŊ‹ÜŪāŊ‹āŽ· āĩĶßąāŊĶ + ቀÜģāŊ–āŊ ÝƒÜŪāŊ  ቁ߱āŊĶÜŪāŊ‹(2 − ÜŪāŊ‹) + ቀÜģāŊ–āŊ ÝƒÜŪāŊ  ቁāĩŠ (5.28) However, a simpler expression can be obtained by neglecting the contribution from the self-weight of the suspension cable,     ÜģāŊ”āŊĢāŊĢāŊĨāŊĒāŊŦ =13ßĐāŊ›ßŠāŊ› ÜŪāŊ āŽķāĩŦßąāŊĶ + ßąāŊĢāĩŊℎāŊāŊ‹ÜŪāŊ‹āŽķ āĩŽ12 − ÜŪāŊ‹āĩ° (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 ÜģāŊ› (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: ÜŪāŊ  = 1000m, ߊāŊ–āŊ  = 800MPa, ߛāŊ–āŊ  = 0.09MN/m3, ßąāŊ‹ = 0.6                   Figure 5.13: Hanger Cable Quantity in Relation to Total Cable Steel Quantity  Parameters: ÜŪāŊ  = 1000m, ßąāŊ‹ = 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 0.500.0000.0200.040āĒŽāĒ€āĄūāĄ―āĒ‡āĒ–āĒ–āĒ˜āĒ•āĒžāĄ― 		ÜŪāŊ‹ = 0.2 ÜŪāŊ‹ = 0.4 ÜŪāŊ‹ = 0.6	āĒŽāĒ€āĄūāĄ―āĒŽāĄ―āĒŽ + āĄ―āĒ‰āĒ“ + āĄ―āĒ‰āĒ™ + āĄ―āĒ™āĒš		ÜŪāŊ‹ = 0.2 ÜŪāŊ‹ = 0.4 ÜŪāŊ‹ = 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,  āŊĐÜ° =ÜģāŊĶāŊ§āŊ Ýƒ2 āĩŽ1 +1 − ÜŪāŊ‹6ÜŪāŊŒāŊ‹ āĩ° +ÜģāŊĶāŊ§āŊĶ݃3 + āŊ–ÜļāŊ  + ÜŠāŊ–āŊ  āĩŽâ„ŽāŊāŊ‹ÜŪāŊŒāŊ‹āĩ° + āĩŦßąāŊĶ + ßąāŊĢāĩŊÜŪāŊ ÜŪāŊŒāŊ‹2+ āĩŦßąāŊĶ + ßąāŊĢāĩŊ(1 − ÜŪāŊ‹)ÜŪāŊ 2 āĩŽ1 +1 − ÜŪāŊ‹4ÜŪāŊŒāŊ‹ āĩ° +ÜģāŊ–āŊĶ݃4  (5.30) Substituting in the horizontal and vertical components of the suspension cable force then gives,   āŊĐÜ°= 124ÜŪāŊŒāŊ‹ āĩĢ3ÜŪāŊ āĩŦßąāŊĶ + ßąāŊĢāĩŊ(2ÜŪāŊŒāŊ‹ + 1)āŽķ + 2ÜģāŊĶāŊ§āŊ Ýƒ(6ÜŪāŊŒāŊ‹ − ÜŪāŊ‹ + 1) + 8ÜģāŊĶāŊ§āŊĶ݃(ÜŪāŊŒāŊ‹)+ 3ÜģāŊ–āŊ Ýƒ(4ÜŪāŊŒāŊ‹ + 1) + 6ÜģāŊ–āŊĶ݃(ÜŪāŊŒāŊ‹)āĩ§ (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,   ÜļāŊ = āŊĐÜ° + āŊ§Üđ(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,  ÜĢāŊ§ = āŊĐÜ°+ āŊ§Üđ(ݖ)ߚāŊ§ßŠāŊ§  (5.33) where ߊāŊ§ is the design stress of the tower and ߚāŊ§ is the reduction coefficient to account for the effect of out-of-plane loading (ߚāŊ§ = 0.7).  Notwithstanding, the self-weight of the tower is also a function of the tower area,  āŊ§Üđ(ݖ) = ÜĢāŊ§ß›āŊ§Ý– (5.34) where ߛāŊ§ is the unit weight of the tower material.  Substituting Equation (5.34) into Equation (5.33) results in,  ÜĢāŊ§ = āŊĐܰߚāŊ§ßŠāŊ§ − ߛāŊ§Ý– (5.35) Accordingly, the total quantity of material required in both towers can be calculated as,   ÜģāŊ§ = 2ßĐāŊ§ āŊĐÜ° āķą āĩŽ1ߚāŊ§ßŠāŊ§ − ߛāŊ§Ý–āĩ°āŊ›āģ…āŽūāŊ›āēģāŽī݀ݖ (5.36)which then simplifies to,  ÜģāŊ§ =2 āŊĐܰ݃ ݈݊ āĩĶ11 − (ℎāŊ + ℎāŪŧ)ߛāŊ§ßšāŊ§ßŠāŊ§āĩŠ (5.37) The only parameter left to assign is ℎāŪŧ 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,    ℎāŪŧ = 50 āĩŽ1 −ÜŪāŊ 4000āĩ° [݅݊ ݉݁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,	ßąāŊĶ, an expression for the superstructure quantity can be readily obtained,    ÜģāŊĶ =ßąāŊĶÜŪāŊ Ýƒß™āŊĶāŊ—āŊŸ (1 + 2ÜŪāŊŒāŊ‹) (5.39) However,	ßąāŊĶ 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, ߙāŊĶāŊ—āŊŸ â‰Ĩ 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 (ßąāŊĶ) and the magnitude of the live load (ßąāŊĢ).  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 ßąāŊĶāŊĒ 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,   Δ߱āŊĶ = ߛāŊĶ[(ÜĢāŊĶāŊ  − ÜĢāŊĶāŊ āŊĒ) + (ÜĢāŊĶāŊ” − ÜĢāŊĶāŊ”āŊĒ) + (ÜĢāŊĶāŊ• − ÜĢāŊĶāŊ•āŊĒ)] (5.40) where ߛāŊĶ is the unit weight of the superstructure material.  In addition, ÜĢāŊĶāŊ /ÜĢāŊĶāŊ āŊĒ is the area required for miscellaneous transverse support members (i.e. floor beams, diaphragms, etc.), ÜĢāŊĶāŊ”/ÜĢāŊĶāŊ”āŊĒ is the area required for longitudinal axial demands, and ÜĢāŊĶāŊ•/ÜĢāŊĶāŊ•āŊĒ is the area required for longitudinal bending demands.  Accordingly, when the longitudinal bridge proportions are varied (ÜĢāŊĶāŊ  − ÜĢāŊĶāŊ āŊĒ) 5.2  Derivation of Bridge Quantities   175 can be assumed equal to zero, upon which,     Δ߱āŊĶ = ߛāŊĶ[(ÜĢāŊĶāŊ” − ÜĢāŊĶāŊ”āŊĒ) + (ÜĢāŊĶāŊ• − ÜĢāŊĶāŊ•āŊĒ)] (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 ÜĻāŊĶāŽĩ and ÜĻāŊĢāŽĩ are, together, equal to the horizontal force in the suspension cable.  Thus,    ÜĻāŊĶāŽĩ + ÜĻāŊĢāŽĩ = ÜŠāŊ–āŊ  (5.42) Substituting in the appropriate expression for  ÜŠāŊ–āŊ  then yields,   ÜĻāŊĶāŽĩ + ÜĻāŊĢāŽĩ =18ℎāŊāŊ‹ āĩĢÜģāŊ–āŊ Ýƒ + āĩŦßąāŊĶ + ßąāŊĢāĩŊÜŪāŊ‹(2 − ÜŪāŊ‹)āĩ§ (5.43) The uniform axial force components in the side span (ÜĻāŊĶāŽķāŊ” and ÜĻāŊĢāŽķ) are a result of the anchorage force required to equilibrate the unbalanced portion of the dead and live load in the stayed regions of ßąāŊĢßąāŊĶ	ÜĻāŊĶāŽ·ÜĻāŊĢāŽ·ÜĻāŊĢāŽķ	ÜĻāŊĢāŽĩ	ÜĻāŊĶāŽĩ	ÜĻāŊĶāŽķāŊ”	ÜĻāŊĶāŽķāŊ•	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,    ÜĻāŊĶāŽķāŊ” + ÜĻāŊĢāŽķ = āŊ–ÜļāŊĶāīĪāīĪāīĪāīĪÜŪāŊŒāŊ‹â„ŽāŊāŊ‹ (5.44) where āŊ–ÜļāŊĶāīĪāīĪāīĪāīĪ is the vertical component of the anchorage force.  āŊ–ÜļāŊĶāīĪāīĪāīĪāīĪ 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,     ÜĻāŊĶāŽķāŊ” + ÜĻāŊĢāŽķ =124ℎāŊāŊ‹ āĩĢ3āĩŦßąāŊĶ + ßąāŊĢāĩŊÜŪāŊ (1 − ÜŪāŊ‹)āŽķ + 2ÜģāŊĶāŊ§āŊ Ýƒ(1 − ÜŪāŊ‹) − 4ÜģāŊĶāŊ§āŊĶ݃ÜŪāŊŒāŊ‹âˆ’ 12ßąāŊĶÜŪāŊ ÜŪāŊŒāŊ‹āŽķāĩ§ (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 ÜĻāŊĶāŽķāŊ•.  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,  ÜĻāŊĶāŽķāŊ•āŊ” = āķąßąāŊĶݔℎāŊ ݀ݔāŊŦāŽī= 12ßąāŊĶݔāŽķℎāŊ  (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,  ÜĻāŊĶāŽ· + ÜĻāŊĢāŽ· = āķąāĩŦßąāŊĶ + ßąāŊĢāĩŊݔℎāŊ ݀ݔāŊŦāŽī= 12āĩŦßąāŊĶ + ßąāŊĢāĩŊݔāŽķℎāŊ  (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,   ÜļÜąÜŪāŊĶāŊ” =1ߊāŊĶ ቎āĩŦÜĻāŊĶāŽĩ + ÜĻāŊĢāŽĩāĩŊÜŪāŊ (1 + 2ÜŪāŊŒāŊ‹) + 2āĩŦÜĻāŊĶāŽķāŊ” + ÜĻāŊĢāŽķāĩŊ(ÜŪāŊŒāŊ‹ÜŪāŊ )+ 2āķą (ÜĻāŊĶāŽķāŊ•āŊ”)݀ݔāŊ…āģ„āģƒāŊ…āģ˜āŽī+ 2āķą āĩŦÜĻāŊĶāŽ· + ÜĻāŊĢāŽ·āĩŊ݀ݔāŊ…āģ˜(āŽĩāŽŋāŊ…āģƒ)āŽķāŽī቏ (5.48) which simplifies to, 5.2  Derivation of Bridge Quantities   177 ÜļÜąÜŪāŊĶāŊ” =ÜŪāŊ 24ℎāŊāŊ‹ßŠāŊĶ ቂÜŪ݉āĩŦ߱ݏ + ߱݌āĩŊāĩŦ1 − ÜŪÜī3 + 3ÜŪÜī + 6ÜŪÜĩÜīāĩŊ − 16ÜŪ݉ÜŪÜĩÜī3߱ݏ+ 4Üģݏݐ݉݃ÜŪÜĩÜī(1 − ÜŪÜī) − 8Üģݏݐݏ݃(ÜŪÜĩÜī)2 + 3ÜģÜŋ݉݃(2ÜŪÜĩÜī + 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),  ÜĢāŊĶāŊ” =ÜļÜąÜŪāŊĶāŊ”ÜŪāŊ (1 + 2ÜŪāŊŒāŊ‹) (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,      Δ߱āŊĶ = ߛݏ(ÜĢÜ―Ý − ÜĢÝÜ―Ý‹) (5.51) and the dead load of the superstructure can be written as,   ßąāŊĶ = ßąāŊĶāŊĒ + ߛݏ(ÜĢÜ―Ý − ÜĢÝÜ―Ý‹) (5.52) However, because ßą āŊĶ depends on ÜĢāŊ”āŊĶ, which is also a function of ßą āŊĶ, 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 ßąāŊĶ, 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 ßą āŊĶ.  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,   ÜģāŊĶ =ÜŪāŊ (1 + 2ÜŪāŊŒāŊ‹)݃ [ßąāŊĶ − (ߙāŊĶāŊ—āŊŸ − 1)ßąāŊĶāŊĒ] (5.53)  5.2  Derivation of Bridge Quantities   178 5.2.8 Summary of Equations ÜŪāŊ 	=	Main	span	length  ßąāŊĶ =	Superstructure	dead	load	   ÜŪāŊ‹	=	Suspension	ratio  ßąāŊĶāŊĒ =	Reference	superstructure	dead	load    ÜŪāŊŒāŊ‹	=	Side-to-main	span	length	ratio  ßąāŊĢ =	Live	load    ℎāŊāŊ‹	=	Tower	height-to-span	ratio  ßĐāŊ  /	ߛāŊ  =	Density/Unit	weight	of	material	‘m’    ߚāŊ§	=	Factor	for	out-of-plane	loading	on	towers  ߊāŊ  =	Design	stress	of	material	‘m’	   ߙāŊĶāŊ—āŊŸ	=	Factor	for	superimposed	dead	load  ݃ =	Standard	acceleration	due	to	gravity	 Stay Cable Steel Quantity  Main	Span	 	 ÜģāŊĶāŊ§āŊ  =ßĐāŊĶāŊ§ßŠāŊĶāŊ§ ÜŪāŊ āŽķāĩŦßąāŊĶ + ßąāŊĢāĩŊ(1 − ÜŪāŊ‹)ℎāŊāŊ‹ ቈ(1 − ÜŪāŊ‹)āŽķ12ℎāŊāŊ‹āŽķ+ 1቉ (5.54) Side	Span	 	 ÜģāŊĶāŊ§āŊĶ =ßĐāŊĶāŊ§ßŠāŊĶāŊ§ ÜŪāŊ āŽķāĩŦßąāŊĶ + ßąāŊĢāĩŊℎāŊāŊ‹ÜŪāŊŒāŊ‹ ቈ23 āĩŽÜŪāŊŒāŊ‹â„ŽāŊāŊ‹āĩ°āŽķ+ 2቉ (5.55) Suspension Cable Quantity  ÜģāŊ–āŊ  =1݃ ÜŪ݉āĩŦßąāŊĶ + ßąāŊĢāĩŊÜŪāŊ‹(2 − ÜŪāŊ‹)ßĶߟ1 − ßĶߟ ቈ(1 − ÜŪāŊ‹)ߟ + ÜŪāŊ‹(ߟāŽķ + 5)6 ቉ (5.56) where,	  ߟ = āķĻ1 + 16 āĩŽ ℎÜķÜī2 − ÜŪāŊ‹āĩ°āŽķ and ßĶ =ߛāŊ–āŊ  ÜŪāŊ 8 ߊāŊ–āŊ â„ŽāŊāŊ‹  Anchor Cable Quantity  ÜģāŊ–āŊĶ =112ßĐāŊ–āŊĶߊāŊ–āŊĶ ÜŪāŊ āĩĢ3ÜŪāŊ āĩŦßąāŊĶ + ßąāŊĢāĩŊ + 3ÜģāŊ–āŊ Ýƒ + 2ÜģāŊĶāŊ§āŊ Ýƒ(1 − ÜŪāŊ‹) − 4ÜģāŊĶāŊ§āŊĶ݃(ÜŪāŊŒāŊ‹)− 12ÜŪāŊ ÜŪāŊŒāŊ‹āŽķßąāŊĶāĩ§ āĩŽÜŪāŊŒāŊ‹â„ŽāŊāŊ‹ +ℎāŊāŊ‹ÜŪāŊŒāŊ‹āĩ° (5.57) Hanger Cable Quantity  ÜģāŊ› =13ßĐāŊ›ßŠāŊ› ÜŪāŊ āŽķāĩŦßąāŊĶ + ßąāŊĢāĩŊℎāŊāŊ‹ÜŪāŊ‹āŽķ āĩŽ12 − ÜŪāŊ‹āĩ° (5.58)5.2  Derivation of Bridge Quantities   179 Tower Quantity  ÜģāŊ§ =2 āŊĐܰ݃ ݈݊ āĩĶ11 − (ℎāŊāŊ‹ÜŪāŊ  + ℎāŪŧ)ߛāŊ§ßšāŊ§ßŠāŊ§āĩŠ (5.59) where,	  āŊĐÜ° =124ÜŪāŊŒāŊ‹ āĩĢ3ÜŪāŊ āĩŦßąāŊĶ + ßąāŊĢāĩŊ(2ÜŪāŊŒāŊ‹ + 1)āŽķ + 2ÜģāŊĶāŊ§āŊ Ýƒ(6ÜŪāŊŒāŊ‹ − ÜŪāŊ‹ + 1) + 8ÜģāŊĶāŊ§āŊĶ݃(ÜŪāŊŒāŊ‹)+ 3ÜģāŊ–āŊ Ýƒ(4ÜŪāŊŒāŊ‹ + 1) + 6ÜģāŊ–āŊĶ݃(ÜŪāŊŒāŊ‹)āĩ§  and,	  ℎāŪŧ = 50 āĩŽ1 −ÜŪāŊ 4000āĩ° [݅݊	݉݁ݐݎ݁ݏ]  Superstructure Quantity  ÜģāŊĶ =ÜŪāŊ (1 + 2ÜŪāŊŒāŊ‹)݃ [ßąāŊĶ − (ߙāŊĶāŊ—āŊŸ − 1)ßąāŊĶāŊĒ] (5.60) Load Correction Equation  ßąāŊĶ = ßąāŊĶāŊĒ + ߛāŊĶ(ÜĢāŊĶāŊ” − ÜĢāŊĶāŊ”āŊĒ) (5.61) where,	  ÜĢāŊĶāŊ” =124ℎāŊāŊ‹ßŠāŊĶ(2ÜŪāŊŒāŊ‹ + 1) āĩĢÜŪāŊ āĩŦßąāŊĶ + ßąāŊĢāĩŊāĩŦ1 − ÜŪāŊ‹āŽ· + 3ÜŪāŊ‹ + 6ÜŪāŊŒāŊ‹āĩŊ − 16ÜŪāŊ ÜŪāŊŒāŊ‹āŽ·ßąāŊĶ+ 4ÜģāŊĶāŊ§āŊ ÝƒÜŪāŊŒāŊ‹(1 − ÜŪāŊ‹) − 8ÜģāŊĶāŊ§āŊĶ݃(ÜŪāŊŒāŊ‹)āŽķ + 3ÜģāŊ–āŊ Ýƒ(2ÜŪāŊŒāŊ‹ + 1)āĩ§  and,	  ÜĢāŊĶāŊ”āŊĒ is computed using the same expression as ÜĢāŊĶāŊ” 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,	ÜĨāŊ, can be expressed as,    ÜĨāŊ = ÜŋāŊĶāŊ§(ÜģāŊĶāŊ§āŊ  + ÜģāŊĶāŊ§āŊĶ) + ÜŋāŊ–āŊ (ÜģāŊ–āŊ  + ÜģāŊ–āŊĶ) + ÜŋāŊ›ÜģāŊ› + ÜŋāŊ§ÜģāŊ§ + ÜŋāŊĶÜģāŊĶ + āŊ™ÜŋÜģāŊ™ (5.62)where ÜŋāŊ  represents the unit cost of component ‘m’ and ÜģāŊ™ 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, Üģ āŊ™ 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,  ݀ÜĨāŊÝ€â„ŽāŊāŊ‹ = 0 (5.63)Neglecting the change in the foundation quantity, Equation (5.63) may also be written as, ݀(ÜģāŊĶāŊ§āŊ  + ÜģāŊĶāŊ§āŊĶ)݀ℎāŊāŊ‹ +ÜŋāŊ–āŊ ÜŋāŊĶāŊ§Ý€(ÜģāŊ–āŊ  + ÜģāŊ–āŊĶ)݀ℎāŊāŊ‹ +ÜŋāŊ›ÜŋāŊĶāŊ§Ý€ÜģāŊ›Ý€â„ŽāŊāŊ‹ +ÜŋāŊ§ÜŋāŊĶāŊ§Ý€ÜģāŊ§Ý€â„ŽāŊāŊ‹ +ÜŋāŊĶÜŋāŊĶāŊ§Ý€ÜģāŊĶ݀ℎāŊāŊ‹ = 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 (ßąāŊĢ) and superstructure dead load (ßąāŊĶ).  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, ßąāŊ‹āŊĒ, which is defined as, 5.3  Span Proportions   181 ßąāŊ‹āŊĒ =߱݌߱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 ߊ (MPa) 25 250 25 250 800 800 800 ߚāŊ§ n/a n/a 0.7 0.7 n/a n/a n/a ߙāŊĶāŊ—āŊŸ 1.1 1.1 n/a n/a n/a n/a n/a Üŋ ÜŋāŊĶāŊ§â„  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 (ÜŪāŊ‹āŊĒ = 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 (ℎāŊāŊ‹āŊĒ) and the live load ratio (ßąāŊ‹āŊĒ).  The tower height-to-span ratio is set at a conventional value for cable-stayed bridges, ℎāŊāŊ‹āŊĒ = 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  ßąāŊ‹āŊĒ = 0.6 and ßąāŊ‹āŊĒ = 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 (ÜŪāŊ‹) and multiple side-to-main span 5.3  Span Proportions   182 length ratios (ÜŪāŊŒāŊ‹).  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: ÜŪāŊ  = 500m,	ÜŪāŊ āŊĒ = ÜŪāŊ , ÜŪāŊŒāŊ‹āŊĒ = ÜŪāŊŒāŊ‹, ℎāŊāŊ‹āŊĒ = 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 āĄļāĄŋāĄū āĄļāĄū 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: ÜŪāŊ  = 500m,	ÜŪāŊ‹ = 0.4, ÜŪāŊŒāŊ‹ = 0.3, ÜŪāŊ āŊĒ = ÜŪāŊ , ÜŪāŊŒāŊ‹āŊĒ = ÜŪāŊŒāŊ‹, ℎāŊāŊ‹āŊĒ = 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 0.500.20.40.60.811.21.4āĒŽāĒ€āĄū āĒŽāĒ€āĄū	āĄŊāĄīāĄŊāĄīāĒ•	āĄŊāĄīāĄŊāĄīāĒ•	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:	ÜŪāŊ‹ = 0.4, ÜŪāŊŒāŊ‹ = 0.3, ÜŪāŊ āŊĒ = ÜŪāŊ , ÜŪāŊŒāŊ‹āŊĒ = ÜŪāŊŒāŊ‹, ℎāŊāŊ‹āŊĒ = 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 (ßąāŊ‹āŊĒ, ℎāŊāŊ‹āŊĒ), and the input factor which accounts for superimposed dead load (ߙāŊĶāŊ—āŊŸ).  The only exception is the design stress and unit cost of the suspension/anchor cable which have a notable 200 400 600 800 10000.200.220.240.260.280.30āĒŽāĒ€āĄū_āĒ•āĒ–āĒš	āĄļāĒ“ (āĒ“)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:   ÜŋāŊ–āŊ  ÜŋāŊĶāŊ§â„   ÜŋāŊ› ÜŋāŊĶāŊ§â„   ÜŋāŊĶ ÜŋāŊĶāŊ§â„   ÜŋāŊ§ ÜŋāŊĶāŊ§â„   ßąāŊ‹āŊĒ  ߊāŊĶāŊ§  ߊāŊ–āŊ   ߊāŊ›  ߊāŊĶ  ߚāŊ§ßŠāŊ§  ℎāŊāŊ‹āŊĒ  ßĐāŊĶāŊ§  ßĐāŊ–āŊ   ßĐāŊ›  ߙāŊĶāŊ—āŊŸ       Figure 5.17: Sensitivity of Optimum Tower Height-to-Span Ratio Parameters: ÜŪāŊ  = 500m,	ÜŪāŊ‹ = 0.4, ÜŪāŊŒāŊ‹ = 0.3, ÜŪāŊ āŊĒ = ÜŪāŊ , ÜŪāŊŒāŊ‹āŊĒ = ÜŪāŊŒāŊ‹, ℎāŊāŊ‹āŊĒ = 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,  āŊ™Ü° = āŊĐÜ° +12ÜģāŊ§Ýƒ (5.66)where Ü° āŊĐ is the vertical force at the top of the towers from the cable system (Equation (5.31)), and Üģ āŊ§ is the tower quantity.  The relationship between āŊ™Ü° 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: ÜŪāŊ  = 500m,	ÜŪāŊ‹ = 0.4, ÜŪāŊŒāŊ‹ = 0.3, ÜŪāŊ āŊĒ = ÜŪāŊ , ÜŪāŊŒāŊ‹āŊĒ = ÜŪāŊŒāŊ‹, ℎāŊāŊ‹āŊĒ = 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 0.300.951.001.05āĄšā·ĄāĒŒ	āĒŽāĒ€āĄū5.3  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,   ÜīāŊĢ ≅߱āŊĶÜŪāŊ 8ÜŪāŊŒāŊ‹ āĩĢ1 + ßąāŊ‹ − 4ÜŪāŊŒāŊ‹āŽķāĩ§ (5.67) The relationship between the side-to-main span ratio (ÜŪāŊŒāŊ‹) and the uplift force (ÜīāŊĢ) is plotted in Figure 5.19.  The ordinates of the plot are normalized so that they are independent of the superstructure dead load (ßąāŊĶ), and the main span length (ÜŪāŊ ).  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 ßą āŊĶāŊĶ and ßą āŊĶāŊ  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 0.500.00.51.01.52.0āĄūāĒ–āĢ“āĒ™āĄļāĒ“	āĄļāĄŋāĄū ßąāŊ‹ = 0.2 ßąāŊ‹ = 0.4 ßąāŊ‹ = 0.6 ßąāŊ‹ = 0 5.3  Span Proportions   189  ÜīāŊĢ ≅߱āŊĶāŊ ÜŪāŊ 8ÜŪāŊŒāŊ‹ āĩĪ1 + ßąāŊ‹āŊ  − 4ÜŪāŊŒāŊ‹āŽķ āĩŽßąāŊĶāŊĶßąāŊĶāŊ āĩ°āĩĻ (5.68) where ßą āŊ‹āŊ  is the unfactored live load ratio for the main span only (ßąāŊ‹āŊ  = ßą āŊĢ ßąāŊĶāŊ â„ ).  Solving for the superstructure dead load ratio (ßąāŊĶāŊĶ ßąāŊĶāŊ â„ ) when Üī āŊĢ equals zero then gives the balancing condition as,    ßąāŊĶāŊĶßąāŊĶāŊ  ≅1 + ßąāŊ‹āŊ 4ÜŪāŊŒāŊ‹āŽķ (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 0.501.02.03.04.05.06.07.08.09.010.0āĢ“āĒ™āĒ™āĢ“āĒ™āĒ“	āĄļāĄŋāĄū ßąāŊ‹āŊ  = 0.2ßąāŊ‹āŊ  = 0.4ßąāŊ‹āŊ  = 0.6ßąāŊ‹āŊ  = 0 ßąāŊĶāŊ ßąāŊĢ ÜŪāŊ ÜŪāŊŒāŊ‹ÜŪāŊ  ÜŪāŊŒāŊ‹ÜŪāŊ 0 0 ßąāŊĶāŊĶ ßąāŊĶāŊĶßąāŊ‹āŊ  = ßąāŊĢ ßąāŊĶāŊ â„   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,    āŊ™Ü° ≅ÜŪāŊ 8ÜŪāŊŒāŊ‹ āĩŦßąāŊĶ + ßąāŊĢāĩŊ(2ÜŪāŊŒāŊ‹ + 1)āŽķΛ (5.70)where,  Λ = 1 + ݈݊ āĩĶ 11 − (ℎāŊāŊ‹ÜŪāŊ  + ℎāŪŧ)ߛāŊ§ßšāŊ§ßŠāŊ§āĩŠ  Since all of the parameters can be assumed independent of ÜŪāŊŒāŊ‹, 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 0.501.01.21.31.51.61.8āĄšāĒŒāĄļāĒ“āĩŦāĢ“āĒ™ + āĢ“āĒ–āĩŊāĒŦ	āĄļāĄŋāĄū5.3  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,   ÜŪāŊŒāŊ‹ÜŪāŊ 	ÜŪāŊŒāŊ‹ÜŪāŊ 	ÜŪāŊ (1 − ÜŪāŊ‹āŽĩ) 2⁄ÜŪāŊ (1 − ÜŪāŊ‹āŽķ) 2⁄=	>	Bridge 1 Bridge 2 5.3  Span Proportions   193   ÜŪāŊ‹ â‰Ī 1 − 2ÜŪāŊŒāŊ‹ (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 â‰Ī ÜŪāŊ‹ â‰Ī 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 â‰Ī ÜŪāŊŒāŊ‹_āŊĒāŊĢāŊ§ â‰Ī 0.4   0.2 â‰Ī ÜŪāŊ‹_āŊĒāŊĢāŊ§ â‰Ī 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,   (ÜŠāŊ–āŊ )āŪ―āŊ… ≅ÜŪāŊ ßąāŊĶ8ÜĩāŊ‹ āĩĪßąāŊ–āŊ ßąāŊĶ + ÜŪāŊ‹(2 − ÜŪāŊ‹)āĩĻ (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 0.301.001.101.201.301.401.50āĩŦāĄīā·ĄāĒ‰āĒ“āĩŊāĄ°āĄļ	āĄļāĄū5.4  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: ℎāŊāŊ‹ = 0.25,	ÜŪāŊ‹ = 0.3, ÜŪāŊŒāŊ‹ = 0.35, ÜŪāŊ āŊĒ = ÜŪāŊ , ÜŪāŊŒāŊ‹āŊĒ = ÜŪāŊŒāŊ‹, ℎāŊāŊ‹āŊĒ = ℎāŊāŊ‹, 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 10000246810āĄļāĒ“(āĒ“) āĄļāĒ“(āĒ“)	%	āĄŊāĒŽāĒ‡āĒ”āĒāĒ‹	%	āĄŊāĒŽāĒ‡āĒ”āĒāĒ‹	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.                           ðļðī𝑔ðļ𝐞𝑔ðļðīð‘Ąðļðžð‘Ą  ðđ𝑖ð‘Ĩ𝑒𝑑 𝑆ð‘Ēð‘ð‘ð‘œð‘Ÿð‘Ą (TYP)  𝑅𝑜𝑙𝑙𝑒𝑟 𝑆ð‘Ēð‘ð‘ð‘œð‘Ÿð‘Ą (TYP) ðŋ𝑚 50  (𝑇𝑌𝑃) ðļðī𝑔,ðļ𝐞𝑔  ðŋ𝑚 500  𝜔𝑝 = 75𝑘𝑁/𝑚 𝑉𝑎𝑟𝑖𝑏𝑙𝑒 37.5𝑚 𝑉𝑎𝑟𝑖𝑏𝑙𝑒 0.36(ðŋ𝑚) 0.36(ðŋ𝑚) 0.28(ðŋ𝑚) 0.225(ðŋ𝑚) 𝜔𝑠 = 𝜔𝑝 0.6  ðļðīð‘Ą,ðļðžð‘Ą   ðŋ𝑚 = 1000𝑚 0.36(ðŋ𝑚) 0.36(ðŋ𝑚) 𝑇𝑎𝑏𝑙𝑒 6.2 ð‘Ĩ ð‘Ķ    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) ðŧð‘Ķ𝑏𝑟𝑖𝑑 ðĩ𝑟𝑖𝑑𝑔𝑒 ðķ𝑜𝑛ð‘Ģð‘’ð‘›ð‘Ąð‘–ð‘œð‘›ð‘Žð‘™ ðķ𝑎𝑏𝑙𝑒 ð‘†ð‘Ąð‘Žð‘Ķ𝑒𝑑 ðĩ𝑟𝑖𝑑𝑔𝑒 Main Span Side Span Deflections (m) Shear Forces (MN) Moments (MN·m) AS1 AS2 AS3 ðķ𝑜𝑛ð‘Ģð‘’ð‘›ð‘Ąð‘–ð‘œð‘›ð‘Žð‘™ ðķ𝑎𝑏𝑙𝑒 ð‘†ð‘Ąð‘Žð‘Ķ𝑒𝑑 ðĩ𝑟𝑖𝑑𝑔𝑒 ðŧð‘Ķ𝑏𝑟𝑖𝑑 ðĩ𝑟𝑖𝑑𝑔𝑒 ðļ𝐞𝑔 ðļðžð‘Ąðļðī𝑐   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)    ð‘Ĩ𝑝𝑖ðŋ𝑠ð‘Ĩ𝑝𝑖 ðŋ𝑠 ð‘Ĩ𝑝𝑖 ðŋ𝑠 ð‘Ĩ𝑝𝑖 ðŋ𝑠 ð‘Ĩ𝑝𝑖 ðŋ𝑠 ð‘Ĩ𝑝𝑖 ðŋ𝑠 ð‘Ĩ𝑝𝑖 ðŋ𝑠 ð‘Ĩ𝑝𝑖 ðŋ𝑠 ƗAS2 Deflections (m) Axial Forces (MN) Shear Forces (MN) Moments (MN·m) Cable Stress Range (MN) Cable Stress Range Ɨ (MN) ð‘Ĩ𝑝𝑖  ðŋ𝑠 = 0  ð‘Ĩ𝑝𝑖  ðŋ𝑠 = 0.5  ð‘Ĩ𝑝𝑖  ðŋ𝑠 = 0.75  ð‘Ĩ𝑝𝑖  ðŋ𝑠 = 0.25  ð‘ģ𝒔 𝒙𝒑𝒊 Main Span Side Span Deflections (m) Shear Forces (MN) Moments (MN·m) AS1 AS2 AS3 ð‘Ĩ𝑝𝑖  ðŋ𝑠 = 0.5  ð‘Ĩ𝑝𝑖  ðŋ𝑠 = 0.75  ð‘Ĩ𝑝𝑖  ðŋ𝑠 = 0.25  ð‘Ĩ𝑝𝑖  ðŋ𝑠 = 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 ðžð‘›ð‘Ą.𝑃𝑖𝑒𝑟 3 ðžð‘›ð‘Ą.𝑃𝑖𝑒𝑟𝑠 4 ðžð‘›ð‘Ą.𝑃𝑖𝑒𝑟𝑠 2 ðžð‘›ð‘Ą.𝑃𝑖𝑒𝑟𝑠 Main Span Side Span Deflections (m) Shear Forces (MN) Moments (MN·m) AS1 AS2 AS3 𝑁𝑜 ðžð‘›ð‘Ą.𝑃𝑖𝑒𝑟 3 ðžð‘›ð‘Ą.𝑃𝑖𝑒𝑟𝑠 4 ðžð‘›ð‘Ą.𝑃𝑖𝑒𝑟𝑠 2 ðžð‘›ð‘Ą.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  (ðŧ𝑐𝑚)𝐷ðŋ ≅ðŋ𝑚𝜔𝑠𝑚8𝑆𝑅[𝜔𝑐𝑚𝜔𝑠𝑚+ ðŋ𝑅(2 − ðŋ𝑅)]𝜔𝑠𝑚 𝜔𝑠𝜔𝑠𝑚 𝜔𝑠𝑠ðŋ𝑚 𝑆𝑅 ðŋ𝑅 𝜔𝑐ðŧ𝑆𝑅 ≅ 𝜇𝑉𝑝𝑓𝜇 𝑉𝑝𝑓𝑉𝑝𝑓ðŋ𝑅 1 − 2ðŋ𝑆𝑅𝑉𝑝𝑓 ≅ (𝜔𝑠𝑠 − 𝜔𝑠𝑚)ðŋ𝑆𝑅ðŋ𝑚 − (ðŧ𝑐𝑚)𝐷ðŋ (𝑆𝑅ðŋ𝑆𝑅)ðŋ𝑆𝑅ðŧ𝑆𝑅(ðŧ𝑐𝑚)𝐷ðŋ= 𝜇2𝑆𝑅(1 − ðŋ𝑅)[𝜔𝑐𝑚𝜔𝑠𝑚+ (1 − ðŋ𝑅)2 (1 − 2𝜔𝑠𝑠𝜔𝑠𝑚) + 1𝜔𝑐𝑚𝜔𝑠𝑚− (1 − ðŋ𝑅)2 + 1]𝜔𝑐Vertical Component of Suspension Cable Force Surplus Dead Load from Side Span Superstructure          ðŋ𝑚 ðŋ𝑆𝑅 ðŋ𝑅  𝜔𝑅 ð›ū𝑐 𝜎𝑎𝑙𝑙𝑜ð‘Ī0.20 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       ð‘ģð‘đ ð‘ģð‘đ ð‘ģð‘đ ð‘Ŋ𝑚ð‘đ(ð‘Ŋ𝒄𝒎)ð‘Ŧð‘ģ ð‘Ŋ𝑚ð‘đ(ð‘Ŋ𝒄𝒎)ð‘Ŧð‘ģ ð‘Ŋ𝑚ð‘đ(ð‘Ŋ𝒄𝒎)ð‘Ŧð‘ģ 𝝁 = 𝟎.𝟓 𝝁 = 𝟎.𝟑 𝝁 = 𝟎.𝟕             ðŋ𝑚 ðŋ𝑆𝑅 ðŋ𝑅  𝜔𝑅 ð›ū𝑐 𝜎𝑎𝑙𝑙𝑜ð‘Ī0.20 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       ð‘ģð‘đ ð‘ģð‘đ ð‘ģð‘đ 𝝎𝒔𝒔𝝎𝒔𝒎 𝝎𝒔𝒔𝝎𝒔𝒎 𝝎𝒔𝒔𝝎𝒔𝒎 𝝁 = 𝟎.𝟓 𝝁 = 𝟎.𝟑 𝝁 = 𝟎.𝟕    ðŋ𝑅     ðŋ𝑚 ð›ū𝑐 𝜎𝑎𝑙𝑙𝑜ð‘Ī0.20 0.22 0.24 0.26 0.28 0.3024578𝑆𝑅 = 0.25 𝑆𝑅 = 0.225       𝑆𝑅 = 0.20       ð‘ģð‘đ ð‘Ļ𝒄 (𝒔𝒖𝒔𝒑𝒆𝒏𝒔𝒊𝒐𝒏 𝒃𝒓𝒊𝒅𝒈𝒆)ð‘Ļ𝒄 (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 Üš (longitudinal), Üŧ (vertical), and Üž (transverse) directions by the vectors ΔܹāŊ, ΔÜŧāŊ , and ΔܾāŊ 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.        āŽĩÜ°â€ĶāŊĄ Nodal Numbering Scheme ΔXā­˜ Projected Length of Cable in ‘x’ DirectionāŽĩÜĩâ€ĶāŊĄ Segment Numbering Scheme ΔYā­˜ Projected Length of Cable in ‘y’ DirectionßąāŊ– ƗSelf-Weight of Suspension Cable ΔZā­˜ Projected Length of Cable in ‘z’ DirectionÜĻℎāŊœ Hanger Force Acting at Node i ݊ Number of Cable Segments     ƗNot Shown for Clarity Figure A1: 3D View of a Suspension Cable  ÜŧÜžÜĻℎāŽķāŽĩܰ߂ܹāŊ āŽĩÜĩ߂ܾāŊß‚ÜŧāŊ  ÜĻℎāŽ·ÜĻℎāŊœāŽŋāŽĩÜĻℎāŊœÜĻℎāŊœāŽūāŽĩÜĻℎāŊĄāŽ·Ü°āŊœÜ°āŽŋāŽĩāŊœÜ°āŊœÜ°āŽūāŽĩāŊĄÜ° ÜĩāŽķÜĩāŽ·āŊœÜĩāŽŋāŽĩāŊœÜĩāŊœÜĩāŽūāŽĩÜĩāŊĄ āŽķÜ° ÜšAppendix 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),    ݕāŊœ =ÜŠāŊœßąāŊ– Üŋ݋ݏℎ āĩŽßąāŊ–Ý”āŊœâˆ—ÜŠāŊœ + ÜĢāŊœāĩ° + ÜĪāŊœ (A1)where, ÜĢāŊœ = Ü―ÝÝ…ÝŠâ„Ž ቎߱āŊ–â„ŽāŊœ2ÜŠāŊœÝÝ…ÝŠâ„Ž ቀ߱āŊ–Ü―āŊœ2ÜŠāŊœ ቁ቏ − ßąāŊ–Ü―āŊœ2ÜŠāŊœ  and ÜĪāŊœ = −ܩāŊœßąāŊ– Üŋ݋ݏℎሹÜĢāŊœáˆŧ    ßąāŊ– ƗSelf-Weight of Suspension Cable ÜŠāŊœ ‘Horizontal’ Cable Force in Segment iሚÜīݔāŊœ, ÜīݕāŊœ, ÜīݖāŊœáˆŧ Cable Force Components at Beginning of Segment iሹ߂ݔāŊœ, ߂ݕāŊœ, ߂ݖāŊœáˆŧ Projected Dimensions of Segment i   ƗNot Shown for Clarity Figure A2: Free Body Diagram of Segment i ݕݖ߂ݔāŊœ ߂ݕāŊœ ݔݔ∗߂ݖāŊœÜīݔāŊœÜīݕāŊœ = −ܩāŊœÝ€Ý•Ý€Ý”∗āļŽāŊŦ∗ā­€āŽī = −ܩāŊœÝÝ…݊ℎሚÜĢāŊœáˆŧ ÜīݖāŊœÜŠāŊœÜīݔāŊœÜīݖāŊœ ÜŠāŊœÝ€Ý•Ý€Ý”∗āļŽāŊŦ∗ā­€āŊ”āģ”= ÜŠāŊœÝÝ…ÝŠâ„Ž āĩŽßąāŊ–Ü―āŊœÜŠāŊœ + ÜĢāŊœāĩ°â„ŽāŊœ 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),    ÜŠāŊœ = āķĨሚÜīݔāŊœáˆŧāŽķ + ሚÜīݖāŊœáˆŧāŽķ (A2)    Ü―āŊœ = āķĨሹΔݔāŊœáˆŧāŽķ + ሹΔݖāŊœáˆŧāŽķ (A3)    ℎāŊœ = ΔyāŊœ (A4)    ݔāŊœâˆ— = ݔāŊœāķĻ1 + āĩŽÎ”Ý–āŊœÎ”Ý”āŊœāĩ°āŽķ (A5) where, the local ݖ and ݔ axes are related through the following relationship,    ݖāŊœ = ݔāŊœ āĩŽÎ”Ý–āŊœÎ”Ý”āŊœāĩ° (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 ÜīݔāŽĩâ€ĶāŊĄ, ÜīݕāŽĩâ€ĶāŊĄ, ÜīݖāŽĩâ€ĶāŊĄ, ΔݕāŽĩâ€ĶāŊĄ, and ΔݖāŽĩâ€ĶāŊĄ.  However, given the support reactions at Node 1 ሚÜīݔāŽĩ, ÜīݕāŽĩ, ÜīݖāŽĩáˆŧ, the transverse and vertical projected dimensions of Segment 1 can be determined from the geometrical conditions upon which,    ΔݖāŊœ = ΔݔāŊœ āĩŽÜīݖāŊœÜīݔāŊœāĩ° (A7)and     ÜīݕāŊœ = −ܩāŊœÝÝ…݊ℎሚÜĢāŊœáˆŧ (A8)where, ÜĢāŊœ is a function of  ΔݕāŊœ.  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    ÜīݔāŊœāŽūāŽĩ = ÜīݔāŊœ + ÜĻℎāŊŦāŊœāŽūāŽĩ (A9)    ÜīݕāŊœāŽūāŽĩ = ÜĻℎāŊŽāŊœāŽūāŽĩ − ÜŠāŊœÝÝ…ÝŠâ„Ž āĩŽßąāŊ–Ü―āŊœÜŠāŊœ + ÜĢāŊœāĩ° (A10)    ÜīݖāŊœāŽūāŽĩ = ÜīݖāŊœ + ÜĻℎāŊ­āŊœāŽūāŽĩ (A11) where, ÜĻℎāŊŦāģ”, ÜĻℎāŊŽāģ”, and ÜĻℎāŊ­āģ” 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 |ÜĻℎ|, the hanger force components at a given node ݅  can be computed as (neglecting the sag effect of the hangers),     ÜĻℎāŊŦāŊœ  = |ÜĻℎāŊœ|ۉۇ ߜℎāŊŦāķ§ßœâ„ŽāŊŦāŽķ + ߜℎāŊŽāŽķ + ߜℎāŊ­āŽķیۊāŊœ(A12)    ÜĻℎāŊŽāŊœ  = |ÜĻℎāŊœ|ۉۇ ߜℎāŊŽāķ§ßœâ„ŽāŊŦāŽķ + ߜℎāŊŽāŽķ + ߜℎāŊ­āŽķیۊāŊœ(A13)    ÜĻℎāŊ­āŊœ  = |ÜĻℎāŊœ|ۉۇ ߜℎāŊ­āķ§ßœâ„ŽāŊŦāŽķ + ߜℎāŊŽāŽķ + ߜℎāŊ­āŽķیۊāŊœ(A14) where,   ߜℎāŊŦāŊœ = ℎāŊŦāŊœ − ā· ΔݔāŊ āŊœāŽŋāŽĩāŊ ā­€āŽĩ   ߜℎāŊŽāŊœ = ℎāŊŽāŊœ − ā· ΔyāŊ āŊœāŽŋāŽĩāŊ ā­€āŽĩ  JOINT EQUILIBRIUM EQUATIONS Appendix A:  Procedure to Determine Cable Shape in Three Dimensions   271  ߜℎāŊ­āŊœ = ℎāŊ­āŊœ − ā· ΔzāŊ āŊœāŽŋāŽĩāŊ ā­€āŽĩ  As shown in Figure A3, ߜℎāŊŦ, ߜℎāŊŽ, and ߜℎāŊ­ represent the projected dimensions of the hanger in the ݔ, ݕ, and ݖ directions, respectively.   And, āĩŦℎāŊŦ, ℎāŊŽ, ℎāŊ­āĩŊāŊœ 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,      ÜĻℎāŊŦāŊœ = ÜĻℎāŊŽāŊœ ቆߜℎāŊŦߜℎāŊŽá‰‡āŊœ (A15)  ÜĻℎāŊ­āŊœ = ÜĻℎāŊŽāŊœ ቆߜℎāŊ­ßœâ„ŽāŊŽá‰‡āŊœ (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 Üī ݔāŽĩ, Üī ݕāŽĩ, and Üī ݖāŽĩ.  However, since these Figure A3: YZ Section at Node i *Nodes (2â€Ķ i-1) not shown for clarityℎāŊ­āŊœBridge Superstructure ÜŧÜžāŽĩÜ°āŊœÜ°â„ŽāŊŽāŊœßœâ„ŽāŊ­āŊœßœâ„ŽāŊŽāŊœ 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 ΔܹāŊ, which yield the specified end offsets of the cable (see Figure A1).      ΔÜŧāŊ  = ā· ΔݕāŊœāŊĄāŊœā­€āŽĩ= Üĩ݌݁Üŋ݂݅݅݁݀ ÜļÜ―ÝˆÝ‘Ý (A17)    ΔܾāŊ  = ā· ΔzāŊœāŊĄāŊœā­€āŽĩ= Üĩ݌݁Üŋ݂݅݅݁݀ ÜļÜ―ÝˆÝ‘Ý (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,  Üš = ΔܹāŊ2  is chosen.  This value, referred to hereon as the vertical cable sag, āŊŽÝ‚, is computed for given set of support reactions as.    ā· ΔݕāŊœāŊĪāŊœā­€āŽĩ  if there is a hanger at midspan  āŊŽÝ‚ =   (A19)   ā· ΔݕāŊœāŊĢāŽŋāŽĩāŊœā­€āŽĩ+ ݕāŊĢ āĩŽÝ”āŊĢ =ΔݔāŊĢ2 āĩ° 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, ßąāŊ– = ߛāŊ–ÜĢāŊ–. 2. An array containing ΔݔāŽĩâ€ĶāŊĄ  3. An array containing ÜĻℎāŽĩâ€ĶāŊĄ, or alternatively ÜĻℎāŊŽāŽĩâ€ĶāŊĄ 4. Target values for āŊŽÝ‚, ΔÜŧāŊ , and ΔZāŊ.  Also, the tolerance accepted in achieving the target values, denoted as ÜķÜąÜŪ.     5. An initial guess for the support reactions at Node 1, denoted as ÜīÝ”Ü―āŽĩ, ÜīÝ•Ü―āŽĩ, and ÜīÝ–Ü―āŽĩ.      Steps: 1. Set the support reactions at Node 1 equal to ሚÜīÝ”Ü―āŽĩ, ÜīÝ•Ü―āŽĩ, ÜīÝ–Ü―āŽĩáˆŧ. 2. Compute ΔzāŊœ using Equation (A7).  3. Solve for ΔݕāŊœ using Equation (A8).  4. Decompose ÜĻℎāŊœ using Equations (A12)-(A14).  5. Compute ÜīݔāŊœāŽūāŽĩ, ÜīݕāŊœāŽūāŽĩ, and ÜīݖāŊœāŽūāŽĩ using the joint equilibrium equations, (A9)-(A11). 6. Repeat Steps 2 through 5 for ݅ = 1 â€Ķ ݊.  7. Compute the vertical cable end offset (denoted as ΔÜŧāŊ āŊ”), the transverse cable end offset (denoted as ΔܾāŊāŊ”), and the vertical cable sag (denoted as āŊŽÝ‚āŊ”) corresponding to ሚÜīÝ”Ü―āŽĩ, ÜīÝ•Ü―āŽĩ, ÜīÝ–Ü―āŽĩáˆŧ using Equations (A17)-(A19) combined with the geometric parameters obtained in Steps 2 through 6. 8. Determine the error in the target parameters,  ΔÜŧāŊ āŪūāŊĨāŊĨāŊĒāŊĨ = ΔÜŧāŊ − ΔÜŧāŊ āŊ”, ΔZāŊāŪūāŊĨāŊĨāŊĒāŊĨ = ΔZāŊ − ΔZāŊāŊ”, and āŊŽÝ‚āŪūāŊĨāŊĨāŊĒāŊĨ =  āŊŽÝ‚ −  āŊŽÝ‚āŊ” 9. Check convergence a. If ቀāļŦΔÜŧāŊ āŪūāŊĨāŊĨāŊĒāŊĨāļŦ  ∧ āļŦΔZāŊāŪūāŊĨāŊĨāŊĒāŊĨāļŦ  ∧ ቚ āŊŽÝ‚āŪūāŊĨāŊĨāŊĒāŊĨቚቁ > ÜķÜąÜŪ advance to Step 10. b. If ቀāļŦΔÜŧāŊ āŪūāŊĨāŊĨāŊĒāŊĨāļŦ  ∧ āļŦΔZāŊāŪūāŊĨāŊĨāŊĒāŊĨāļŦ  ∧ ቚ āŊŽÝ‚āŪūāŊĨāŊĨāŊĒāŊĨቚቁ â‰Ī ÜķÜąÜŪ advance to Step 13.  See Figure A1 Appendix A:  Procedure to Determine Cable Shape in Three Dimensions   274 10. Numerically compute the Jacobian Matrix, áˆūÜŽáˆŋ =ۏێێێۍ āŊ—āŊ™āģĪāģŒāģāŊ—āŊ‹āŊŦāģŒāģāŊ—āŊ™āģĪāģŒāģŽāŊ—āŊ‹āŊŽāģŒāģŽāŊ—āŊ™āģĪāģŒāģāŊ—āŊ‹āŊ­āģŒāģā­Ēā­žā­āģ…āģŒāģāŊ—āŊ‹āŊŦāģŒāģā­Ēā­žā­āģ…āģŒāģŽāŊ—āŊ‹āŊŽāģŒāģŽā­Ēā­žā­āģ…āģŒāģāŊ—āŊ‹āŊ­āģŒāģā­Ēā­žā­žāģ…āģŒāģāŊ—āŊ‹āŊŦāģŒāģā­Ēā­žā­žāģ…āģŒāģŽāŊ—āŊ‹āŊŽāģŒāģŽā­Ēā­žā­žāģ…āģŒāģāŊ—āŊ‹āŊ­āģŒāģ ےۑۑۑې  a. Set, ÜīݔÜūāŽĩ = ሚ1 − ÜķÜąÜŪáˆŧÜīÝ”Ü―āŽĩ, ÜīݕÜŋāŽĩ = ሚ1 − ÜķÜąÜŪáˆŧÜīÝ•Ü―āŽĩ, and Üīݖ݀āŽĩ =ሚ1 − ÜķÜąÜŪáˆŧÜīÝ–Ü―āŽĩ. b. Repeat Steps 2 through 6 except with the support reactions at Node 1 equal to ሚÜīݔÜūāŽĩ, ÜīÝ•Ü―āŽĩ, ÜīÝ–Ü―āŽĩáˆŧ and label the cable parameters in Step 7 āĩŦΔÜŧāŊ āŊ•, ΔܾāŊāŊ•, āŊŽÝ‚āŊ•āĩŊ.  c. Compute first column of the Jacobian Matrix,  āŊ—āŊ™āģĪāģŒāģāŊ—āŊ‹āŊŦāģŒāģ =āŊ™āģĪāģŒāŽŋāŊ™āģĪāģāŊ‹āŊŦāŊ”ā°­āŽŋāŊ‹āŊŦāŊ•ā°­  ;  ā­Ēā­žā­āģ…āģŒāģāŊ—āŊ‹āŊŦāģŒāģ =ā­žā­āģ…āģŒāŽŋā­žā­āģ…āģāŊ‹āŊŦāŊ”ā°­āŽŋāŊ‹āŊŦāŊ•ā°­  ;  ā­Ēā­žā­žāģ…āģŒāģāŊ—āŊ‹āŊŦāģŒāģ =ā­žā­žāģ…āģŒāŽŋā­žā­žāģ…āģāŊ‹āŊŦāŊ”ā°­āŽŋāŊ‹āŊŦāŊ•ā°­   d. Repeat Steps 2 through 6 except with the support reactions at Node 1 equal to ሚÜīÝ”Ü―āŽĩ, ÜīݕÜŋāŽĩ, ÜīÝ–Ü―āŽĩáˆŧ and label the cable parameters in Step 7 āĩŦΔÜŧāŊ āŊ–, ΔܾāŊāŊ–, āŊŽÝ‚āŊ–āĩŊ. e. Compute second column of the Jacobian Matrix,  āŊ—āŊ™āģĪāģŒāģŽāŊ—āŊ‹āŊŽāģŒāģŽ =āŊ™āģĪāģŒāŽŋāŊ™āģĪāģŽāŊ‹āŊŽāŊ”ā°­āŽŋāŊ‹āŊŽāŊ–ā°­  ;  ā­Ēā­žā­āģ…āģŒāģŽāŊ—āŊ‹āŊŽāģŒāģŽ =ā­žā­āģ…āģŒāŽŋā­žā­āģ…āģŽāŊ‹āŊŽāŊ”ā°­āŽŋāŊ‹āŊŽāŊ–ā°­  ;  ā­Ēā­žā­žāģ…āģŒāģŽāŊ—āŊ‹āŊŽāģŒāģŽ =ā­žā­žāģ…āģŒāŽŋā­žā­žāģ…āģŽāŊ‹āŊŽāŊ”ā°­āŽŋāŊ‹āŊŽāŊ–ā°­   f. Repeat Steps 2 through 6 except with the support reactions at Node 1 equal to ሚÜīÝ”Ü―āŽĩ, ÜīÝ•Ü―āŽĩ, Üīݖ݀āŽĩáˆŧ and label the cable parameters in Step 7 āĩŦΔÜŧāŊ āŊ—, ΔܾāŊāŊ—, āŊŽÝ‚āŊ—āĩŊ. g. Compute third column of Jacobian Matrix, āŊ—āŊ™āģĪāģŒāģāŊ—āŊ‹āŊ­āģŒāģ =āŊ™āģĪāģŒāŽŋāŊ™āģĪāģāŊ‹āŊ­āŊ”ā°­āŽŋāŊ‹āŊ­āŊ—ā°­  ;  ā­Ēā­žā­āģ…āģŒāģāŊ—āŊ‹āŊ­āģŒāģ =ā­žā­āģ…āģŒāŽŋā­žā­āģ…āģāŊ‹āŊ­āŊ”ā°­āŽŋāŊ‹āŊ­āŊ—ā°­  ;  ā­Ēā­žā­žāģ…āģŒāģāŊ—āŊ‹āŊ­āģŒāģ =ā­žā­žāģ…āģŒāŽŋā­žā­žāģ…āģāŊ‹āŊ­āŊ”ā°­āŽŋāŊ‹āŊ­āŊ—ā°­   11. Update the initial guess values for the support reactions at Node 1, áˆūÜīÜ―áˆŋāŊ‡āŪūāŊ = áˆūÜīÜ―áˆŋ +áˆūΔÜīÜ―áˆŋ a. Set,  áˆūÜīÜ―áˆŋ = āĩĨÜīÝ”Ü―āŽĩÜīÝ•Ü―āŽĩÜīÝ–Ü―āŽĩāĩĐ  b. Compute the requisite change in the support reactions,   áˆūΔÜīÜ―áˆŋ = áˆūÜŽáˆŋāŽŋāŽĩ ቎āŊŽÝ‚āŪūāŊĨāŊĨāŊĒāŊĨΔÜŧāŊ āŪūāŊĨāŊĨāŊĒāŊĨΔZāŊāŪūāŊĨāŊĨāŊĒāŊĨ቏  12. Repeat Steps 1 through 11 until the convergence criterion in Step 9b is met. 13. With ሚÜīÝ”Ü―āŽĩ, ÜīÝ•Ü―āŽĩ, ÜīÝ–Ü―āŽĩáˆŧ 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 ÜĻℎāŽĩ 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 ÜīÝ”Ü―āŽĩ, ÜīÝ•Ü―āŽĩ, and ÜīÝ–Ü―āŽĩ.   If the transverse force component of all hangers is zero āĩŦÜĻℎāŊ­āŽĩâ€ĶāŊĄ = 0āĩŊ, 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, āļŦáˆūΔÜīÜ―áˆŋāŊāļŦ â‰Ī 0.5āļŦáˆūÜīÜ―áˆŋāŊāļŦ be placed on Step 11 for ݆ = 1 â€Ķ ݎ݋ݓݏሹáˆūΔÜīÜ―áˆŋáˆŧ. 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).    āŽĩÜķ = āķ§ÜīݔāŽĩāŽķ + ÜīݕāŽĩāŽķ + ÜīݕāŽĩāŽķ (A20)    āŊĄÜķāŽūāŽĩ = āķ§ÜīݔāŊĄāŽūāŽĩāŽķ + ÜīݕāŊĄāŽūāŽĩāŽķ + ÜīݕāŊĄāŽūāŽĩāŽķ (A21)Also, the angles formed between the ends of the cable and the ݔ, ݕ, and ݖ axes are given by, ß āŊŦāŽĩ = Ü―Üŋ݋ݏ ቆ|ÜīݔāŽĩ|āŽĩÜķቇ and ß āŊŦāŊĄāŽūāŽĩ = Ü―Üŋ݋ݏ ቆ|ÜīݔāŊĄāŽūāŽĩ|āŊĄÜķāŽūāŽĩቇ (A22) ß āŊŽāŽĩ = Ü―Üŋ݋ݏ ቆ|ÜīݔāŽĩ|āŽĩÜķቇ and ß āŊŽāŊĄāŽūāŽĩ = Ü―Üŋ݋ݏ ቆ|ÜīݕāŊĄāŽūāŽĩ|āŊĄÜķāŽūāŽĩቇ (A23) ß āŊ­āŽĩ = Ü―Üŋ݋ݏ ቆ|ÜīݖāŽĩ|āŽĩÜķቇ and ß āŊ­āŊĄāŽūāŽĩ = Ü―Üŋ݋ݏ ቆ|ÜīݖāŊĄāŽūāŽĩ|āŊĄÜķāŽūāŽĩቇ (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)),  Ü·ÜĩÜŪ = āĩĨā· ÜĨāŊœ − ΔāŊœāŊĄāŊœā­€āŽĩāĩĐ (A25)where,    ÜĨāŊœ =ÜŠāŊœßąāŊ– āĩĪݏ݅݊ℎ āĩŽßąāŊ–Ü―āŊœÜŠāŊœ + ÜĢāŊœāĩ° − ݏ݅݊ℎሹÜĢāŊœáˆŧāĩĻ (A26)and,    ΔāŊœ =ÜŠāŊœÜ―āŊœÜ§āŊ–ÜĢāŊ– ቈ߱āŊ–ΔݕāŊœāŽķ2ÜŠāŊœÜ―āŊœ Üŋ݋ݐℎ āĩŽßąāŊ–Ü―āŊœ2ÜŠāŊœ āĩ° +12 +ÜŠāŊœ2ßąāŊ–Ü―āŊœ ݏ݅݊ℎ āĩŽßąāŊ–Ü―āŊœÜŠāŊœ āĩ°á‰‰ (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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