{"Affiliation":[{"label":"Affiliation","value":"Applied Science, Faculty of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."},{"label":"Affiliation","value":"Civil Engineering, Department of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."}],"AggregatedSourceRepository":[{"label":"AggregatedSourceRepository","value":"DSpace","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","classmap":"ore:Aggregation","property":"edm:dataProvider"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","explain":"A Europeana Data Model Property; The name or identifier of the organization who contributes data indirectly to an aggregation service (e.g. Europeana)"}],"Campus":[{"label":"Campus","value":"UBCV","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","classmap":"oc:ThesisDescription","property":"oc:degreeCampus"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","explain":"UBC Open Collections Metadata Components; Local Field; Identifies the name of the campus from which the graduate completed their degree."}],"Creator":[{"label":"Creator","value":"Sauer, Devin James","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/creator","classmap":"dpla:SourceResource","property":"dcterms:creator"},"iri":"http:\/\/purl.org\/dc\/terms\/creator","explain":"A Dublin Core Terms Property; An entity primarily responsible for making the resource.; Examples of a Contributor include a person, an organization, or a service."}],"DateAvailable":[{"label":"DateAvailable","value":"2017-04-20T18:40:11Z","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"edm:WebResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"DateIssued":[{"label":"DateIssued","value":"2017","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"oc:SourceResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"Degree":[{"label":"Degree","value":"Doctor of Philosophy - PhD","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","classmap":"vivo:ThesisDegree","property":"vivo:relatedDegree"},"iri":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","explain":"VIVO-ISF Ontology V1.6 Property; The thesis degree; Extended Property specified by UBC, as per https:\/\/wiki.duraspace.org\/display\/VIVO\/Ontology+Editor%27s+Guide"}],"DegreeGrantor":[{"label":"DegreeGrantor","value":"University of British Columbia","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","classmap":"oc:ThesisDescription","property":"oc:degreeGrantor"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the institution where thesis was granted."}],"Description":[{"label":"Description","value":"The self-anchored discontinuous hybrid cable bridge (SDHCB) is a novel type of bridge system which has the potential to overcome many of the deficiencies of conventional cable bridge structures while preserving their advantages. To date, research on the system is extremely limited. Accordingly, this thesis examines the structural and economic attributes of the system, as well its constructability. These areas of research are vital in evaluating the utility of the system and in advancing its development. \r\n\r\nThe structural attributes of SDHCBs were studied in this thesis using a systematic approach. First, the behaviour of each of the two basic cable types found in hybrid cable bridges was studied under a wide range of parameters. Then, starting with a bare model of a SDHCB, a series of analyses was performed while progressively expanding the model so that the influence of various parameters and bridge components could be isolated, and accurately assessed. The model parameters were further refined through a cost analysis. Thereafter, upon reaching a complete and detailed model, the influence of various structural parameters was re-assessed, the structural benefits of employing various supplemental design components were appraised, and the constructability of the system was addressed. \r\n\r\nThis work is significant in that it has provided a highly generalized and robust model of a SDHCB. Using this model, it is possible to ascertain how various design parameters such as geometric factors, material properties, and loading conditions affect the structural behaviour, cost, and constructability of the system. Many insights were also obtained from this research which led to the formation of a recommended, universal, design space for the system. In addition, the practicality and adaptability of the system were demonstrated through the development of several innovative construction schemes which aim to reduce construction duration and costs through the creation of multiple work fronts and the elimination of large temporary works.","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/description","classmap":"dpla:SourceResource","property":"dcterms:description"},"iri":"http:\/\/purl.org\/dc\/terms\/description","explain":"A Dublin Core Terms Property; An account of the resource.; Description may include but is not limited to: an abstract, a table of contents, a graphical representation, or a free-text account of the resource."}],"DigitalResourceOriginalRecord":[{"label":"DigitalResourceOriginalRecord","value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/61293?expand=metadata","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","classmap":"ore:Aggregation","property":"edm:aggregatedCHO"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","explain":"A Europeana Data Model Property; The identifier of the source object, e.g. the Mona Lisa itself. This could be a full linked open date URI or an internal identifier"}],"FullText":[{"label":"FullText","value":" \ud835\udc4e\ud835\udc4e\ud835\udc5c\ud835\udc34\ud835\udc34\ud835\udc60\ud835\udc4e\ud835\udc34\ud835\udc60\ud835\udc4e\ud835\udc5c\ud835\udc34\ud835\udc50\ud835\udc34\ud835\udc50\ud835\udc5a\ud835\udc34\ud835\udc54\ud835\udc34\u210e\ud835\udc34\ud835\udc60\ud835\udc4f\ud835\udc34\ud835\udc60\ud835\udc4f\ud835\udc5c\ud835\udc34\ud835\udc60\ud835\udc5a\ud835\udc34\ud835\udc60\ud835\udc5a\ud835\udc5c\ud835\udc34\ud835\udc61\ud835\udc4f\ud835\udc35\ud835\udc50\ud835\udc50\ud835\udc50\ud835\udc5a\ud835\udc50\ud835\udc53\ud835\udc50\u210e\ud835\udc50\ud835\udc60\ud835\udc50\ud835\udc60\ud835\udc61\ud835\udc50\ud835\udc61\ud835\udc36\ud835\udc36\ud835\udc3b\ud835\udc36\ud835\udc3b\ud835\udc5c\ud835\udc36\ud835\udc5c, \ud835\udc36\ud835\udc53\ud835\udc51\ud835\udc38\ud835\udc38\ud835\udc50\ud835\udc38\ud835\udc52\ud835\udc53\ud835\udc53\ud835\udc38\ud835\udc60\ud835\udc52\ud835\udc50\ud835\udc38\ud835\udc61\ud835\udc4e\ud835\udc5b\ud835\udc53\ud835\udc53\u210e\ud835\udc53\ud835\udc5c\ud835\udc53\ud835\udc5d\ud835\udc53\ud835\udc66\ud835\udc39\ud835\udc50\ud835\udc39\u210e\ud835\udc39\u210e\ud835\udc65, \ud835\udc39\u210e\ud835\udc66, \ud835\udc39\u210e\ud835\udc67\ud835\udc39\ud835\udc60\ud835\udc39\ud835\udc5d\ud835\udc54\u210e\u210e\ud835\udc35\u210e\ud835\udc5c\u210e\ud835\udc47\u210e\ud835\udc47\ud835\udc45 \u210e\ud835\udc47: \ud835\udc3f\ud835\udc5a\u210e\ud835\udc47\ud835\udc45\ud835\udc5c\u210e\ud835\udc65, \u210e\ud835\udc66, \u210e\ud835\udc67\ud835\udc3b\ud835\udc3b\ud835\udc50\ud835\udc5a, \ud835\udc3b\ud835\udc50\ud835\udc60\ud835\udc3b\ud835\udc46\ud835\udc45\ud835\udc3c\ud835\udc54\ud835\udc3c\ud835\udc61\ud835\udc3e\ud835\udc3e\ud835\udc5f\ud835\udc3f\ud835\udc50\ud835\udc3f\ud835\udc50\ud835\udc5a _1\ud835\udc3f\ud835\udc50\ud835\udc5a_2\ud835\udc3f\ud835\udc50\ud835\udc5c\ud835\udc3f\u210e\ud835\udc3f\ud835\udc5a\ud835\udc3f\ud835\udc5a\ud835\udc5c\ud835\udc3f\ud835\udc5d\ud835\udc3f\ud835\udc45 \ud835\udc3f\u210e: \ud835\udc3f\ud835\udc5a\ud835\udc3f\ud835\udc60\ud835\udc3f\ud835\udc46\ud835\udc45 \ud835\udc3f\ud835\udc60: \ud835\udc3f\ud835\udc5a\ud835\udc3f\ud835\udc46\ud835\udc45\ud835\udc5c\ud835\udc40\ud835\udc40\ud835\udc46\ud835\udc42\ud835\udc5b\ud835\udc41\ud835\udc63\ud835\udc41\ud835\udc53\ud835\udc5d, \ud835\udc5e\ud835\udc44\ud835\udc44\ud835\udc50\ud835\udc5a, \ud835\udc44\ud835\udc50\ud835\udc60\ud835\udc44\ud835\udc53\ud835\udc44\ud835\udc39\ud835\udc4e\ud835\udc5b, \ud835\udc44\ud835\udc3b\ud835\udc4e\ud835\udc5f\ud835\udc5d\ud835\udc44\u210e\ud835\udc44\ud835\udc60\ud835\udc44\ud835\udc60\ud835\udc61\ud835\udc44\ud835\udc60\ud835\udc61\ud835\udc5a , \ud835\udc44\ud835\udc60\ud835\udc61\ud835\udc60\ud835\udc44\ud835\udc61\ud835\udc45\ud835\udc5d\ud835\udc45\ud835\udc65, \ud835\udc45\ud835\udc66, \ud835\udc45\ud835\udc67\ud835\udc46\ud835\udc45 \ud835\udc53: \ud835\udc3f\ud835\udc5a\ud835\udc47\ud835\udc47\ud835\udc50\ud835\udc60\ud835\udc47\ud835\udc34, \ud835\udc47\ud835\udc35\ud835\udc48\ud835\udc46\ud835\udc3f\ud835\udc48\ud835\udc46\ud835\udc3f\ud835\udc50 , \ud835\udc48\ud835\udc46\ud835\udc3f\ud835\udc5d\ud835\udc48\ud835\udc46\ud835\udc3f\ud835\udc5c, \ud835\udc48\ud835\udc46\ud835\udc3f\ud835\udc53\ud835\udc49\ud835\udc34, \ud835\udc49\ud835\udc35\ud835\udc49\ud835\udc50\ud835\udc5a,\ud835\udc49\ud835\udc50\ud835\udc60\ud835\udc49\ud835\udc5d\ud835\udc53\ud835\udc49\ud835\udc60\ud835\udc49\ud835\udc47\ud835\udc49\ud835\udc42\ud835\udc3f\ud835\udc60\ud835\udc4e\ud835\udc4a\ud835\udc61\ud835\udc65\ud835\udc5d\ud835\udc65\ud835\udc5d\ud835\udc56\ud835\udc66\ud835\udc35\ud835\udc66\ud835\udc50 , \ud835\udc66\ud835\udc5d\ud835\udc67\ud835\udefc\ud835\udc5d\ud835\udefc\ud835\udc60\ud835\udc51\ud835\udc59\ud835\udefd\ud835\udc61\ud835\udefe\ud835\udc50\ud835\udefe\ud835\udc50\ud835\udc5a\ud835\udefe\ud835\udc60\ud835\udefe\ud835\udc60\ud835\udc61\ud835\udefe\ud835\udc61\ud835\udeff\ud835\udeff\ud835\udc46\ud835\udc42(\ud835\udeff)\ud835\udc39\ud835\udc3f, (\ud835\udeff)\ud835\udc43\ud835\udc3f\ud835\udeff\ud835\udc50\ud835\udeff\ud835\udc52\ud835\udeff\u210e, \ud835\udeff\ud835\udc63\ud835\udeff\ud835\udc5f\ud835\udeff\ud835\udc60\ud835\udc53\ud835\udeff\ud835\udc4e\ud835\udeff\ud835\udc36\ud835\udeff\ud835\udc53\ud835\udeff\u210e\ud835\udeff\u210e\ud835\udc65, \ud835\udeff\u210e\ud835\udc66, \ud835\udeff\u210e\ud835\udc67\ud835\udeff\ud835\udc3f\ud835\udc50\ud835\udeff\ud835\udf03\ud835\udc50\ud835\udeff\ud835\udf0e\ud835\udc50\ud835\udee5\ud835\udc52\ud835\udee5\ud835\udc52\ud835\udc5c , \ud835\udee5\ud835\udc52\ud835\udc53\ud835\udee5\ud835\udc65, \ud835\udee5\ud835\udc66, \ud835\udee5\ud835\udc67\ud835\udee5\ud835\udc4b\ud835\udc47 , \ud835\udee5\ud835\udc4c\ud835\udc47 , \ud835\udee5\ud835\udc4d\ud835\udc47\ud835\udee5\ud835\udf14\ud835\udc60\ud835\udf16\ud835\udc50\ud835\udf02\ud835\udf03\ud835\udc34, \ud835\udf03\ud835\udc35\ud835\udf03\ud835\udc50\ud835\udf03\ud835\udc65, \ud835\udf03\ud835\udc66 , \ud835\udf03\ud835\udc67\ud835\udf06\u039b\u03bc\ud835\udf09\ud835\udf0c\ud835\udc50\ud835\udc5a, \ud835\udf0c\ud835\udc50\ud835\udc60\ud835\udf0c\u210e\ud835\udf0c\ud835\udc60\ud835\udc61\ud835\udf0c\ud835\udc61\ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64\ud835\udf0e\ud835\udc4e,\ud835\udc37\ud835\udc3f, \ud835\udf0e\ud835\udc4e,\ud835\udc3f\ud835\udc3f\ud835\udf0e\ud835\udc4f,\ud835\udc3f\ud835\udc3f\ud835\udf0e\ud835\udc50 \ud835\udc39\ud835\udc50 \ud835\udc34\ud835\udc50\u2044\ud835\udf0e\ud835\udc50\ud835\udc5c, \ud835\udf0e\ud835\udc50\ud835\udc53\ud835\udf0e\ud835\udc50\ud835\udc5a, \ud835\udf0e\ud835\udc50\ud835\udc60\ud835\udf0e\u210e\ud835\udf0e\ud835\udc60\ud835\udc61\ud835\udf0e\ud835\udc60\ud835\udf0e\ud835\udc61\ud835\udef4\ud835\udc3b\ud835\udc60\ud835\udc61\ud835\udc60, \ud835\udef4\ud835\udc49\ud835\udc60\ud835\udc61\ud835\udc60\ud835\udef9\ud835\udc50 \ud835\udf14\ud835\udc50\ud835\udc4e\/\ud835\udc3b\ud835\udef9\ud835\udc50\ud835\udc5d \ud835\udf14\ud835\udc50\ud835\udc5d\ud835\udc4e\/\ud835\udc3b?\u0305?\ud835\udc5d (\ud835\udf14\ud835\udc50 + \ud835\udf14\ud835\udc60)\ud835\udc3f\ud835\udc5a\/\ud835\udc3b\ud835\udf14\ud835\udc50\ud835\udf14\ud835\udc50\ud835\udc5a\ud835\udf14\ud835\udc50\ud835\udc5d\ud835\udf14\ud835\udc5d\ud835\udf14\ud835\udc60\ud835\udf14\ud835\udc60\ud835\udc5a, \ud835\udf14\ud835\udc60\ud835\udc60\ud835\udf14\ud835\udc60\ud835\udc5c\ud835\udf14\ud835\udc45 \ud835\udf14\ud835\udc5d: \ud835\udf14\ud835\udc60\ud835\udf14\ud835\udc45\ud835\udc5a \ud835\udf14\ud835\udc5d: \ud835\udf14\ud835\udc60\ud835\udc5a\ud835\udf14\ud835\udc45\ud835\udc5c \ud835\udf14\ud835\udc5d: \ud835\udf14\ud835\udc60\ud835\udc5c\ud835\udefaSuperstructure 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 \uf0a7 \uf0a7 \uf0a7 \uf0a7 \uf0a7 \uf0a7 \uf0a7 \uf0a7 \uf0a7 \ud835\udf14\ud835\udc50 \ud835\udc53\ud835\udc3b\ud835\udc512\ud835\udc66\ud835\udc51\ud835\udc652=\ud835\udf14\ud835\udc50\ud835\udc3b\u221a1+ (\ud835\udc51\ud835\udc66\ud835\udc51\ud835\udc65)2\ud835\udf03\ud835\udc35 \ud835\udf03\ud835\udc34 \ud835\udc53 \ud835\udc9a \ud835\udc99 \ud835\udc4e2\u2044 \ud835\udc4e2\u2044 \u210e \ud835\udc49\ud835\udc34 \ud835\udc49\ud835\udc35 \ud835\udc47\ud835\udc35 \ud835\udc47\ud835\udc34 \ud835\udc3b \ud835\udc3b \ud835\udc39\ud835\udc50 \ud835\udc39\ud835\udc50 \u03a8\ud835\udc50 =\ud835\udf14\ud835\udc50\ud835\udc4e\ud835\udc3b\u03a9 =\u210e\ud835\udc4e\ud835\udc51\ud835\udc66\ud835\udc51\ud835\udc65= \ud835\udc60\ud835\udc56\ud835\udc5b\u210e (\u03a8\ud835\udc50\ud835\udc65\ud835\udc4e+ \ud835\udc34) \ud835\udc66 =\ud835\udc4e\u03a8\ud835\udc50\ud835\udc50\ud835\udc5c\ud835\udc60\u210e (\u03a8\ud835\udc50\ud835\udc65\ud835\udc4e+ \ud835\udc34) + \ud835\udc35\ud835\udc66(\ud835\udc65 = 0) = 0 \ud835\udc66(\ud835\udc65 = \ud835\udc4e) = \u210e \ud835\udc34 = \ud835\udc4e\ud835\udc60\ud835\udc56\ud835\udc5b\u210e [\u03a8\ud835\udc50\u03a92\ud835\udc60\ud835\udc56\ud835\udc5b\u210e (\u03a8\ud835\udc502 )]\u2212\u03a8\ud835\udc502\ud835\udc35 = \u2212\ud835\udc4e\u03a8\ud835\udc50\ud835\udc50\ud835\udc5c\ud835\udc60\u210e (\ud835\udc34)\ud835\udc49\ud835\udc34 = \ud835\udc3b\ud835\udc51\ud835\udc66\ud835\udc51\ud835\udc65|\ud835\udc65=0=\ud835\udf14\ud835\udc50\ud835\udc4e\u03a8\ud835\udc50\ud835\udc60\ud835\udc56\ud835\udc5b\u210e (\ud835\udc34)\ud835\udc49\ud835\udc35 = \ud835\udc3b\ud835\udc51\ud835\udc66\ud835\udc51\ud835\udc65|\ud835\udc65=\ud835\udc4e=\ud835\udf14\ud835\udc50\ud835\udc4e\u03a8\ud835\udc50\ud835\udc60\ud835\udc56\ud835\udc5b\u210e(\u03a8\ud835\udc50 + \ud835\udc34)\u210e \ud835\udc47\ud835\udc34 \ud835\udc47\ud835\udc35\u210e\ud835\udc47\ud835\udc34 = \u221a\ud835\udc3b2 + \ud835\udc49\ud835\udc342 =\ud835\udf14\ud835\udc50\ud835\udc4e\u03a8\ud835\udc50\ud835\udc50\ud835\udc5c\ud835\udc60\u210e(\ud835\udc34)\ud835\udc47\ud835\udc35 = \u221a\ud835\udc3b2 + \ud835\udc49\ud835\udc352 =\ud835\udf14\ud835\udc50\ud835\udc4e\u03a8\ud835\udc50\ud835\udc50\ud835\udc5c\ud835\udc60\u210e(\u03a8\ud835\udc50 + \ud835\udc34)\ud835\udf03\ud835\udc34 = \ud835\udc4e\ud835\udc61\ud835\udc4e\ud835\udc5b\ud835\udc51\ud835\udc66\ud835\udc51\ud835\udc65|\ud835\udc65=0= \ud835\udc4e\ud835\udc61\ud835\udc4e\ud835\udc5b[\ud835\udc60\ud835\udc56\ud835\udc5b\u210e(\ud835\udc34)]\ud835\udf03\ud835\udc35 = \ud835\udc4e\ud835\udc61\ud835\udc4e\ud835\udc5b\ud835\udc51\ud835\udc66\ud835\udc51\ud835\udc65|\ud835\udc65=\ud835\udc4e= \ud835\udc4e\ud835\udc61\ud835\udc4e\ud835\udc5b[\ud835\udc60\ud835\udc56\ud835\udc5b\u210e(\u03a8\ud835\udc50 +\ud835\udc34)]\ud835\udc36\ud835\udc36 = \u222b \u221a1 + (\ud835\udc51\ud835\udc66\ud835\udc51\ud835\udc65)2\ud835\udc51\ud835\udc65\ud835\udc4e0=\ud835\udc4e\u03a8\ud835\udc50[\ud835\udc60\ud835\udc56\ud835\udc5b\u210e(\u03a8\ud835\udc50 + \ud835\udc34) \u2212 \ud835\udc60\ud835\udc56\ud835\udc5b\u210e (\ud835\udc34)]\ud835\udc48\ud835\udc46\ud835\udc3f \u0394\ud835\udc52\ud835\udc36 = \ud835\udc48\ud835\udc46\ud835\udc3f + \u0394\ud835\udc52\u2206\ud835\udc52 =\ud835\udc3b\ud835\udc38\ud835\udc50\ud835\udc34\ud835\udc50\u222b [1 + (\ud835\udc51\ud835\udc66\ud835\udc51\ud835\udc65)2] \ud835\udc51\ud835\udc65\ud835\udc4e0=\ud835\udefe\ud835\udc50\ud835\udc4e2\u03a8\ud835\udc50\ud835\udc38\ud835\udc50[\u03a8\ud835\udc50\u03a922\ud835\udc50\ud835\udc5c\ud835\udc61\u210e (\u03a8\ud835\udc502) +12+12\u03a8\ud835\udc50\ud835\udc60\ud835\udc56\ud835\udc5b\u210e(\u03a8\ud835\udc50)]\ud835\udc34\ud835\udc50 \ud835\udc38\ud835\udc50 \ud835\udefe\ud835\udc50\ud835\udc36 \u0394\ud835\udc52\ud835\udc48\ud835\udc46\ud835\udc3f = \ud835\udc36 \u2212 \u0394\ud835\udc52\u03a8\ud835\udc50\ud835\udc39\ud835\udc50\ud835\udc39\ud835\udc50 = \ud835\udc3b\ud835\udc3f\ud835\udc50\ud835\udc4e= \ud835\udc3b\u221a1 + \u03a92\u03a8\ud835\udc50\u03a8\ud835\udc50 =\ud835\udf14\ud835\udc50\ud835\udc4e\ud835\udc39\ud835\udc50\u221a1 + \u03a92\ud835\udf0e\ud835\udc50\u03a8\ud835\udc50 =\ud835\udefe\ud835\udc50\ud835\udc4e\ud835\udf0e\ud835\udc50\u221a1 + \u03a92\ud835\udf14\ud835\udc50\ud835\udc5d \ud835\udf14\ud835\udc50\ud835\udc5d = \ud835\udf14\ud835\udc50\u221a1+ \u03a92\u03a9\ud835\udc512\ud835\udc66\ud835\udc51\ud835\udc652=\ud835\udf14\ud835\udc50\ud835\udc5d\ud835\udc3b\ud835\udc51\ud835\udc66\ud835\udc51\ud835\udc65=\u03a8\ud835\udc5d2\ud835\udc4e(2\ud835\udc65 \u2212 \ud835\udc4e) + \u03a9\ud835\udc66 =\u03a8\ud835\udc5d\ud835\udc652\ud835\udc4e(\ud835\udc65 \u2212 \ud835\udc4e) + \u03a9\ud835\udc65\ud835\udf14\ud835\udc50 \ud835\udf14\ud835\udc50 \ud835\udf14\ud835\udc50\ud835\udc5d \u03a8\ud835\udf14\ud835\udc50\ud835\udc5d \u03a8\ud835\udc5d =\ud835\udf14\ud835\udc50\ud835\udc5d\ud835\udc4e\ud835\udc3b\ud835\udc49\ud835\udc34 = \ud835\udc3b\ud835\udc51\ud835\udc66\ud835\udc51\ud835\udc65|\ud835\udc65=0=\ud835\udf14\ud835\udc50\ud835\udc5d\ud835\udc4e\u03a8\ud835\udc5d(\u03a9 \u2212\u03a8\ud835\udc5d2)\ud835\udc49\ud835\udc35 = \ud835\udc3b\ud835\udc51\ud835\udc66\ud835\udc51\ud835\udc65|\ud835\udc65=\ud835\udc4e=\ud835\udf14\ud835\udc50\ud835\udc5d\ud835\udc4e\u03a8\ud835\udc5d(\u03a9 +\u03a8\ud835\udc5d2)\ud835\udc47\ud835\udc34 = \u221a\ud835\udc3b2 + \ud835\udc49\ud835\udc342 =\ud835\udf14\ud835\udc50\ud835\udc5d\ud835\udc4e\u03a8\ud835\udc5d\u221a1 + (\u03a9 \u2212\u03a8\ud835\udc5d2)2\ud835\udc47\ud835\udc35 = \u221a\ud835\udc3b2 + \ud835\udc49\ud835\udc352 =\ud835\udf14\ud835\udc50\ud835\udc5d\ud835\udc4e\u03a8\ud835\udc5d\u221a1 + (\u03a9 +\u03a8\ud835\udc5d2)2\ud835\udf03\ud835\udc34 = \ud835\udc4e\ud835\udc61\ud835\udc4e\ud835\udc5b\ud835\udc51\ud835\udc66\ud835\udc51\ud835\udc65|\ud835\udc65=0= \ud835\udc4e\ud835\udc61\ud835\udc4e\ud835\udc5b(\u03a9 \u2212\u03a8\ud835\udc5d2)\ud835\udf03\ud835\udc35 = \ud835\udc4e\ud835\udc61\ud835\udc4e\ud835\udc5b\ud835\udc51\ud835\udc66\ud835\udc51\ud835\udc65|\ud835\udc65=\ud835\udc4e= \ud835\udc4e\ud835\udc61\ud835\udc4e\ud835\udc5b(\u03a9 +\u03a8\ud835\udc5d2)\ud835\udc36 = \u222b \u221a[1 + (\ud835\udc51\ud835\udc66\ud835\udc51\ud835\udc65)2] \ud835\udc51\ud835\udc65\ud835\udc4e0\ud835\udc36 =\ud835\udc4e2\u03a8\ud835\udc5d[(\u03a9 +\u03a8\ud835\udc5d2)\u221a1 + \u03a92 + \u03a9\u03a8\ud835\udc5d +\u03a8\ud835\udc5d24\u2212 (\u03a9 \u2212\u03a8\ud835\udc5d2)\u221a1 + \u03a92 \u2212 \u03a9\u03a8\ud835\udc5d +\u03a8\ud835\udc5d24+ \ud835\udc4e\ud835\udc60\ud835\udc56\ud835\udc5b\u210e (\u03a9 +\u03a8\ud835\udc5d2)\u2212 \ud835\udc4e\ud835\udc60\ud835\udc56\ud835\udc5b\u210e (\u03a9 \u2212\u03a8\ud835\udc5d2)]\u221a1 + \ud835\udc58 \ud835\udc58\u221a1 + \ud835\udc58 = \u2211(\u22121)\ud835\udc5a(2\ud835\udc5a)!(1 \u2212 2\ud835\udc5a)(\ud835\udc5a!)2(4\ud835\udc5a)\ud835\udc58\ud835\udc5a\u221e\ud835\udc5a=0\ud835\udc58 = 0 |\ud835\udc58| \u2264 1\u03a9\u03a9 = 0 \ud835\udc36\ud835\udc4e\ud835\udc5d\ud835\udc5d\ud835\udc5f\ud835\udc5c\ud835\udc65|\u03a9=0 = \u222b [1 +12(\ud835\udc51\ud835\udc66\ud835\udc51\ud835\udc65)2]\ud835\udc51\ud835\udc65 = \ud835\udc3f\ud835\udc50 [1 +124(\ud835\udf14\ud835\udc50\ud835\udc3f\ud835\udc50\ud835\udc3b\u210e\ud835\udc5c\ud835\udc5f\ud835\udc56\ud835\udc67)2]\ud835\udc3f\ud835\udc500\ud835\udc3b\u210e\ud835\udc5c\ud835\udc5f\ud835\udc56\ud835\udc67 \ud835\udc3f\ud835\udc50\u03a9\ud835\udc36\ud835\udc60\ud835\udc56\ud835\udc5a\ud835\udc5d\ud835\udc59\ud835\udc56\ud835\udc53\ud835\udc56\ud835\udc52\ud835\udc51 = \ud835\udc4e\u221a1 + \u03a92 [1 +124(\u03a8\ud835\udc5d1 + \u03a92)2]\u2206\ud835\udc52 =\ud835\udc3b\ud835\udc38\ud835\udc50\ud835\udc34\ud835\udc50\u222b [1 + (\ud835\udc51\ud835\udc66\ud835\udc51\ud835\udc65)2] \ud835\udc51\ud835\udc65\ud835\udc4e0=\ud835\udefe\ud835\udc50\ud835\udc4e2\u221a1 + \u03a92\u03a8\ud835\udc5d\ud835\udc38\ud835\udc50[1 + \u03a92 +112\u03a8\ud835\udc5d2]\u03a8\ud835\udc5d\u03a8\ud835\udc5d =\ud835\udf14\ud835\udc50\ud835\udc4e\ud835\udc39\ud835\udc50(1 + \u03a92)\u03a8\ud835\udc5d =\ud835\udefe\ud835\udc50\ud835\udc4e\ud835\udf0e\ud835\udc50(1 + \u03a92) \ud835\udc3f\ud835\udc50 \ud835\udc53\u210e\ud835\udc5c\ud835\udc5f\ud835\udc56\ud835\udc67 \ud835\udc4e2\u2044 \ud835\udc4e2\u2044 \ud835\udc3b\u210e\ud835\udc5c\ud835\udc5f\ud835\udc56\ud835\udc67 =\ud835\udf14\ud835\udc50\ud835\udc3f\ud835\udc5028\ud835\udc53\u210e\ud835\udc5c\ud835\udc5f\ud835\udc56\ud835\udc67 \ud835\udc3b\u210e\ud835\udc5c\ud835\udc5f\ud835\udc56\ud835\udc67 \ud835\udc3b \ud835\udc3b =\ud835\udf14\ud835\udc50\ud835\udc4e28\ud835\udc53\u210e\ud835\udc5c\ud835\udc5f\ud835\udc56\ud835\udc67= \ud835\udc3b\u210e\ud835\udc5c\ud835\udc5f\ud835\udc56\ud835\udc67 (\ud835\udc4e\ud835\udc3f\ud835\udc50)2 \u03a8\ud835\udc5d\ud835\udc53\ud835\udc5d\u03a8\ud835\udc5d =8\ud835\udc53\ud835\udc5d\ud835\udc4e\ud835\udf0e\ud835\udc50\ud835\udc53\ud835\udc5d\ud835\udec0 = \ud835\udfce \ud835\udec0 = \ud835\udfce. \ud835\udfd3 \ud835\udec0 = \ud835\udfcf \ud835\udec0 = \ud835\udfcf. \ud835\udfd3 \ud835\udec0 = \ud835\udfd00 0.02 0.04 0.06 0.08 0.1 0.12 0.1401 \uf02e1042 \uf02e1043 \uf02e1044 \uf02e1045 \uf02e104\ud835\udf38\ud835\udc84\ud835\udc82\ud835\udf48\ud835\udc84 \ud835\udc76\ud835\udc79 \ud835\udfd6\ud835\udc87\ud835\udc91\ud835\udc82(\ud835\udfcf + \ud835\udf34\ud835\udfd0) |(\ud835\udc9a\ud835\udc84 \u2212 \ud835\udc9a\ud835\udc91)\ud835\udc8e\ud835\udc82\ud835\udc99|\ud835\udc82 \ud835\udf0e\ud835\udc50 \ud835\udc4e \ud835\udc53\ud835\udc5d \ud835\udf0e\ud835\udc50 \ud835\udc4e < \ud835\udc36\ud835\udefe\ud835\udc50 \ud835\udc4e \ud835\udefa \ud835\udf0e\ud835\udc50\ud835\udf48\ud835\udc84 \ud835\udc82 \ud835\udf48\ud835\udc84 \ud835\udf48\ud835\udc84 \ud835\udc82 \ud835\udf48\ud835\udc84 (\ud835\udc9a\ud835\udc84 \u2212 \ud835\udc9a\ud835\udc91)\ud835\udc8e\ud835\udc82\ud835\udc99 (\ud835\udc9a\ud835\udc84 \u2212 \ud835\udc9a\ud835\udc91)\ud835\udc8e\ud835\udc82\ud835\udc99 \u03a9\ud835\udc48\ud835\udc46\ud835\udc3f\ud835\udc5d\ud835\udc48\ud835\udc46\ud835\udc3f\ud835\udc50\ud835\udc38\ud835\udc50 \ud835\udc3f\ud835\udc5a \ud835\udefe\ud835\udc50 \ud835\udf0e\ud835\udc50\ud835\udc3f\ud835\udc5a 1.0 0.5 0.0 \ud835\udc3f\ud835\udc5a4 1.0 0.5 0.0 \ud835\udec0 = \ud835\udfce\ud835\udec0 = \ud835\udfce. \ud835\udfd3\ud835\udec0 = \ud835\udfcf\ud835\udec0 = \ud835\udfcf. \ud835\udfd3\ud835\udec0 = \ud835\udfd0\ud835\udc38\ud835\udc500 0.02 0.04 0.06 0.08 0.1 0.12 0.1402 \uf02e1064 \uf02e1066 \uf02e1068 \uf02e1060 0.02 0.04 0.06 0.08 0.1 0.12 0.1402 \uf02e1064 \uf02e1066 \uf02e1068 \uf02e106|\ud835\udc7c\ud835\udc7a\ud835\udc73\ud835\udc84 \u2212 \ud835\udc7c\ud835\udc7a\ud835\udc73\ud835\udc91|\ud835\udc82 \ud835\udf38\ud835\udc84\ud835\udc82\ud835\udf48\ud835\udc84 \ud835\udc76\ud835\udc79 \ud835\udfd6\ud835\udc87\ud835\udc91\ud835\udc82(\ud835\udfcf + \ud835\udf34\ud835\udfd0) \ud835\udf0e\ud835\udc50 \ud835\udc4e \ud835\udf0e\ud835\udc50 \ud835\udc53\ud835\udc5d \ud835\udf0e\ud835\udc50\ud835\udc5c \ud835\udf0e\ud835\udc50\ud835\udc53\ud835\udc36\ud835\udc5c = \ud835\udc3f\ud835\udc50\ud835\udc5c [1 +124(1 + \u03a92)(\ud835\udefe\ud835\udc50\ud835\udc3f\ud835\udc50\ud835\udc5c\ud835\udf0e\ud835\udc50\ud835\udc5c)2]\u0394\ud835\udc52\ud835\udc5c =\ud835\udc3f\ud835\udc50\ud835\udc5c\ud835\udf0e\ud835\udc50\ud835\udc5c\ud835\udc38\ud835\udc50+112\ud835\udc38\ud835\udc50(\ud835\udefe\ud835\udc502\ud835\udc3f\ud835\udc50\ud835\udc5c3\ud835\udf0e\ud835\udc50\ud835\udc5c)\ud835\udc36\ud835\udc53 = \ud835\udc3f\ud835\udc50\ud835\udc5c + \u03b4\ud835\udc3f\ud835\udc50 +124(1 + \u03a92)(\ud835\udefe\ud835\udc50\ud835\udf0e\ud835\udc50\ud835\udc53)2(\ud835\udc3f\ud835\udc50\ud835\udc5c + \ud835\udeff\ud835\udc3f\ud835\udc50)3\u0394\ud835\udc52\ud835\udc53 =(\ud835\udc3f\ud835\udc50\ud835\udc5c + \u03b4\ud835\udc3f\ud835\udc50)\ud835\udf0e\ud835\udc50\ud835\udc53\ud835\udc38\ud835\udc50++112\ud835\udc38\ud835\udc50\ud835\udefe\ud835\udc502\ud835\udf0e\ud835\udc50\ud835\udc53(\ud835\udc3f\ud835\udc50\ud835\udc5c + \ud835\udeff\ud835\udc3f\ud835\udc50)3 \ud835\udc53\ud835\udc5c \ud835\udc53\ud835\udc5c \u2212 \ud835\udeff\ud835\udc53 \ud835\udc4e\ud835\udc5c \ud835\udf0e\ud835\udc50\ud835\udc53 = \ud835\udf0e\ud835\udc50\ud835\udc5c + \ud835\udeff\ud835\udf0e\ud835\udc50 + \ud835\udf0e\ud835\udc50\ud835\udc5c \ud835\udf0e\ud835\udc50\ud835\udc5c \ud835\udf0e\ud835\udc50\ud835\udc53 \u210e\ud835\udc5c \u210e\ud835\udc5c + \ud835\udeff\u210e \ud835\udc4e\ud835\udc5c + \ud835\udeff\ud835\udc4e \ud835\udc48\ud835\udc46\ud835\udc3f\ud835\udc5c = \ud835\udc48\ud835\udc46\ud835\udc3f\ud835\udc53\u03b4\ud835\udc3f\ud835\udc50\ud835\udc3f\ud835\udc50\ud835\udc5c\u2212\u03b4\ud835\udc3f\ud835\udc50\ud835\udc3f\ud835\udc50\ud835\udc5c\ud835\udf0e\ud835\udc50\ud835\udc53\ud835\udc38\ud835\udc50=(\ud835\udf0e\ud835\udc50\ud835\udc53 \u2212 \ud835\udf0e\ud835\udc50\ud835\udc5c)\ud835\udc38\ud835\udc50\u2212\ud835\udefe\ud835\udc502\ud835\udc4e224[1\ud835\udf0e\ud835\udc50\ud835\udc532(\ud835\udc3f\ud835\udc50\ud835\udc5c + \u03b4\ud835\udc3f\ud835\udc50)3\ud835\udc3f\ud835\udc50\ud835\udc5c3 \u22121\ud835\udf0e\ud835\udc50\ud835\udc5c2]+\ud835\udefe\ud835\udc502\ud835\udc3f\ud835\udc50\ud835\udc5c212\ud835\udc38\ud835\udc50\ud835\udf0e\ud835\udc50\ud835\udc53[(\ud835\udc3f\ud835\udc50\ud835\udc5c + \u03b4\ud835\udc3f\ud835\udc50)3\ud835\udc3f\ud835\udc50\ud835\udc5c3 \u2212\ud835\udf0e\ud835\udc50\ud835\udc53\ud835\udf0e\ud835\udc50\ud835\udc5c](\ud835\udeff\ud835\udc3f\ud835\udc50 \u226a \ud835\udc3f\ud835\udc50\ud835\udc5c)\u03b4\ud835\udc3f\ud835\udc50\ud835\udc3f\ud835\udc50\ud835\udc5c=(\ud835\udf0e\ud835\udc50\ud835\udc53 \u2212 \ud835\udf0e\ud835\udc50\ud835\udc5c)\ud835\udc38\ud835\udc50+\ud835\udefe\ud835\udc502\ud835\udc4e224(1\ud835\udf0e\ud835\udc50\ud835\udc5c2\u22121\ud835\udf0e\ud835\udc50\ud835\udc532)Linear Term Nonlinear Term ~1~0~0\ud835\udc38\ud835\udc50\ud835\udc38\ud835\udc60\ud835\udc52\ud835\udc50\ud835\udc38\ud835\udc60\ud835\udc52\ud835\udc50 = (\ud835\udf0e\ud835\udc50\ud835\udc53 \u2212 \ud835\udf0e\ud835\udc50\ud835\udc5c)\ud835\udc3f\ud835\udc50\ud835\udc5c\u03b4\ud835\udc3f\ud835\udc50\ud835\udc38\ud835\udc60\ud835\udc52\ud835\udc50 =11\ud835\udc38\ud835\udc50+\ud835\udefe\ud835\udc502\ud835\udc4e2(\ud835\udf0e\ud835\udc50\ud835\udc53 + \ud835\udf0e\ud835\udc50\ud835\udc5c)24\ud835\udf0e\ud835\udc50\ud835\udc532\ud835\udf0e\ud835\udc50\ud835\udc5c2 \ud835\udf48\ud835\udc84\ud835\udc87 (0,0) \ud835\udc3f\ud835\udc50\ud835\udc5c \ud835\udf39\ud835\udc73\ud835\udc84\ud835\udc73\ud835\udc84\ud835\udc90 \ud835\udf0e\ud835\udc50\ud835\udc5c \ud835\udc6c\ud835\udc84 \ud835\udfcf \ud835\udeff\ud835\udf0e\ud835\udc50 \ud835\udeff\ud835\udf0e\ud835\udc50\ud835\udeff\ud835\udf0e\ud835\udc50 \ud835\udeff\ud835\udf0e\ud835\udc50\ud835\udeff\ud835\udf0e\ud835\udc50\ud835\udeff\ud835\udf0e\ud835\udc50 \u226a \ud835\udf0e\ud835\udc50\ud835\udc5c\ud835\udf0e\ud835\udc50\ud835\udc53 = \ud835\udf0e\ud835\udc50\ud835\udc5c \ud835\udf0e\ud835\udc50\ud835\udc53 \ud835\udf0e\ud835\udc50\ud835\udc5c\ud835\udc38\ud835\udc61\ud835\udc4e\ud835\udc5b =11\ud835\udc38\ud835\udc50+\ud835\udefe\ud835\udc502\ud835\udc4e212\ud835\udf0e\ud835\udc50\ud835\udc5c3 \ud835\udeff\ud835\udc3f\ud835\udc50\ud835\udc3f\ud835\udc50\ud835\udc5c \ud835\udf0e\ud835\udc50\ud835\udc5c \ud835\udc6c\ud835\udc94\ud835\udc86\ud835\udc84 \ud835\udfcf \ud835\udf0e\ud835\udc50\ud835\udc53 \ud835\udeff\ud835\udf0e\ud835\udc50 \ud835\udf0e\ud835\udc50\ud835\udc5c\ud835\udc38\ud835\udc50\ud835\uded4\ud835\udc84\ud835\udc90 = \ud835\udfcf\ud835\udfce\ud835\udfce\ud835\udc74\ud835\udc77\ud835\udc82\ud835\uded4\ud835\udc84\ud835\udc90 = \ud835\udfd0\ud835\udfce\ud835\udfce\ud835\udc74\ud835\udc77\ud835\udc82\ud835\uded4\ud835\udc84\ud835\udc90 = \ud835\udfd1\ud835\udfce\ud835\udfce\ud835\udc74\ud835\udc77\ud835\udc82\ud835\uded4\ud835\udc84\ud835\udc90 = \ud835\udfd2\ud835\udfce\ud835\udfce\ud835\udc74\ud835\udc77\ud835\udc82\ud835\uded4\ud835\udc84\ud835\udc90 = \ud835\udfd3\ud835\udfce\ud835\udfce\ud835\udc74\ud835\udc77\ud835\udc82\ud835\uded4\ud835\udc84\ud835\udc90 = \ud835\udfd4\ud835\udfce\ud835\udfce\ud835\udc74\ud835\udc77\ud835\udc82\ud835\uded4\ud835\udc84\ud835\udc90 = \ud835\udfd5\ud835\udfce\ud835\udfce\ud835\udc74\ud835\udc77\ud835\udc82\ud835\uded4\ud835\udc84\ud835\udc90 = \ud835\udfd6\ud835\udfce\ud835\udfce\ud835\udc74\ud835\udc77\ud835\udc82\ud835\uded4\ud835\udc84\ud835\udc90 = \ud835\udfd7\ud835\udfce\ud835\udfce\ud835\udc74\ud835\udc77\ud835\udc82\ud835\uded4\ud835\udc84\ud835\udc90 = \ud835\udfcf\ud835\udfce\ud835\udfce\ud835\udfce\ud835\udc74\ud835\udc77\ud835\udc82\ud835\udc38\ud835\udc50 \ud835\udefe\ud835\udc50\ud835\udc82 (\ud835\udc8e) \ud835\udc6c\ud835\udc95\ud835\udc82\ud835\udc8f\ud835\udc6c\ud835\udc84 \ud835\udf0e\ud835\udc50\ud835\udc5c \ud835\udf0e\ud835\udc50\ud835\udc5c \ud835\udc4e \ud835\udc43\ud835\udeff\ud835\udf03\ud835\udc50 \u226a \ud835\udf03\ud835\udc50\u03b4\ud835\udc50 = \ud835\udf16\ud835\udc50\ud835\udc3f\ud835\udc50 =\ud835\udf0e\ud835\udc50\ud835\udc38\ud835\udc52\ud835\udc53\ud835\udc53\ud835\udc4e\ud835\udc50\ud835\udc5c\ud835\udc60\ud835\udf03\ud835\udc50\ud835\udc38\ud835\udc52\ud835\udc53\ud835\udc53 \ud835\udc38\ud835\udc60\ud835\udc52\ud835\udc50 \ud835\udc38\ud835\udc61\ud835\udc4e\ud835\udc5b\ud835\udeff\ud835\udc63 =\u03b4\ud835\udc50\ud835\udc60\ud835\udc56\ud835\udc5b\ud835\udf03\ud835\udc50=\ud835\udf0e\ud835\udc50\ud835\udc38\ud835\udc52\ud835\udc53\ud835\udc53(1\ud835\udc60\ud835\udc56\ud835\udc5b\ud835\udf03\ud835\udc501\ud835\udc50\ud835\udc5c\ud835\udc60\ud835\udf03\ud835\udc50)\ud835\udc4e\ud835\udc4e\ud835\udf03\ud835\udc50 = 45\u00b0\ud835\udf03\ud835\udc50 = 45\u00b021.5\u00b0 \u2264 \ud835\udf03\ud835\udc50 \u2264 26.5\u00b0\ud835\udf03\ud835\udc50 = 26.5\u00b0\u2018\ud835\udc4e\u2019 0 7.5 15 22.5 30 37.5 45 52.5 60 67.5 75 82.5 9000.511.522.533.544.55\ud835\udeff\ud835\udc63 \ud835\udc4e \u210e \ud835\udc43 Conventional Inclination of Longest Stay in a Cable-Stayed Bridge \ud835\udf3d\ud835\udc84 (\ud835\udc85\ud835\udc86\ud835\udc88\ud835\udc93\ud835\udc86\ud835\udc86\ud835\udc94) \ud835\udf39?\u0302? \ud835\udeff\ud835\udf03\ud835\udc50 \ud835\udf03\ud835\udc50 \ud835\udc56 \ud835\udc66\ud835\udc56 =\ud835\udc3b\ud835\udf14\ud835\udc50\ud835\udc50\ud835\udc5c\ud835\udc60\u210e (\ud835\udf14\ud835\udc50\ud835\udc65\ud835\udc56\ud835\udc3b+ \ud835\udc34\ud835\udc56) + \ud835\udc35\ud835\udc56 \ud835\udc411\u2026\ud835\udc5b \ud835\udc47\ud835\udc5a\ud835\udc4e\ud835\udc65\ud835\udc461\u2026\ud835\udc5b \ud835\udc3b\ud835\udc39\u210e\ud835\udc56 {\ud835\udc65\ud835\udc56 , \ud835\udc66\ud835\udc56}\ud835\udf14\ud835\udc50 \u0394\ud835\udc65\ud835\udc56\ud835\udc53 \u0394y\ud835\udc56\ud835\udc3f\ud835\udc5a\ud835\udc53 \ud835\udc3f\ud835\udc5a2 \ud835\udee5\ud835\udc65\ud835\udc56 \ud835\udee5\ud835\udc66\ud835\udc56 \ud835\udc66\ud835\udc56 \ud835\udc39\u210e\ud835\udc56 \ud835\udc65\ud835\udc56 \ud835\udc3b \ud835\udc47\ud835\udc5a\ud835\udc4e\ud835\udc65 \ud835\udf14\ud835\udc50 \ud835\udf14\ud835\udc50 \ud835\udc3b\ud835\udc51\ud835\udc66\ud835\udc56\ud835\udc51\ud835\udc65\ud835\udc56|\ud835\udc65\ud835\udc56=\ud835\udee5\ud835\udc65\ud835\udc56 \ud835\udc3b \ud835\udc3b\ud835\udc51\ud835\udc66\ud835\udc56\ud835\udc51\ud835\udc65\ud835\udc56|\ud835\udc65\ud835\udc56=0 \ud835\udc3b \ud835\udc39\u210e1 \ud835\udc39\u210e\ud835\udc56\u22121 \ud835\udc39\u210e\ud835\udc56+1 \ud835\udc39\u210e\ud835\udc5b \ud835\udc461 \ud835\udc46\ud835\udc56\u22121 \ud835\udc46\ud835\udc56 \ud835\udc46\ud835\udc56+1 \ud835\udc46\ud835\udc5b \ud835\udc411 \ud835\udc41\ud835\udc56\u22121 \ud835\udc41\ud835\udc56 \ud835\udc41\ud835\udc56+1 \ud835\udc41\ud835\udc5b Free Body Diagram of Segment i \ud835\udc34\ud835\udc56 = \ud835\udc4e\ud835\udc60\ud835\udc56\ud835\udc5b\u210e [\ud835\udf14\ud835\udc50\u0394\ud835\udc66\ud835\udc562\ud835\udc3b\ud835\udc60\ud835\udc56\ud835\udc5b\u210e (\ud835\udf14\ud835\udc50\u0394\ud835\udc65\ud835\udc562\ud835\udc3b )]\u2212\ud835\udf14\ud835\udc50\u0394\ud835\udc65\ud835\udc562\ud835\udc3b\ud835\udc35\ud835\udc56 = \u2212\ud835\udc3b\ud835\udf14\ud835\udc50\ud835\udc50\ud835\udc5c\ud835\udc60\u210e (\ud835\udc34\ud835\udc56) \ud835\udc5b + 1 \u0394\ud835\udc661\u2026\ud835\udc5b \ud835\udc3b\ud835\udc3b \u0394\ud835\udc661\u2026\ud835\udc5b\ud835\udc56 = 12\ud835\udc3b\ud835\udc51\ud835\udc661\ud835\udc51\ud835\udc651|\ud835\udc651=0= \ud835\udc39\u210e1\ud835\udc5f\ud835\udc52\ud835\udc51\ud835\udc62\ud835\udc50\ud835\udc52\ud835\udc60 \ud835\udc61\ud835\udc5c\u2192 \ud835\udc3b\ud835\udc60\ud835\udc56\ud835\udc5b\u210e(\ud835\udc341) =\ud835\udc39\u210e12\ud835\udc56 = 2\u2026\ud835\udc5b \ud835\udc3b\ud835\udc51\ud835\udc66\ud835\udc56\ud835\udc51\ud835\udc65\ud835\udc56|\ud835\udc65\ud835\udc56=0= \ud835\udc3b\ud835\udc51\ud835\udc66\ud835\udc56\u22121\ud835\udc51\ud835\udc65\ud835\udc56\u22121|\ud835\udc65\ud835\udc56\u22121=\u0394\ud835\udc65\ud835\udc56\u22121+ \ud835\udc39\u210e\ud835\udc56\ud835\udc5f\ud835\udc52\ud835\udc51\ud835\udc62\ud835\udc50\ud835\udc52\ud835\udc60 \ud835\udc61\ud835\udc5c\u2192 \ud835\udc3b [\ud835\udc60\ud835\udc56\ud835\udc5b\u210e(\ud835\udc34\ud835\udc56) \u2212 \ud835\udc60\ud835\udc56\ud835\udc5b\u210e (\ud835\udf14\ud835\udc50\u0394\ud835\udc65\ud835\udc56\u22121\ud835\udc3b+ \ud835\udc34\ud835\udc56\u22121)] = \ud835\udc39\u210e\ud835\udc56\u0394\ud835\udc661\u2026\ud835\udc5b \ud835\udc53\ud835\udc3b \ud835\udc53 =\u2211\u0394y\ud835\udc56\ud835\udc5b\ud835\udc56=1\ud835\udc3b \ud835\udf14\ud835\udc50 = \ud835\udefe\ud835\udc50\ud835\udc34\ud835\udc50 \u0394\ud835\udc651\u2026\ud835\udc5b \ud835\udc39\u210e1\u2026\ud835\udc5b \ud835\udc53\ud835\udc47\ud835\udc47\ud835\udc42\ud835\udc3f \ud835\udc3b\ud835\udc34 \u0394\ud835\udc661\u2026\ud835\udc5b \ud835\udc3b\ud835\udc34 \ud835\udc53\ud835\udc34 \ud835\udc3b\ud835\udc34\u0394\ud835\udc661\u2026\ud835\udc5b \ud835\udc53\ud835\udc38 = \ud835\udc53\ud835\udc47 \u2212 \ud835\udc53\ud835\udc34 \ud835\udc53\ud835\udc38 > \ud835\udc47\ud835\udc42\ud835\udc3f \ud835\udc53\ud835\udc38 \u2264 \ud835\udc47\ud835\udc42\ud835\udc3f \ud835\udc51\ud835\udc3b\ud835\udc34\ud835\udc35\ud835\udc51\ud835\udc53\ud835\udc34\ud835\udc35= (\ud835\udc3b\ud835\udc34\u2212\ud835\udc3b\ud835\udc35\ud835\udc53\ud835\udc34\u2212\ud835\udc53\ud835\udc35) \ud835\udc3b\ud835\udc35 = (1 \u2212 \ud835\udc47\ud835\udc42\ud835\udc3f)\ud835\udc3b\ud835\udc34 \u0394\ud835\udc661\u2026\ud835\udc5b \ud835\udc3b\ud835\udc35 \ud835\udc53\ud835\udc35 \ud835\udc3b\ud835\udc35\u0394\ud835\udc661\u2026\ud835\udc5b \ud835\udc3b\ud835\udc34 \ud835\udc41\ud835\udc38\ud835\udc4a = \ud835\udc3b\ud835\udc34 \ud835\udc42\ud835\udc3f\ud835\udc37 + \ud835\udc53\ud835\udc38 (\ud835\udc51\ud835\udc3b\ud835\udc34\ud835\udc35\ud835\udc51\ud835\udc53\ud835\udc34\ud835\udc35) \ud835\udc3b\ud835\udc34\u0394\ud835\udc661\u2026\ud835\udc5b \uf0a7 \uf0a7 \u0394\ud835\udc651\u2026\ud835\udc5b \ud835\udc39\u210e1\u2026\ud835\udc5b\ud835\udc39\u210e1 \u0394\ud835\udc651\uf0a7 \ud835\udc3b\ud835\udc3b\ud835\udc34\uf0a7 \ud835\udc3b\ud835\udc34 \ud835\udc41\ud835\udc38\ud835\udc4a\ud835\udc3b\ud835\udc34 \ud835\udc41\ud835\udc38\ud835\udc4a \u2265 0.5\ud835\udc3b\ud835\udc34 \ud835\udc42\ud835\udc3f\ud835\udc37\uf0a7 \ud835\udc48\ud835\udc46\ud835\udc3f = 2 [\u2211\ud835\udc36\ud835\udc56 \u2212 \u0394\ud835\udc56\ud835\udc5b\ud835\udc56=1] \ud835\udc36\ud835\udc56 =\ud835\udc3b\ud835\udf14\ud835\udc50[\ud835\udc60\ud835\udc56\ud835\udc5b\u210e (\ud835\udf14\ud835\udc50\u0394\ud835\udc65\ud835\udc56\ud835\udc3b+ \ud835\udc34\ud835\udc56) \u2212 \ud835\udc60\ud835\udc56\ud835\udc5b\u210e (\ud835\udc34\ud835\udc56)] \u0394\ud835\udc52\ud835\udc56 =\ud835\udc3b\u0394\ud835\udc65\ud835\udc56\ud835\udc38\ud835\udc50\ud835\udc34\ud835\udc50[\ud835\udf14\ud835\udc50\u0394\ud835\udc66\ud835\udc5622\ud835\udc3b\u0394\ud835\udc65\ud835\udc56\ud835\udc50\ud835\udc5c\ud835\udc61\u210e (\ud835\udf14\ud835\udc50\u0394\ud835\udc65\ud835\udc562\ud835\udc3b) +12+\ud835\udc3b2\ud835\udf14\ud835\udc50\u0394\ud835\udc65\ud835\udc56\ud835\udc60\ud835\udc56\ud835\udc5b\u210e (\ud835\udf14\ud835\udc50\u0394\ud835\udc65\ud835\udc56\ud835\udc3b)]\uf0a7 \ud835\udc46\ud835\udc5a,\ud835\udc66(\ud835\udc65)|\ud835\udc46\ud835\udc5a =1\ud835\udc3b[\u2211 \ud835\udc39\u210e\ud835\udc57 (\ud835\udc65 \u2212\u2211\ud835\udf06\ud835\udc56\ud835\udc57\ud835\udc56=1)+\ud835\udf14\ud835\udc50\ud835\udc6522\u2212 \ud835\udc49\ud835\udc60\ud835\udc65\ud835\udc5a\u22121\ud835\udc57=1]\ud835\udc65 \ud835\udc66\ud835\udc49\ud835\udc60 =12\u2211 \ud835\udc39\u210e\ud835\udc56 +\ud835\udf14\ud835\udc50\ud835\udc3f\ud835\udc5a2\ud835\udc41\u210e\ud835\udc4e\ud835\udc5b\ud835\udc54\ud835\udc52\ud835\udc5f\ud835\udc60\ud835\udc56=1\ud835\udc3b =1\ud835\udc53[\ud835\udc49\ud835\udc60\ud835\udc3f\ud835\udc5a2\u2212 \ud835\udf14\ud835\udc50\ud835\udc3f\ud835\udc5a28\u2212\u2211\ud835\udc39\u210e\ud835\udc57 (\ud835\udc3f\ud835\udc5a2\u2212\u2211\ud835\udf06\ud835\udc56\ud835\udc57\ud835\udc56=1)\ud835\udc41\u2217\ud835\udc57=1]\ud835\udc41\u210e\ud835\udc4e\ud835\udc5b\ud835\udc54\ud835\udc52\ud835\udc5f\ud835\udc60\u221212, \ud835\udc41\u2217 =\ud835\udc41\u210e\ud835\udc4e\ud835\udc5b\ud835\udc54\ud835\udc52\ud835\udc5f\ud835\udc602\uf0a7 \uf0a7 \uf0a7 \ud835\udf14\ud835\udc50\ud835\udf06 \ud835\udf14\ud835\udc60 \ud835\udc411\u2026 \ud835\udc3f\ud835\udc5a\ud835\udc461\u2026 \ud835\udc3b\ud835\udc41\u210e\ud835\udc4e\ud835\udc5b\ud835\udc54\ud835\udc52\ud835\udc5f\ud835\udc60 \ud835\udc49\ud835\udc60\ud835\udf14\ud835\udc50 \u03bb\ud835\udc5a\ud835\udc53 \ud835\udc39\u210e\ud835\udc5a\ud835\udc66 \ud835\udc39\u210e\ud835\udc5a+1 \ud835\udc65 \ud835\udc3b \ud835\udc39\u210e1 \ud835\udc39\u210e\ud835\udc5a \ud835\udc39\u210e\ud835\udc5a\u22122 \ud835\udc49\ud835\udc60 \ud835\udc39\u210e\ud835\udc5a\u22121 \ud835\udf061 \ud835\udf06\ud835\udc5a\u22122 \ud835\udf06\ud835\udc5a\u22121 \ud835\udf06\ud835\udc5a \ud835\udf06\ud835\udc5a+1 \ud835\udc411 \ud835\udc41\ud835\udc5a\u22122 \ud835\udc41\ud835\udc5a\u22121 \ud835\udc41\ud835\udc5a \ud835\udc41\ud835\udc5a+1 \ud835\udc3b \ud835\udc3f\ud835\udc5a2 \ud835\udc39\u210e2 \ud835\udf062 \ud835\udc53 \ud835\udc412 \ud835\udc461 \ud835\udc462 \ud835\udc46\ud835\udc5a\u22122 \ud835\udc46\ud835\udc5a\u22121 \ud835\udc46\ud835\udc5a \ud835\udc46\ud835\udc5a+1 (\ud835\udf14\ud835\udc50+\ud835\udf14\ud835\udc60)\ud835\udc3f\ud835\udc5a28\ud835\udc53, \ud835\udc3b =(\ud835\udf14\ud835\udc50+\ud835\udf14\ud835\udc60)\ud835\udc3f\ud835\udc5a28\ud835\udc53\u2212\ud835\udf14\ud835\udc60\ud835\udf0628\ud835\udc53,\ud835\udc66(\ud835\udc65) =(\ud835\udf14\ud835\udc50 +\ud835\udf14\ud835\udc60)\ud835\udc652\ud835\udc3b[\ud835\udc65 \u2212 \ud835\udc3f\ud835\udc5a]\ud835\udc65\ud835\udc36\ud835\udc5a = \ud835\udf06\ud835\udc5a\u221a1+ \u03a9\ud835\udc5a2 [1 +124\u03a8\ud835\udc50\ud835\udc5a2(1 + \u03a9\ud835\udc5a2)]\u2206\ud835\udc52\ud835\udc5a =\ud835\udc3b\ud835\udf06\ud835\udc5a\ud835\udc38\ud835\udc50\ud835\udc34\ud835\udc50[1 + \u03a9\ud835\udc5a2 +112\u03a8\ud835\udc50\ud835\udc5a2(1 + \u03a9\ud835\udc5a2)]\u03a8\ud835\udc50\ud835\udc5a =\ud835\udf14\ud835\udc50\ud835\udf06\ud835\udc5a\ud835\udc3b\u03a9\ud835\udc5a =\u210e\ud835\udc5a\ud835\udf06\ud835\udc5a \ud835\udf06\ud835\udc5a\ud835\udf06 \ud835\udc3b\u210e\ud835\udc5a\u210e\ud835\udc5a = \ud835\udc66\ud835\udc5a \u2212 \ud835\udc66\ud835\udc5a\u22121\ud835\udc66\ud835\udc5a \ud835\udc5a\ud835\udc66\ud835\udc5a \ud835\udc3b\ud835\udc48\ud835\udc46\ud835\udc3f\ud835\udc47\ud835\udc5c\ud835\udc61\ud835\udc4e\ud835\udc59 = \u2211 (\ud835\udc48\ud835\udc46\ud835\udc3f\ud835\udc5a)\ud835\udc41\u210e\ud835\udc4e\ud835\udc5b\ud835\udc54\ud835\udc52\ud835\udc5f\ud835\udc60+1\ud835\udc5a=1 = \u2211 (\ud835\udc36\ud835\udc5a \u2212 \u0394\ud835\udc52\ud835\udc5a)\ud835\udc41\u210e\ud835\udc4e\ud835\udc5b\ud835\udc54\ud835\udc52\ud835\udc5f\ud835\udc60+1\ud835\udc5a=1\ud835\udc65\ud835\udc66\ud835\udc50 \ud835\udc66\ud835\udc5d\u03bb \ud835\udc3f\ud835\udc5a \ud835\udf14\ud835\udc50 \ud835\udf14\ud835\udc60\ud835\udf14\ud835\udc5d\ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64 0 0.1 0.2 0.3 0.4 0.505 \uf02e1051 \uf02e1041.5 \uf02e1042 \uf02e104\ud835\udf4e\ud835\udc84\ud835\udf4e\ud835\udc94\u2044 (\ud835\udc9a\ud835\udc91 \u2212 \ud835\udc9a\ud835\udc84)\ud835\udc8e\ud835\udc82\ud835\udc99\ud835\udc73\ud835\udc8e \ud835\udc53 =\ud835\udc3f\ud835\udc5a10\ud835\udc48\ud835\udc46\ud835\udc3f\ud835\udc50\ud835\udc48\ud835\udc46\ud835\udc3f\ud835\udc5d \ud835\udefe\ud835\udc50 \ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60250 500 750 1000 1250 1500 1750 200005 \uf02e1051 \uf02e1041.5 \uf02e1042 \uf02e104(\ud835\udc9a\ud835\udc91 \u2212 \ud835\udc9a\ud835\udc84)\ud835\udc8e\ud835\udc82\ud835\udc99\ud835\udc73\ud835\udc8e \ud835\udc73\ud835\udc8e(\ud835\udc8e) \ud835\udc53 =\ud835\udc3f\ud835\udc5a10\ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60\ud835\udc36 = \u222b \u221a[1 + (\ud835\udc51\ud835\udc66\ud835\udc51\ud835\udc65)2] \ud835\udc51\ud835\udc65\ud835\udc3f\ud835\udc5a0=\ud835\udc3f\ud835\udc5a2\u03a8\u0305\ud835\udc5d[\u03a8\u0305\ud835\udc5d\u221a1+\u03a8\u0305\ud835\udc5d24+ 2\ud835\udc60\ud835\udc56\ud835\udc5b\u210e (\u03a8\u0305\ud835\udc5d2)]\ud835\udefe\ud835\udc50 \ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udc38\ud835\udc50\ud835\udc73\ud835\udc8e (\ud835\udc8e) (\ud835\udc7c\ud835\udc7a\ud835\udc73\ud835\udc84 \u2212 \ud835\udc7c\ud835\udc7a\ud835\udc73\ud835\udc91)\ud835\udc73\ud835\udc8e \ud835\udc53 =\ud835\udc3f\ud835\udc5a10\u0394\ud835\udc52 =\ud835\udc3b\ud835\udc38\ud835\udc50\ud835\udc34\ud835\udc50\u222b [1 + (\ud835\udc51\ud835\udc66\ud835\udc51\ud835\udc65)2] \ud835\udc51\ud835\udc65\ud835\udc3f\ud835\udc5a0=\ud835\udc3b\ud835\udc3f\ud835\udc5a\ud835\udc38\ud835\udc50\ud835\udc34\ud835\udc50[1 +112\u03a8\u0305\ud835\udc5d2]\u03a8\u0305\ud835\udc5d =(\ud835\udf14\ud835\udc50 +\ud835\udf14\ud835\udc60)\ud835\udc3f\ud835\udc5a\ud835\udc3b= 8(\ud835\udc53\ud835\udc3f\ud835\udc5a)\uf0a7 \ud835\udf14\ud835\udc50 , \ud835\udf14\ud835\udc60\uf0a7 \uf0a7 \ud835\udf14\ud835\udc5d\uf0a7 \ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64\u03c9\ud835\udc50 =(\u03c9\ud835\udc60 +\u03c9\ud835\udc5d) \ud835\udf09 \u221a1 + 16 \ud835\udc46\ud835\udc4521 \u2212 \ud835\udf09 \u221a1 + 16 \ud835\udc46\ud835\udc452\ud835\udc46\ud835\udc45 \ud835\udc53: \ud835\udc3f\ud835\udc5a\ud835\udf09\ud835\udf09 =\ud835\udefe\ud835\udc50 \ud835\udc3f\ud835\udc5a8 \ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64 \ud835\udc46\ud835\udc45\ud835\udf14\ud835\udc60 \ud835\udf14\ud835\udc5d\ud835\udf14\ud835\udc5d:\ud835\udf14\ud835\udc60 \ud835\udefe\ud835\udc50 \ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc640 250 500 750 1000 1250 1500 1750 200000.050.10.150.20.250.30.350.40.450.5\ud835\udc73\ud835\udc8e (\ud835\udc8e) \ud835\udf4e\ud835\udc84\ud835\udf4e\ud835\udc94 +\ud835\udf4e\ud835\udc91 \ud835\udc46\ud835\udc45 = 0.20 \ud835\udc46\ud835\udc45 = 0.15 \ud835\udc46\ud835\udc45 = 0.10 \ud835\udf14\ud835\udc5d:\ud835\udf14\ud835\udc60 = 0.6\ud835\udc65\ud835\udc5d \ud835\udc3f\ud835\udc5d\ud835\udc3f\ud835\udc5d2 \u2264 \ud835\udc65\ud835\udc5d \u2264 \ud835\udc3f\ud835\udc5a \u2212\ud835\udc3f\ud835\udc5d2 \ud835\udc3f\ud835\udc5a \ud835\udefe\ud835\udc50 \ud835\udc38\ud835\udc50 \ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udc46\ud835\udc45\ud835\udf14\ud835\udc60 \ud835\udf14\ud835\udc5d \ud835\udc65\ud835\udc5d \ud835\udc3f\ud835\udc5d \ud835\udeff\ud835\udc5a\ud835\udc4e\ud835\udc65 \ud835\udc3f\ud835\udc5a \ud835\udc73\ud835\udc91\ud835\udc73\ud835\udc8e\u2044 \ud835\udc99\ud835\udc91\ud835\udc73\ud835\udc8e\u2044 \ud835\udf39\ud835\udc8e\ud835\udc82\ud835\udc99\ud835\udc73\ud835\udc8e \ud835\udc65\ud835\udc5d \ud835\udc3f\ud835\udc5a\u2044 = 0.21 & 0.79\ud835\udc3f\ud835\udc5d \ud835\udc3f\ud835\udc5a\u2044 = 0.4\ud835\udc65\ud835\udc5d \ud835\udc3f\ud835\udc5a\u2044 = 0.5\ud835\udc3f\ud835\udc5d \ud835\udc3f\ud835\udc5a\u2044 = 0.38 \ud835\udc3f\ud835\udc5a \ud835\udefe\ud835\udc50 \ud835\udc38\ud835\udc50 \ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udc46\ud835\udc45\ud835\udc73\ud835\udc91\ud835\udc73\ud835\udc8e\u2044 \ud835\udc99\ud835\udc91\ud835\udc73\ud835\udc8e\u2044 \ud835\udc65\ud835\udc5d \ud835\udc3f\ud835\udc5d \ud835\udeff\ud835\udc5a\ud835\udc4e\ud835\udc65 \ud835\udc3f\ud835\udc5a \ud835\udf14\ud835\udc60 \ud835\udf14\ud835\udc5d \ud835\udf39\ud835\udc8e\ud835\udc82\ud835\udc99\ud835\udc73\ud835\udc8e \u03b4\ud835\udc60\ud835\udc53 \ud835\udc3f\ud835\udc5a \ud835\udefe\ud835\udc50 \ud835\udc38\ud835\udc50 \ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udc46\ud835\udc45\ud835\udf14\ud835\udc60 \ud835\udf14\ud835\udc5d \ud835\udc65\ud835\udc5d \ud835\udc3f\ud835\udc5d \ud835\udeff\ud835\udc5a\ud835\udc4e\ud835\udc65 \ud835\udc3f\ud835\udc5a \ud835\udc73\ud835\udc91\ud835\udc73\ud835\udc8e\u2044 \ud835\udc99\ud835\udc91\ud835\udc73\ud835\udc8e\u2044 \ud835\udf39\ud835\udc94\ud835\udc87\ud835\udf39\ud835\udc8e\ud835\udc82\ud835\udc99 (%) \ud835\udeff\ud835\udc60\ud835\udc53 \ud835\udc38\ud835\udc50\ud835\udc34\ud835\udc50 \ud835\udc38\ud835\udc50\ud835\udc34\ud835\udc50 = \u221e \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udc3f\ud835\udc5a \ud835\udefe\ud835\udc50 \ud835\udc38\ud835\udc50 \ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udc46\ud835\udc45\ud835\udf4e\ud835\udc94 \ud835\udf14\ud835\udc5d \ud835\udc65\ud835\udc5d \ud835\udc3f\ud835\udc5d \ud835\udeff\ud835\udc5a\ud835\udc4e\ud835\udc65 \ud835\udc3f\ud835\udc5a \ud835\udc73\ud835\udc91\ud835\udc73\ud835\udc8e\u2044 \ud835\udc99\ud835\udc91\ud835\udc73\ud835\udc8e\u2044 \ud835\udf39\ud835\udc8e\ud835\udc82\ud835\udc99\ud835\udc73\ud835\udc8e \ud835\udfd0\ud835\udf4e\ud835\udc94 \ud835\udf14\ud835\udc5d \ud835\udc65\ud835\udc5d \ud835\udc3f\ud835\udc5d \ud835\udeff\ud835\udc5a\ud835\udc4e\ud835\udc65 \ud835\udc3f\ud835\udc5a \uf0a7 \uf0a7 \uf0a7 Suspension Cables Superstructure Hangers Transitory Load 0 100 200 300 400 500 6001020304050607080\ud835\udc7a\ud835\udc91\ud835\udc82\ud835\udc8f\u2212 \ud835\udc95\ud835\udc90 \u2212 \ud835\udc6b\ud835\udc86\ud835\udc91\ud835\udc95\ud835\udc89 \ud835\udc79\ud835\udc82\ud835\udc95\ud835\udc8a\ud835\udc90 \ud835\udc7a\ud835\udc91\ud835\udc82\ud835\udc8f\u2212\ud835\udc95\ud835\udc90\u2212\ud835\udc7e\ud835\udc8a\ud835\udc85\ud835\udc95\ud835\udc89 \ud835\udc79\ud835\udc82\ud835\udc95\ud835\udc8a\ud835\udc90 \uf076 \uf076 \uf076 \ud835\udc99\ud835\udc91 \ud835\udc73\ud835\udc8e\u2044 = \ud835\udfce. \ud835\udfd6, \ud835\udc73\ud835\udc91 \ud835\udc73\ud835\udc8e\u2044 = \ud835\udfce. \ud835\udfd2 \ud835\udc99\ud835\udc91 \ud835\udc73\ud835\udc8e\u2044 = \ud835\udfce. \ud835\udfd3, \ud835\udc73\ud835\udc91 \ud835\udc73\ud835\udc8e\u2044 = \ud835\udfce. \ud835\udfd2 \ud835\udc99\ud835\udc91 \ud835\udc73\ud835\udc8e\u2044 = \ud835\udfce. \ud835\udfd3, \ud835\udc73\ud835\udc91 \ud835\udc73\ud835\udc8e\u2044 = \ud835\udfcf. \ud835\udfce \ud835\udefe\ud835\udc50 \ud835\udc38\ud835\udc50 \ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udc46\ud835\udc45250 500 750 100000.0010.0020.0030.0040.0050.0060.007\ud835\udf39\ud835\udc8e\ud835\udc82\ud835\udc99\ud835\udc73\ud835\udc8e \ud835\udc73\ud835\udc8e (\ud835\udc8e) LS3\ud835\udeff\ud835\udc5a\ud835\udc4e\ud835\udc65 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udeff\ud835\udc5a\ud835\udc4e\ud835\udc65 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udeff\ud835\udc5a\ud835\udc4e\ud835\udc65 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 LS2LS1\ud835\udefc\ud835\udc5d \ud835\udefc\ud835\udc5d \ud835\udc99\ud835\udc91 \ud835\udc73\ud835\udc8e\u2044 = \ud835\udfce. \ud835\udfd6, \ud835\udc73\ud835\udc91 \ud835\udc73\ud835\udc8e\u2044 = \ud835\udfce. \ud835\udfd2 \ud835\udc99\ud835\udc91 \ud835\udc73\ud835\udc8e\u2044 = \ud835\udfce. \ud835\udfd3, \ud835\udc73\ud835\udc91 \ud835\udc73\ud835\udc8e\u2044 = \ud835\udfce. \ud835\udfd2 \ud835\udc99\ud835\udc91 \ud835\udc73\ud835\udc8e\u2044 = \ud835\udfce. \ud835\udfd3, \ud835\udc73\ud835\udc91 \ud835\udc73\ud835\udc8e\u2044 = \ud835\udfcf. \ud835\udfce \ud835\udc3f\ud835\udc5a \ud835\udefe\ud835\udc50 \ud835\udc38\ud835\udc50 \ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64 \ud835\udc46\ud835\udc45 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60\ud835\udefc\ud835\udc5d = 10 0.002 0.004 0.006 0.008 0.01 0.012 0.01400.250.50.7511.251.51.752\ud835\udf36\ud835\udc91 \ud835\udf39\ud835\udc8e\ud835\udc82\ud835\udc99\ud835\udc73\ud835\udc8e \ud835\udeff\ud835\udc5a\ud835\udc4e\ud835\udc65 \ud835\udefc\ud835\udc5d\ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udeff\ud835\udc5a\ud835\udc4e\ud835\udc65 \ud835\udefc\ud835\udc5d\ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udeff\ud835\udc5a\ud835\udc4e\ud835\udc65 \ud835\udefc\ud835\udc5d\ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 LS1 LS2 LS3\ud835\udeff\ud835\udc5a\ud835\udc4e\ud835\udc65 SO (\ud835\udc3f\ud835\udc5a = 250\ud835\udc5a) (\ud835\udc3f\ud835\udc5a = 500\ud835\udc5a) (\ud835\udc3f\ud835\udc5a = 750\ud835\udc5a) (\ud835\udc3f\ud835\udc5a = 1000\ud835\udc5a)\ud835\udefe\ud835\udc50 \ud835\udc38\ud835\udc50 \ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64 \ud835\udc46\ud835\udc45 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc600 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 1.75 211.11.21.31.41.51.6\ud835\udf39\ud835\udc8e\ud835\udc82\ud835\udc99 \ud835\udc7a\ud835\udc76\ud835\udf39\ud835\udc8e\ud835\udc82\ud835\udc99 \ud835\udf39\ud835\udc8e\ud835\udc82\ud835\udc99 \ud835\udc7a\ud835\udc76\ud835\udf39\ud835\udc8e\ud835\udc82\ud835\udc99 \ud835\udf36\ud835\udc91 \ud835\udf36\ud835\udc91 \ud835\udf36\ud835\udc91 \ud835\udf36\ud835\udc91 LS1LS2LS3LS1LS2LS3LS1LS2LS3LS1LS2LS3\ud835\udf14\ud835\udc5d: \ud835\udf14\ud835\udc50\ud835\udf14\ud835\udc5d: (\ud835\udf14\ud835\udc50 +\ud835\udf14\ud835\udc60 +\ud835\udf14\ud835\udc5d)\ud835\udf14\ud835\udc50 = \ud835\udf14\ud835\udc60 (1 + \ud835\udf14\ud835\udc45) \u00b7 \ud835\udeeb\u03c9\ud835\udc45 \ud835\udf14\ud835\udc5d: \ud835\udf14\ud835\udc60\u039a\ud835\udeeb = \ud835\udf09 \u221a1 + 16 \ud835\udc46\ud835\udc4521 \u2212 \ud835\udf09 \u221a1 + 16 \ud835\udc46\ud835\udc452\ud835\udf09 =\ud835\udefe\ud835\udc50 \ud835\udc3f\ud835\udc5a8 \ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64 \ud835\udc46\ud835\udc45\ud835\udf14\ud835\udc5d\ud835\udf14\ud835\udc5d\ud835\udf14\ud835\udc50=\ud835\udf14\ud835\udc45 (1 + \ud835\udf14\ud835\udc45) \u00b7 \ud835\udeeb\ud835\udeff\ud835\udc522\ud835\udeff\ud835\udc521\u2245 (\ud835\udf14\ud835\udc452 1 + \ud835\udf14\ud835\udc452) (1 + \ud835\udf14\ud835\udc451 \ud835\udf14\ud835\udc451)\ud835\udeff\ud835\udc521 \ud835\udeff\ud835\udc522\ud835\udf14\ud835\udc5d\ud835\udf14\ud835\udc50 +\ud835\udf14\ud835\udc60 +\ud835\udf14\ud835\udc5d=\ud835\udf14\ud835\udc45 (1 + \ud835\udf14\ud835\udc45)(\ud835\udeeb + 1)\ud835\udeff\ud835\udc60\ud835\udc532\ud835\udeff\ud835\udc60\ud835\udc531\u2245 (\ud835\udf14\ud835\udc452 1 + \ud835\udf14\ud835\udc452) (1 + \ud835\udf14\ud835\udc451 \ud835\udf14\ud835\udc451)\ud835\udeff\ud835\udc5a\ud835\udc4e\ud835\udc652\ud835\udeff\ud835\udc5a\ud835\udc4e\ud835\udc651=\ud835\udeff\ud835\udc60\ud835\udc532 + \ud835\udeff\ud835\udc522\ud835\udeff\ud835\udc60\ud835\udc531 + \ud835\udeff\ud835\udc521\ud835\udeff\ud835\udc5a\ud835\udc4e\ud835\udc652\ud835\udeff\ud835\udc5a\ud835\udc4e\ud835\udc651\u2245\ud835\udeff\ud835\udc60\ud835\udc532\ud835\udeff\ud835\udc60\ud835\udc531\u2245\ud835\udeff\ud835\udc522\ud835\udeff\ud835\udc521\u2245 (\ud835\udf14\ud835\udc452 1 + \ud835\udf14\ud835\udc452) (1 + \ud835\udf14\ud835\udc451 \ud835\udf14\ud835\udc451)\ud835\udeff\ud835\udc5a\ud835\udc4e\ud835\udc65 \ud835\udeff\ud835\udc60\ud835\udc53\ud835\udc65\ud835\udc5d\ud835\udc3f\ud835\udc5d\ud835\udc65\ud835\udc5d \ud835\udc3f\ud835\udc5a \ud835\udc3f\ud835\udc5d \ud835\udc3f\ud835\udc5a\ud835\udefc\ud835\udc5d = 1\ud835\udeda\ud835\udc79\ud835\udfd0 = \ud835\udfce.\ud835\udfd0 \ud835\udeda\ud835\udc79\ud835\udfd0 = \ud835\udfce.\ud835\udfd1 \ud835\udeda\ud835\udc79\ud835\udfd0 = \ud835\udfce.\ud835\udfd2 \ud835\udeda\ud835\udc79\ud835\udfd0 = \ud835\udfce. \ud835\udfd3 \ud835\udeda\ud835\udc79\ud835\udfd0 = \ud835\udfce. \ud835\udfd4 \ud835\udc3f\ud835\udc5a \ud835\udefe\ud835\udc50 \ud835\udc38\ud835\udc50 \ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64 \ud835\udc46\ud835\udc450.2 0.3 0.4 0.5 0.600.250.50.7511.251.51.7522.252.50.2 0.3 0.4 0.5 0.600.250.50.7511.251.51.7522.252.5\ud835\udf4e\ud835\udc79\ud835\udfcf \ud835\udf4e\ud835\udc79\ud835\udfcf \ud835\udf39\ud835\udc86\ud835\udfd0\ud835\udf39\ud835\udc86\ud835\udfcf \ud835\udf39\ud835\udc94\ud835\udc87\ud835\udfd0\ud835\udf39\ud835\udc94\ud835\udc87\ud835\udfcf (\ud835\udc3f\ud835\udc5a = 250\ud835\udc5a) (\ud835\udc3f\ud835\udc5a = 1000\ud835\udc5a)\ud835\udefe\ud835\udc50 \ud835\udc38\ud835\udc50 \ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64 \ud835\udc46\ud835\udc450 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 0.611.051.11.151.21.251.3\ud835\udf39\ud835\udc8e\ud835\udc82\ud835\udc99 \ud835\udc7a\ud835\udc76\ud835\udf39\ud835\udc8e\ud835\udc82\ud835\udc99 \ud835\udf4e\ud835\udc79 \ud835\udf4e\ud835\udc79 LS1LS2LS3LS1LS2LS3 \ud835\udc99\ud835\udc91 \ud835\udc73\ud835\udc8e\u2044 = \ud835\udfce. \ud835\udfd6, \ud835\udc73\ud835\udc91 \ud835\udc73\ud835\udc8e\u2044 = \ud835\udfce. \ud835\udfd2 \ud835\udc99\ud835\udc91 \ud835\udc73\ud835\udc8e\u2044 = \ud835\udfce. \ud835\udfd3, \ud835\udc73\ud835\udc91 \ud835\udc73\ud835\udc8e\u2044 = \ud835\udfce. \ud835\udfd2 \ud835\udc99\ud835\udc91 \ud835\udc73\ud835\udc8e\u2044 = \ud835\udfce. \ud835\udfd3, \ud835\udc73\ud835\udc91 \ud835\udc73\ud835\udc8e\u2044 = \ud835\udfcf. \ud835\udfce \ud835\udc3f\ud835\udc5a \ud835\udefe\ud835\udc50 \ud835\udc38\ud835\udc50 \ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60\ud835\udc36\ud835\udc4e\ud835\udc5d\ud835\udc5d\ud835\udc5f\ud835\udc5c\ud835\udc65 = \ud835\udc3f\ud835\udc5a [1 +83(\ud835\udc53\ud835\udc3f\ud835\udc5a)2\u2212325(\ud835\udc53\ud835\udc3f\ud835\udc5a)4+\u22ef ]\ud835\udc530.06 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 \ud835\udf39\ud835\udc8e\ud835\udc82\ud835\udc99\ud835\udc73\ud835\udc8e \ud835\udc7a\ud835\udc79 \ud835\udeff\ud835\udc5a\ud835\udc4e\ud835\udc65 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udf14\ud835\udc60 \ud835\udeff\ud835\udc5a\ud835\udc4e\ud835\udc65 \ud835\udf14\ud835\udc5d \ud835\udeff\ud835\udc5a\ud835\udc4e\ud835\udc65 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 LS1LS3LS2\ud835\udeff\ud835\udc53 = [1516 \ud835\udc46\ud835\udc45 (5 \u2212 24\ud835\udc46\ud835\udc452)]\u03b4\ud835\udc36\ud835\udc4e\ud835\udc5d\ud835\udc5d\ud835\udc5f\ud835\udc5c\ud835\udc65\ud835\udeff\ud835\udc36\ud835\udeff\ud835\udc53 \u0394\ud835\udc521 = \u0394\ud835\udc522 = 0.0325 \u00b7 \ud835\udc3f\ud835\udc5a1 (\ud835\udc41\ud835\udc5c\ud835\udc61\ud835\udc52: \ud835\udc47\u210e\ud835\udc52 \ud835\udc4e\ud835\udc5a\ud835\udc5c\ud835\udc62\ud835\udc5b\ud835\udc61 \ud835\udc5c\ud835\udc53 \ud835\udc52\ud835\udc59\ud835\udc4e\ud835\udc60\ud835\udc61\ud835\udc56\ud835\udc50 \ud835\udc52\ud835\udc59\ud835\udc5c\ud835\udc5b\ud835\udc54\ud835\udc4e\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\udc56\ud835\udc60 \ud835\udc61\u210e\ud835\udc52 \ud835\udc60\ud835\udc4e\ud835\udc5a\ud835\udc52 \ud835\udc56\ud835\udc5b \ud835\udc4f\ud835\udc5c\ud835\udc61\u210e \ud835\udc50\ud835\udc4e\ud835\udc4f\ud835\udc59\ud835\udc52\ud835\udc60)\ud835\udeff\ud835\udc5a\ud835\udc4e\ud835\udc652 > \ud835\udeff\ud835\udc5a\ud835\udc4e\ud835\udc651 \ud835\udeff\ud835\udc5a\ud835\udc4e\ud835\udc651 \ud835\udc7a\ud835\udc79\ud835\udfcf = \ud835\udfce.\ud835\udfcf \ud835\udc7a\ud835\udc79\ud835\udfd0 = \ud835\udfce. \ud835\udfce\ud835\udfd3 \ud835\udc37\ud835\udc52\ud835\udc53\ud835\udc5c\ud835\udc5f\ud835\udc5a\ud835\udc52\ud835\udc51 \ud835\udc43\ud835\udc5c\ud835\udc60\ud835\udc56\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\udc3f\ud835\udc5a1 \ud835\udc3f\ud835\udc5a2 = \ud835\udc3f\ud835\udc5a1 \ud835\udc3f\ud835\udc5a \ud835\udc38\ud835\udc34\ud835\udc54 \ud835\udc38\ud835\udc3c\ud835\udc54 \ud835\udc53 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udf14\ud835\udc5d\ud835\udc53 \ud835\udc38\ud835\udc34\ud835\udc54, \ud835\udc38\ud835\udc3c\ud835\udc54 \ud835\udc3f\ud835\udc5a \ud835\udc65\ud835\udc5d \ud835\udc3f\ud835\udc5d \ud835\udc45\ud835\udc5c\ud835\udc59\ud835\udc59\ud835\udc52\ud835\udc5f \ud835\udc46\ud835\udc62\ud835\udc5d\ud835\udc5d\ud835\udc5c\ud835\udc5f\ud835\udc61 (TYP) \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udc3f\ud835\udc5a 50\u2044 (\ud835\udc47\ud835\udc4c\ud835\udc43) \ud835\udc45\ud835\udc56\ud835\udc54\ud835\udc56\ud835\udc51 \ud835\udc47\ud835\udc5f\ud835\udc62\ud835\udc60\ud835\udc60 \ud835\udc38\ud835\udc59\ud835\udc52\ud835\udc5a\ud835\udc52\ud835\udc5b\ud835\udc61\ud835\udc60 \ud835\udc3f\ud835\udc5a \ud835\udc38\ud835\udc34\ud835\udc54,\ud835\udc38\ud835\udc3c\ud835\udc54 \ud835\udc53 \ud835\udc45\ud835\udc56\ud835\udc54\ud835\udc56\ud835\udc51 \ud835\udc47\ud835\udc5f\ud835\udc62\ud835\udc60\ud835\udc60 \ud835\udc38\ud835\udc59\ud835\udc52\ud835\udc5a\ud835\udc52\ud835\udc5b\ud835\udc61\ud835\udc60 \ud835\udc3f\ud835\udc5d \ud835\udc65\ud835\udc5d \ud835\udc43\ud835\udc56\ud835\udc5b \ud835\udc46\ud835\udc62\ud835\udc5d\ud835\udc5d\ud835\udc5c\ud835\udc5f\ud835\udc61 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udc3f\ud835\udc5d \ud835\udc3f\ud835\udc5d\ud835\udc3f\ud835\udc5a\ud835\udc38\ud835\udc4e\ud835\udc5f\ud835\udc61\u210e \u2212 \ud835\udc34\ud835\udc5b\ud835\udc50\u210e\ud835\udc5c\ud835\udc5f\ud835\udc52\ud835\udc51 \ud835\udc46\ud835\udc66\ud835\udc60\ud835\udc61\ud835\udc52\ud835\udc5a \ud835\udc46\ud835\udc52\ud835\udc59\ud835\udc53 \u2212 \ud835\udc34\ud835\udc5b\ud835\udc50\u210e\ud835\udc5c\ud835\udc5f\ud835\udc52\ud835\udc51 \ud835\udc46\ud835\udc66\ud835\udc60\ud835\udc61\ud835\udc52\ud835\udc5a 0 2 4 6012340 2 4 601234trace 1111trace 2trace 30 2 4 602550751000 2 4 60255075100trace 1111trace 2trace 3\ud835\udc6c\ud835\udc70\ud835\udc88 (\ud835\udc74\ud835\udc75\ud835\udc8e\ud835\udfd0 \u00d7 \ud835\udfcf\ud835\udfce\ud835\udfd3) \ud835\udf39\ud835\udc91\ud835\udc86\ud835\udc82\ud835\udc8c (\ud835\udc8e) \ud835\udc7a\ud835\udc79 = \ud835\udfce. \ud835\udfcf\ud835\udfce \ud835\udc7a\ud835\udc79 = \ud835\udfce. \ud835\udfcf\ud835\udfd3 \ud835\udc7a\ud835\udc79 = \ud835\udfce. \ud835\udfd0\ud835\udfce \ud835\udc6c\ud835\udc70\ud835\udc88 (\ud835\udc74\ud835\udc75\ud835\udc8e\ud835\udfd0 \u00d7 \ud835\udfcf\ud835\udfce\ud835\udfd3) \ud835\udc74\ud835\udc91\ud835\udc86\ud835\udc82\ud835\udc8c (\ud835\udc74\ud835\udc75\ud835\udc8e) \ud835\udc6c\ud835\udc70\ud835\udc88 (\ud835\udc74\ud835\udc75\ud835\udc8e\ud835\udfd0 \u00d7 \ud835\udfcf\ud835\udfce\ud835\udfd3) \ud835\udc6c\ud835\udc70\ud835\udc88 (\ud835\udc74\ud835\udc75\ud835\udc8e\ud835\udfd0 \u00d7 \ud835\udfcf\ud835\udfce\ud835\udfd3) \ud835\udc3f\ud835\udc5d\ud835\udc3f\ud835\udc5a\ud835\udc65\ud835\udc5d\ud835\udc3f\ud835\udc5a\ud835\udc7a\ud835\udc79 = \ud835\udfce. \ud835\udfcf\ud835\udfce \ud835\udc7a\ud835\udc79 = \ud835\udfce. \ud835\udfcf\ud835\udfd3 \ud835\udc7a\ud835\udc79 = \ud835\udfce. \ud835\udfd0\ud835\udfce \ud835\udc3f\ud835\udc5d\ud835\udc3f\ud835\udc5a\ud835\udc65\ud835\udc5d\ud835\udc3f\ud835\udc5a\ud835\udc38\ud835\udc4e\ud835\udc5f\ud835\udc61\u210e \u2212 \ud835\udc34\ud835\udc5b\ud835\udc50\u210e\ud835\udc5c\ud835\udc5f\ud835\udc52\ud835\udc51 \ud835\udc46\ud835\udc66\ud835\udc60\ud835\udc61\ud835\udc52\ud835\udc5a \ud835\udc46\ud835\udc52\ud835\udc59\ud835\udc53 \u2212 \ud835\udc34\ud835\udc5b\ud835\udc50\u210e\ud835\udc5c\ud835\udc5f\ud835\udc52\ud835\udc51 \ud835\udc46\ud835\udc66\ud835\udc60\ud835\udc61\ud835\udc52\ud835\udc5a 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 3\ud835\udc6c\ud835\udc70\ud835\udc88 (\ud835\udc74\ud835\udc75\ud835\udc8e\ud835\udfd0 \u00d7 \ud835\udfcf\ud835\udfce\ud835\udfd3) \ud835\udf39\ud835\udc91\ud835\udc86\ud835\udc82\ud835\udc8c \ud835\udc7a\ud835\udc76\ud835\udf39\ud835\udc91\ud835\udc86\ud835\udc82\ud835\udc8c \ud835\udc7a\ud835\udc79 = \ud835\udfce. \ud835\udfcf\ud835\udfce \ud835\udc7a\ud835\udc79 = \ud835\udfce. \ud835\udfcf\ud835\udfd3 \ud835\udc7a\ud835\udc79 = \ud835\udfce. \ud835\udfd0\ud835\udfce \ud835\udc6c\ud835\udc70\ud835\udc88 (\ud835\udc74\ud835\udc75\ud835\udc8e\ud835\udfd0 \u00d7 \ud835\udfcf\ud835\udfce\ud835\udfd3) \ud835\udc74\ud835\udc91\ud835\udc86\ud835\udc82\ud835\udc8c \ud835\udc7a\ud835\udc76\ud835\udc74\ud835\udc91\ud835\udc86\ud835\udc82\ud835\udc8c \ud835\udc6c\ud835\udc70\ud835\udc88 (\ud835\udc74\ud835\udc75\ud835\udc8e\ud835\udfd0 \u00d7 \ud835\udfcf\ud835\udfce\ud835\udfd3) \ud835\udc6c\ud835\udc70\ud835\udc88 (\ud835\udc74\ud835\udc75\ud835\udc8e\ud835\udfd0 \u00d7 \ud835\udfcf\ud835\udfce\ud835\udfd3) \ud835\udc7a\ud835\udc79 = \ud835\udfce. \ud835\udfcf\ud835\udfce \ud835\udc7a\ud835\udc79 = \ud835\udfce. \ud835\udfcf\ud835\udfd3 \ud835\udc7a\ud835\udc79 = \ud835\udfce. \ud835\udfd0\ud835\udfce \u1d07\u1d07 \ud835\udc3f\ud835\udc5a \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udf14\ud835\udc5d0.1 0.15 0.211.21.41.61.80 0.5 1 1.5 211.21.41.61.80.1 0.15 0.21821.52528.532\ud835\udc53 \ud835\udc3f\ud835\udc5a \ud835\udc53 \ud835\udc3f\ud835\udc5a \ud835\udc7a\ud835\udc79 = \ud835\udc87 \ud835\udc73\ud835\udc8e\u2044 \ud835\udf39\ud835\udc91\ud835\udc86\ud835\udc82\ud835\udc8c (\ud835\udc8e) \ud835\udc38\ud835\udc3c\ud835\udc54 = 25000\ud835\udc40\ud835\udc41\ud835\udc5a2 \ud835\udc38\ud835\udc34\ud835\udc54 = 200000\ud835\udc40\ud835\udc41 \ud835\udc79\ud835\udc90\ud835\udc8d\ud835\udc8d\ud835\udc86\ud835\udc93 \ud835\udc7a\ud835\udc96\ud835\udc91\ud835\udc91\ud835\udc90\ud835\udc93\ud835\udc95 (TYP) \ud835\udc38\ud835\udc34\ud835\udc54, \ud835\udc38\ud835\udc3c\ud835\udc54 \ud835\udc77\ud835\udc8a\ud835\udc8f \ud835\udc7a\ud835\udc96\ud835\udc91\ud835\udc91\ud835\udc90\ud835\udc93\ud835\udc95 \ud835\udc38\ud835\udc34\ud835\udc54, \ud835\udc38\ud835\udc3c\ud835\udc54 \ud835\udc6c\ud835\udc68\ud835\udc88 (\ud835\udc74\ud835\udc75 \u00d7 \ud835\udfcf\ud835\udfce\ud835\udfd3) \ud835\udc38\ud835\udc3c\ud835\udc54 = 25000\ud835\udc40\ud835\udc41\ud835\udc5a2 \ud835\udc46\ud835\udc45 = 0.2 \ud835\udc7a\ud835\udc79 = \ud835\udc87 \ud835\udc73\ud835\udc8e\u2044 \ud835\udc74\ud835\udc91\ud835\udc86\ud835\udc82\ud835\udc8c (\ud835\udc74\ud835\udc75\ud835\udc8e) \ud835\udc38\ud835\udc3c\ud835\udc54 = 25000\ud835\udc40\ud835\udc41\ud835\udc5a2 \ud835\udc38\ud835\udc34\ud835\udc54 = 200000\ud835\udc40\ud835\udc41 \ud835\udc3f\ud835\udc5a \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udf14\ud835\udc5d \ud835\udc38\ud835\udc34\ud835\udc540.1 0.15 0.20.911.11.21.31.41.50.1 0.15 0.20.911.11.21.31.41.5\ud835\udf39\ud835\udc91\ud835\udc86\ud835\udc82\ud835\udc8c(\ud835\udf39\ud835\udc91\ud835\udc86\ud835\udc82\ud835\udc8c)\ud835\udc79 \ud835\udc74\ud835\udc91\ud835\udc86\ud835\udc82\ud835\udc8c(\ud835\udc74\ud835\udc91\ud835\udc86\ud835\udc82\ud835\udc8c)\ud835\udc79 \ud835\udc7a\ud835\udc79 = \ud835\udc87 \ud835\udc73\ud835\udc8e\u2044 \ud835\udc7a\ud835\udc79 = \ud835\udc87 \ud835\udc73\ud835\udc8e\u2044 Flexible Superstructure (\ud835\udc38\ud835\udc3c\ud835\udc54 = 25000\ud835\udc40\ud835\udc41\ud835\udc5a2) Stiff Superstructure (\ud835\udc38\ud835\udc3c\ud835\udc54 = 600000\ud835\udc40\ud835\udc41\ud835\udc5a2) Sag Ratio = 0.1 Sag Ratio = 0.2 \ud835\udc3f\ud835\udc5a \ud835\udc38\ud835\udc3c\ud835\udc54 \u1d07 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udf14\ud835\udc5d4.5\uf02d3\uf02d1.5\uf02d01.534.54.5\uf02d3\uf02d1.5\uf02d01.534.5300\uf02d200\uf02d100\uf02d0100200300300\uf02d200\uf02d100\uf02d010020030060\uf02d40\uf02d20\uf02d020406060\uf02d40\uf02d20\uf02d0204060300\uf02d200\uf02d100\uf02d0100200300300\uf02d200\uf02d100\uf02d0100200300Deflection Envelope (m) Bending Moment Envelope (MN\u00b7m) Percent Change (%) Percent Change (%) System 2 System 1 \uf0a7 \ud835\udc3f\u210e\uf0a7 \ud835\udc53\u210e\ud835\udc3f\ud835\udc5a \ud835\udc53\ud835\udc53 \ud835\udc53\u210e\ud835\udc53\u210e \ud835\udc3f\u210e \ud835\udc3f\ud835\udc5a \ud835\udc53\ud835\udc3f\u210e \ud835\udc53\u210e\ud835\udc3f\ud835\udc5a 2\u2044 \ud835\udc3f\u210e \ud835\udc53\u210e \ud835\udc53 Suspended Region Stayed Region Stayed Region \ud835\udc3f\ud835\udc5a 2\u2044 \ud835\udc411\u2026\ud835\udc5b \ud835\udc3f\u210e\ud835\udc461\u2026\ud835\udc5b \ud835\udc47\ud835\udc5a\ud835\udc4e\ud835\udc65\ud835\udc39\u210e\ud835\udc56 \ud835\udc3b\ud835\udf14\ud835\udc50 {\ud835\udc65\ud835\udc56 , \ud835\udc66\ud835\udc56}\ud835\udc53 \u0394\ud835\udc65\ud835\udc56\ud835\udc3f\ud835\udc5a \u0394y\ud835\udc56\u0197\ud835\udc53 \ud835\udc3f\ud835\udc5a 2\u2044 \ud835\udee5\ud835\udc65\ud835\udc56 \ud835\udee5\ud835\udc66\ud835\udc56 \ud835\udc66\ud835\udc56 \ud835\udc39\u210e\ud835\udc56 \ud835\udc65\ud835\udc56 \ud835\udc3b \ud835\udc47\ud835\udc5a\ud835\udc4e\ud835\udc65 \ud835\udf14\ud835\udc50 \ud835\udf14\ud835\udc50 \ud835\udc3b\ud835\udc51\ud835\udc66\ud835\udc56\ud835\udc51\ud835\udc65\ud835\udc56|\ud835\udc65\ud835\udc56=\ud835\udee5\ud835\udc65\ud835\udc56 \ud835\udc3b \ud835\udc3b\ud835\udc51\ud835\udc66\ud835\udc56\ud835\udc51\ud835\udc65\ud835\udc56|\ud835\udc65\ud835\udc56=0 \ud835\udc3b \ud835\udc39\u210e1 \ud835\udc39\u210e\ud835\udc56\u22121 \ud835\udc39\u210e\ud835\udc5b \ud835\udc461 \ud835\udc46\ud835\udc56\u22121 \ud835\udc46\ud835\udc56 \ud835\udc46\ud835\udc5b \ud835\udc411 \ud835\udc41\ud835\udc56\u22121 \ud835\udc41\ud835\udc56 \ud835\udc41\ud835\udc5b Free Body Diagram of Segment i \u0197Same as in fully-laden suspension cable \u0197Similar for Segment n, except \ud835\udee5\ud835\udc65\ud835\udc5b = \ud835\udc3f\ud835\udc5a \u2212 \ud835\udc3f\u210e 2 \u2044 \ud835\udc3f\u210e 2\u2044 Stayed Region \ud835\udc3f\ud835\udc5a \u2212 \ud835\udc3f\u210e 2\u2044 \ud835\udc53\ud835\udf14\ud835\udc50\ud835\udc5d\ud835\udf14\ud835\udc50\ud835\udc5d = \ud835\udf14\ud835\udc50\u221a1+ \u03a92 \u03a9\u03a9 =2 y\ud835\udc35\ud835\udc3f\ud835\udc5a \u2212 \ud835\udc3f\u210e \ud835\udc3f\ud835\udc5a \u2212 \ud835\udc3f\u210e 2\u2044 \ud835\udc3f\u210e \ud835\udc53\u210e \ud835\udc53 \ud835\udc34 \ud835\udc35 \ud835\udc36 \ud835\udc37 \ud835\udf14\ud835\udc50\ud835\udc5d \ud835\udf14\ud835\udc50\ud835\udc5d \ud835\udf14\ud835\udc50 \ud835\udc66\ud835\udc35 \ud835\udc3f\ud835\udc5a \u2212 \ud835\udc3f\u210e 2\u2044 \ud835\udc6d\ud835\udc90\ud835\udc93 \ud835\udfce \u2264 \ud835\udc99 < \ud835\udc73\ud835\udc8e \u2212 \ud835\udc73\ud835\udc89 \ud835\udfd0\u2044\ud835\udc66 \ud835\udc65 =1\ud835\udc3b[\ud835\udf14\ud835\udc50\ud835\udc5d\ud835\udc6522\u2212 \ud835\udc49\ud835\udc60\ud835\udc65]\ud835\udc6d\ud835\udc90\ud835\udc93 \ud835\udc73\ud835\udc8e \u2212 \ud835\udc73\ud835\udc89 \ud835\udfd0\u2044 \u2264 \ud835\udc99 \u2264 \ud835\udc73\ud835\udc8e + \ud835\udc73\ud835\udc89 \ud835\udfd0\u2044\ud835\udc66 \ud835\udc65 |\ud835\udc46\ud835\udc5a =1\ud835\udc3b[\u2211 \ud835\udc39\u210e\ud835\udc57 (\ud835\udc65 \u2212 \u2211\ud835\udf06\ud835\udc56\ud835\udc57\ud835\udc56=1) +\ud835\udf14\ud835\udc50\ud835\udc5d\ud835\udf0612 2\ud835\udc65 \u2212 \ud835\udf061 +\ud835\udf14\ud835\udc502 \ud835\udc65 \u2212 \ud835\udf061 2 \u2212 \ud835\udc49\ud835\udc60\ud835\udc65\ud835\udc5a\u22121\ud835\udc57=1]\ud835\udc6d\ud835\udc90\ud835\udc93 \ud835\udc73\ud835\udc8e + \ud835\udc73\ud835\udc89 \ud835\udfd0\u2044 \u2264 \ud835\udc99 < \ud835\udc73\ud835\udc8e\ud835\udc66 \ud835\udc65 =1\ud835\udc3b[\ud835\udf14\ud835\udc50\ud835\udc5d2 2\ud835\udf061 + \ud835\udc3f\u210e \u2212 \ud835\udc65 2 \u2212 \ud835\udc49\ud835\udc60\ud835\udc65]\ud835\udc65 \ud835\udc66\ud835\udc49\ud835\udc60 =12\u2211 \ud835\udc39\u210e\ud835\udc56 + \ud835\udf14\ud835\udc50\ud835\udc5d\ud835\udf061 +\ud835\udf14\ud835\udc50\ud835\udc3f\u210e2\ud835\udc41\u210e\ud835\udc4e\ud835\udc5b\ud835\udc54\ud835\udc52\ud835\udc5f\ud835\udc60\ud835\udc56=1\ud835\udc3b =1\ud835\udc53[\ud835\udc49\ud835\udc60\ud835\udc3f\ud835\udc5a2\u2212\ud835\udf14\ud835\udc50\ud835\udc5d\ud835\udf0612 \ud835\udc3f\ud835\udc5a \u2212 \ud835\udf061 \u2212\ud835\udf14\ud835\udc50\ud835\udc3f\u210e28\u2212 \u2211\ud835\udc39\u210e\ud835\udc57 (\ud835\udc3f\ud835\udc5a2\u2212 \u2211\ud835\udf06\ud835\udc56\ud835\udc57\ud835\udc56=1)\ud835\udc41\u2217\ud835\udc57=1]\ud835\udc3b =1\ud835\udc53\u210e[(\ud835\udc49\ud835\udc60 \u2212 \ud835\udf14\ud835\udc50\ud835\udc5d\ud835\udf061)\ud835\udc3f\u210e2\u2212\ud835\udf14\ud835\udc50\ud835\udc3f\u210e28\u2212 \u2211\ud835\udc39\u210e\ud835\udc57 (\ud835\udc3f\ud835\udc5a2\u2212 \u2211\ud835\udf06\ud835\udc56\ud835\udc57\ud835\udc56=1)\ud835\udc41\u2217\ud835\udc57=1]\ud835\udc41\u210e\ud835\udc4e\ud835\udc5b\ud835\udc54\ud835\udc52\ud835\udc5f\ud835\udc60\u221212, \ud835\udc41\u2217 =\ud835\udc41\u210e\ud835\udc4e\ud835\udc5b\ud835\udc54\ud835\udc52\ud835\udc5f\ud835\udc602\uf0a7 \uf0a7 \uf0a7 \ud835\udf14\ud835\udc50 \ud835\udc411\u2026 \ud835\udc3f\ud835\udc5a\ud835\udc461\u2026 \ud835\udc3f\u210e\ud835\udc41\u210e\ud835\udc4e\ud835\udc5b\ud835\udc54\ud835\udc52\ud835\udc5f\ud835\udc60 \u0197 \ud835\udc3b\ud835\udf14\ud835\udc50 \u0197 \ud835\udc49\ud835\udc60\ud835\udc53 \u03bb\ud835\udc5a\ud835\udc53\u210e \u0197 \ud835\udc39\u210e\ud835\udc5a\u0197\ud835\udc66 \ud835\udc39\u210e\ud835\udc5a+1 \ud835\udc65 \ud835\udc3b \ud835\udc39\u210e1 \ud835\udc39\u210e\ud835\udc5a \ud835\udc49\ud835\udc60 \ud835\udc39\u210e\ud835\udc5a\u22121 \ud835\udf061 = \ud835\udc3f\ud835\udc5a \u2212 \ud835\udc3f\u210e 2\u2044 \ud835\udf06\ud835\udc5a\u22121 \ud835\udf06\ud835\udc5a \ud835\udf06\ud835\udc5a+1 \ud835\udc411 \ud835\udc41\ud835\udc5a\u22121 \ud835\udc41\ud835\udc5a \ud835\udc41\ud835\udc5a+1 \ud835\udc3b \ud835\udc3f\ud835\udc5a 2\u2044 \ud835\udc39\u210e2 \ud835\udf062 \ud835\udc53 \ud835\udc412 \ud835\udc461 \ud835\udc462 \ud835\udc46\ud835\udc5a\u22121 \ud835\udc46\ud835\udc5a \ud835\udc46\ud835\udc5a+1 \ud835\udc3f\u210e 2\u2044 \ud835\udc510\ud835\udc510 = 0\ud835\udf06 \ud835\udf14\ud835\udc60\ud835\udf14\ud835\udc50\ud835\udc5d \ud835\udc3f\ud835\udc5a\u2212\ud835\udc3f\u210e 2+\ud835\udc3f\u210e 2\ud835\udc3f\ud835\udc5a\u2212\ud835\udc3f\u210e \ud835\udf14\ud835\udc50+\ud835\udf14\ud835\udc60 8\ud835\udc53,\ud835\udc3b =\ud835\udf14\ud835\udc50\ud835\udc5d \ud835\udc3f\ud835\udc5a\u2212\ud835\udc3f\u210e 2+\ud835\udc3f\u210e 2\ud835\udc3f\ud835\udc5a\u2212\ud835\udc3f\u210e \ud835\udf14\ud835\udc50+\ud835\udf14\ud835\udc60 \u2212\ud835\udf14\ud835\udc60\ud835\udf0628\ud835\udc53, \ud835\udf14\ud835\udc50+\ud835\udf14\ud835\udc60 \ud835\udc3f\u210e28\ud835\udc53\u210e,\ud835\udc3b = \ud835\udf14\ud835\udc50+\ud835\udf14\ud835\udc60 \ud835\udc3f\u210e2\u2212\ud835\udf14\ud835\udc60\ud835\udf0628\ud835\udc53\u210e, \ud835\udc3f\ud835\udc5a 2\u2044 \ud835\udc3f\ud835\udc5a \u2212 \ud835\udc3f\u210e 2\u2044 \ud835\udc510 \ud835\udc3f\u210e 2\u2044 Stay Cables \ud835\udc65[\ud835\udf14\ud835\udc50\ud835\udc5d \ud835\udc65\u2212\ud835\udc3f\ud835\udc5a \u2212\ud835\udc3f\u210e(\ud835\udf14\ud835\udc50+\ud835\udf14\ud835\udc60\u2212\ud835\udf14\ud835\udc50\ud835\udc5d)]2\ud835\udc3b, \ud835\udc53\ud835\udc5c\ud835\udc5f 0 \u2264 \ud835\udc65 < \ud835\udc3f\ud835\udc5a \u2212 \ud835\udc3f\u210e 2\u2044\ud835\udc66 \ud835\udc65 =[ \ud835\udc3f\ud835\udc5a\u2212\ud835\udc3f\u210e 2(\ud835\udf14\ud835\udc50+\ud835\udf14\ud835\udc60\u2212\ud835\udf14\ud835\udc50\ud835\udc5d)+4\ud835\udc65 \ud835\udf14\ud835\udc50+\ud835\udf14\ud835\udc60 \ud835\udc65\u2212\ud835\udc3f\ud835\udc5a ]8\ud835\udc3b, \ud835\udc53\ud835\udc5c\ud835\udc5f \ud835\udc3f\ud835\udc5a \u2212 \ud835\udc3f\u210e 2\u2044 \u2264 \ud835\udc65 \u2264 \ud835\udc3f\ud835\udc5a + \ud835\udc3f\u210e 2\u2044 \ud835\udc65\u2212\ud835\udc3f\ud835\udc5a [\ud835\udf14\ud835\udc50\ud835\udc5d \ud835\udc65 +\ud835\udc3f\u210e(\ud835\udf14\ud835\udc50+\ud835\udf14\ud835\udc60\u2212\ud835\udf14\ud835\udc50\ud835\udc5d)]2\ud835\udc3b, \ud835\udc53\ud835\udc5c\ud835\udc5f \ud835\udc3f\ud835\udc5a + \ud835\udc3f\u210e 2\u2044 \u2264 \ud835\udc65 < \ud835\udc3f\ud835\udc5a \ud835\udc3f\ud835\udc5a \u2212 \ud835\udc3f\u210e 2\u2044 \u2264 \ud835\udc65 \u2264 \ud835\udc3f\ud835\udc5a + \ud835\udc3f\u210e 2\u2044 \ud835\udc65\ud835\udc510 < 0\ud835\udc36\ud835\udc5a = \ud835\udf06\ud835\udc5a\u221a1+ \ud835\udefa\ud835\udc5a2 [1 +124\ud835\udef9\ud835\udc50\ud835\udc5a2(1 + \ud835\udefa\ud835\udc5a2)]\u2206\ud835\udc52\ud835\udc5a =\ud835\udc3b\ud835\udf06\ud835\udc5a\ud835\udc38\ud835\udc50\ud835\udc34\ud835\udc50[1 + \ud835\udefa\ud835\udc5a2 +112\ud835\udef9\ud835\udc50\ud835\udc5a2(1 + \ud835\udefa\ud835\udc5a2)]\ud835\udef9\ud835\udc50\ud835\udc5a =\ud835\udf14\ud835\udc50\ud835\udf06\ud835\udc5a\ud835\udc3b\ud835\udefa\ud835\udc5a =\u210e\ud835\udc5a\ud835\udf06\ud835\udc5a\u210e\ud835\udc5a\u210e\ud835\udc5a = \ud835\udc66\ud835\udc5a \u2212 \ud835\udc66\ud835\udc5a\u22121\ud835\udc66\ud835\udc5a \ud835\udc5a\ud835\udc3b\ud835\udc48\ud835\udc46\ud835\udc3f\ud835\udc47\ud835\udc5c\ud835\udc61\ud835\udc4e\ud835\udc59 = \u2211 \ud835\udc48\ud835\udc46\ud835\udc3f\ud835\udc5a \ud835\udc41\u210e\ud835\udc4e\ud835\udc5b\ud835\udc54\ud835\udc52\ud835\udc5f\ud835\udc60+1\ud835\udc5a=1 = \u2211 (\ud835\udc36\ud835\udc5a \u2212 \ud835\udee5\ud835\udc52\ud835\udc5a)\ud835\udc41\u210e\ud835\udc4e\ud835\udc5b\ud835\udc54\ud835\udc52\ud835\udc5f\ud835\udc60+1\ud835\udc5a=1\ud835\udf14\ud835\udc50\ud835\udc5d = \ud835\udf14\ud835\udc50\u221a1 + (2 \ud835\udc66\ud835\udc35\ud835\udc3f\ud835\udc5a \u2212 \ud835\udc3f\u210e)2\ud835\udc66\ud835\udc35 = \ud835\udc66 (\ud835\udc65 =\ud835\udc3f\ud835\udc5a \u2212 \ud835\udc3f\u210e2)\ud835\udf14\ud835\udc50\ud835\udc5d \ud835\udc66\ud835\udc35\ud835\udf14\ud835\udc50\ud835\udc5d\ud835\udf14\ud835\udc50 \ud835\udf14\ud835\udc50\ud835\udc5d \ud835\udc66\ud835\udc35\ud835\udf14\ud835\udc50\ud835\udc5d\ud835\udf14\ud835\udc50\ud835\udc5d \u2245 \ud835\udf14\ud835\udc50\u221a1+ 4\ud835\udc532 [\ud835\udf14\ud835\udc50 \ud835\udc3f\ud835\udc5a + \ud835\udc3f\u210e + 2\ud835\udf14\ud835\udc60\ud835\udc3f\u210e\ud835\udf14\ud835\udc50\ud835\udc3f\ud835\udc5a2 + \ud835\udf14\ud835\udc60\ud835\udc3f\u210e 2\ud835\udc3f\ud835\udc5a \u2212 \ud835\udc3f\u210e ]2\ud835\udc53\u210e = \ud835\udc53 \u2212 \ud835\udc66\ud835\udc35\ud835\udc66\ud835\udc35\ud835\udc53\u210e\ud835\udc3f\u210e\u2245 \ud835\udc3f\ud835\udc45 (\ud835\udc53\ud835\udc3f\ud835\udc5a) [\ud835\udf14\ud835\udc50 + \ud835\udf14\ud835\udc60\ud835\udf14\ud835\udc50\ud835\udc5d 1 \u2212 \ud835\udc3f\ud835\udc45 2 + \ud835\udc3f\ud835\udc45 \ud835\udf14\ud835\udc50 + \ud835\udf14\ud835\udc60 2 \u2212 \ud835\udc3f\ud835\udc45 ]\ud835\udc3f\ud835\udc45\ud835\udc3f\ud835\udc45 =\ud835\udc3f\u210e\ud835\udc3f\ud835\udc5a\uf0d8 \uf0d8 \ud835\udf14\ud835\udc50:\ud835\udf14\ud835\udc60\uf0d8 \ud835\udeda\ud835\udc84 \ud835\udeda\ud835\udc94\u2044 = \ud835\udfce. \ud835\udfce\ud835\udfce\ud835\udeda\ud835\udc84 \ud835\udeda\ud835\udc94\u2044 = \ud835\udfce. \ud835\udfce\ud835\udfd3 \ud835\udc0b\ud835\udc04\ud835\udc06\ud835\udc04\ud835\udc0d\ud835\udc03\ud835\udeda\ud835\udc84 \ud835\udeda\ud835\udc94\u2044 = \ud835\udfce. \ud835\udfcf\ud835\udfce\ud835\udeda\ud835\udc84 \ud835\udeda\ud835\udc94\u2044 = \ud835\udfce. \ud835\udfcf\ud835\udfd30.1 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 0.30.040.060.080.10.120.140.160.180.2\ud835\udc73\ud835\udc89 = \ud835\udfce. \ud835\udfd0 \ud835\udc73\ud835\udc8e \ud835\udc73\ud835\udc89 = \ud835\udfce. \ud835\udfd1 \ud835\udc73\ud835\udc8e \ud835\udc73\ud835\udc89 = \ud835\udfce. \ud835\udfd2 \ud835\udc73\ud835\udc8e \ud835\udc73\ud835\udc89 = \ud835\udfce. \ud835\udfd3 \ud835\udc73\ud835\udc8e \ud835\udc87 \ud835\udc73\ud835\udc8e\u2044 \ud835\udc87\ud835\udc89 \ud835\udc73\ud835\udc89\u2044 \ud835\udc87 \ud835\udc73\ud835\udc8e\u2044 \ud835\udc87 \ud835\udc73\ud835\udc8e\u2044 \ud835\udc87 \ud835\udc73\ud835\udc8e\u2044 \ud835\udc87\ud835\udc89 \ud835\udc73\ud835\udc89\u2044 \ud835\udc87\ud835\udc89 \ud835\udc73\ud835\udc89\u2044 \ud835\udc87\ud835\udc89 \ud835\udc73\ud835\udc89\u2044 \ud835\udc73\ud835\udc89 = \ud835\udfce. \ud835\udfd4 \ud835\udc73\ud835\udc8e \ud835\udc87 \ud835\udc73\ud835\udc8e\u2044 \ud835\udc87\ud835\udc89 \ud835\udc73\ud835\udc89\u2044 \ud835\udc87 \ud835\udc87\ud835\udc89 \ud835\udc73\ud835\udc89 \ud835\udc73\ud835\udc8e \ud835\udc66\ud835\udc50 \ud835\udc66\ud835\udc5d\ud835\udc65\ud835\udc65\ud835\udf14\ud835\udc50: \ud835\udf14\ud835\udc60 \ud835\udc73\ud835\udc89 = \ud835\udfce. \ud835\udfd0 \ud835\udc73\ud835\udc8e\ud835\udc73\ud835\udc89 = \ud835\udfce. \ud835\udfd2 \ud835\udc73\ud835\udc8e \ud835\udc0b\ud835\udc04\ud835\udc06\ud835\udc04\ud835\udc0d\ud835\udc03\ud835\udc73\ud835\udc89 = \ud835\udfce. \ud835\udfd4 \ud835\udc73\ud835\udc8e\ud835\udc73\ud835\udc89 = \ud835\udfcf. \ud835\udfce \ud835\udc73\ud835\udc8e\ud835\udf14\ud835\udc50: \ud835\udf14\ud835\udc600 0.05 0.1 0.15 0.201.25 \uf02e1052.5 \uf02e1053.75 \uf02e1055 \uf02e1050 0.05 0.1 0.15 0.205 \uf02e1051 \uf02e1041.5 \uf02e1042 \uf02e1040 0.05 0.1 0.15 0.202 \uf02e1044 \uf02e1046 \uf02e1048 \uf02e104\ud835\udf4e\ud835\udc84 = \ud835\udc62\ud835\udc5b\ud835\udc56\ud835\udc53\ud835\udc5c\ud835\udc5f\ud835\udc5a \ud835\udc64\ud835\udc52\ud835\udc56\ud835\udc54\u210e\ud835\udc61 \ud835\udc5c\ud835\udc53 \ud835\udc60\ud835\udc62\ud835\udc60\ud835\udc5d\ud835\udc52\ud835\udc5b\ud835\udc60\ud835\udc56\ud835\udc5c\ud835\udc5b \ud835\udc50\ud835\udc4e\ud835\udc4f\ud835\udc59\ud835\udc52 \ud835\udf4e\ud835\udc94 = \ud835\udc62\ud835\udc5b\ud835\udc56\ud835\udc53\ud835\udc5c\ud835\udc5f\ud835\udc5a \ud835\udc64\ud835\udc52\ud835\udc56\ud835\udc54\u210e\ud835\udc61 \ud835\udc5c\ud835\udc53 \ud835\udc60\ud835\udc62\ud835\udc5d\ud835\udc52\ud835\udc5f\ud835\udc60\ud835\udc61\ud835\udc5f\ud835\udc62\ud835\udc50\ud835\udc61\ud835\udc62\ud835\udc5f\ud835\udc52 \ud835\udf4e\ud835\udc84 \ud835\udf4e\ud835\udc94\u2044 \ud835\udf4e\ud835\udc84 \ud835\udf4e\ud835\udc94\u2044 \ud835\udc87 = \ud835\udfce. \ud835\udfd1 \ud835\udc73\ud835\udc8e \ud835\udf4e\ud835\udc84 \ud835\udf4e\ud835\udc94\u2044 |(\ud835\udc9a\ud835\udc91 \u2212 \ud835\udc9a\ud835\udc84)\ud835\udc8e\ud835\udc82\ud835\udc99|\ud835\udc73\ud835\udc8e \ud835\udc87 \ud835\udc87\ud835\udc89 \ud835\udc73\ud835\udc89 \ud835\udc73\ud835\udc8e \ud835\udc87 = \ud835\udfce. \ud835\udfcf \ud835\udc73\ud835\udc8e \ud835\udc87 = \ud835\udfce. \ud835\udfd0 \ud835\udc73\ud835\udc8e |(\ud835\udc9a\ud835\udc91 \u2212 \ud835\udc9a\ud835\udc84)\ud835\udc8e\ud835\udc82\ud835\udc99|\ud835\udc73\ud835\udc8e \ud835\udc73\ud835\udc89 = \ud835\udfce. \ud835\udfd0 \ud835\udc73\ud835\udc8e\ud835\udc73\ud835\udc89 = \ud835\udfce. \ud835\udfd2 \ud835\udc73\ud835\udc8e \ud835\udc0b\ud835\udc04\ud835\udc06\ud835\udc04\ud835\udc0d\ud835\udc03\ud835\udc73\ud835\udc89 = \ud835\udfce. \ud835\udfd4 \ud835\udc73\ud835\udc8e\ud835\udc73\ud835\udc89 = \ud835\udfcf. \ud835\udfce \ud835\udc73\ud835\udc8e\ud835\udefe\ud835\udc50 \ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60200 400 600 800 100001.25 \uf02e1052.5 \uf02e1053.75 \uf02e1055 \uf02e105200 400 600 800 100005 \uf02e1051 \uf02e1041.5 \uf02e1042 \uf02e104200 400 600 800 100002 \uf02e1044 \uf02e1046 \uf02e1048 \uf02e104\ud835\udc73\ud835\udc8e \ud835\udc8e \ud835\udc73\ud835\udc8e \ud835\udc8e \ud835\udc87 = \ud835\udfce. \ud835\udfd1 \ud835\udc73\ud835\udc8e \ud835\udc73\ud835\udc8e \ud835\udc8e |(\ud835\udc9a\ud835\udc91 \u2212 \ud835\udc9a\ud835\udc84)\ud835\udc8e\ud835\udc82\ud835\udc99|\ud835\udc73\ud835\udc8e \ud835\udc87 \ud835\udc87\ud835\udc89 \ud835\udc73\ud835\udc89 \ud835\udc73\ud835\udc8e \ud835\udc87 = \ud835\udfce. \ud835\udfcf \ud835\udc73\ud835\udc8e \ud835\udc87 = \ud835\udfce. \ud835\udfd0 \ud835\udc73\ud835\udc8e |(\ud835\udc9a\ud835\udc91 \u2212 \ud835\udc9a\ud835\udc84)\ud835\udc8e\ud835\udc82\ud835\udc99|\ud835\udc73\ud835\udc8e \ud835\udf14\ud835\udc5d: \ud835\udf14\ud835\udc60 \ud835\udc73\ud835\udc89 = \ud835\udfce. \ud835\udfd0 \ud835\udc73\ud835\udc8e\ud835\udc73\ud835\udc89 = \ud835\udfce. \ud835\udfd2 \ud835\udc73\ud835\udc8e \ud835\udc0b\ud835\udc04\ud835\udc06\ud835\udc04\ud835\udc0d\ud835\udc03\ud835\udc73\ud835\udc89 = \ud835\udfce. \ud835\udfd4 \ud835\udc73\ud835\udc8e\ud835\udc73\ud835\udc89 = \ud835\udfcf. \ud835\udfce \ud835\udc73\ud835\udc8e\ud835\udefe\ud835\udc50 \ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udc38\ud835\udc50 \ud835\udf06 \u00b7\ud835\udc3f\ud835\udc5a200 400 600 800 100002 \uf02e1064 \uf02e1066 \uf02e1068 \uf02e1061 \uf02e105200 400 600 800 100001 \uf02e1052 \uf02e1053 \uf02e1054 \uf02e1055 \uf02e105200 400 600 800 100005 \uf02e1051 \uf02e1041.5 \uf02e1042 \uf02e1042.5 \uf02e104\ud835\udc73\ud835\udc8e \ud835\udc8e \ud835\udc73\ud835\udc8e \ud835\udc8e \ud835\udc87 = \ud835\udfce. \ud835\udfd1 \ud835\udc73\ud835\udc8e \ud835\udc73\ud835\udc8e \ud835\udc8e |(\ud835\udc7c\ud835\udc7a\ud835\udc73\ud835\udc84 \u2212 \ud835\udc7c\ud835\udc7a\ud835\udc73\ud835\udc91)|\ud835\udc73\ud835\udc8e \ud835\udc87 \ud835\udc87\ud835\udc89 \ud835\udc73\ud835\udc89 \ud835\udc73\ud835\udc8e \ud835\udc87 = \ud835\udfce. \ud835\udfcf \ud835\udc73\ud835\udc8e \ud835\udc87 = \ud835\udfce. \ud835\udfd0 \ud835\udc73\ud835\udc8e |(\ud835\udc7c\ud835\udc7a\ud835\udc73\ud835\udc84 \u2212 \ud835\udc7c\ud835\udc7a\ud835\udc73\ud835\udc91)|\ud835\udc73\ud835\udc8e \ud835\udc87\ud835\udc73\ud835\udc8e\ud835\udc73\ud835\udc89\ud835\udc73\ud835\udc8e\ud835\udf4e\ud835\udc91\ud835\udf4e\ud835\udc94\ud835\udc73\ud835\udc8e \ud835\udc8e |(\ud835\udc66\ud835\udc5d \u2212 \ud835\udc66\ud835\udc50)\ud835\udc5a\ud835\udc4e\ud835\udc65| \ud835\udc3f\ud835\udc5a\u2044\ud835\udefe\ud835\udc50 \ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64\ud835\udc87\ud835\udc73\ud835\udc8e\ud835\udc73\ud835\udc89\ud835\udc73\ud835\udc8e\ud835\udf4e\ud835\udc91\ud835\udf4e\ud835\udc94\ud835\udc73\ud835\udc8e \ud835\udc8e |(\ud835\udc48\ud835\udc46\ud835\udc3f\ud835\udc50 \u2212 \ud835\udc48\ud835\udc46\ud835\udc3f\ud835\udc5d)\ud835\udc5a\ud835\udc4e\ud835\udc65| \ud835\udc3f\ud835\udc5a\u2044 \ud835\udefe\ud835\udc50 \ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64 \ud835\udc38\ud835\udc50 \ud835\udf06 \u00b7\ud835\udc3f\ud835\udc5a\uf0a7 \ud835\udf14\ud835\udc50 , \ud835\udf14\ud835\udc60\uf0a7 \uf0a7 \ud835\udf14\ud835\udc5d\uf0a7 \uf0a7 \uf0a7 \uf0a7 \uf0a7 \ud835\udc47\ud835\udc5a\ud835\udc4e\ud835\udc65 = \ud835\udc3b\u221a1 + (\ud835\udc51\ud835\udc66\ud835\udc51\ud835\udc65|\ud835\udc65=0)2\ud835\udc34\ud835\udc50_\ud835\udc5f\ud835\udc52\ud835\udc5e =\ud835\udc47\ud835\udc5a\ud835\udc4e\ud835\udc65\ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64=\ud835\udc3b\ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64\u221a1 + (\ud835\udc51\ud835\udc66\ud835\udc51\ud835\udc65|\ud835\udc65=0)2\ud835\udc34\ud835\udc50_\ud835\udc5f\ud835\udc52\ud835\udc5e\ud835\udf14\ud835\udc50 =\ud835\udefe\ud835\udc50\ud835\udc3b\ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64\u221a1+ (\ud835\udc51\ud835\udc66\ud835\udc51\ud835\udc65|\ud835\udc65=0)2\ud835\udf14\ud835\udc50 =\ud835\udefe\ud835\udc50\ud835\udc3f\ud835\udc5a[\ud835\udf14\ud835\udc50 + \ud835\udf14\ud835\udc60\ud835\udc3f\ud835\udc45 2 \u2212 \ud835\udc3f\ud835\udc45 ]8 \ud835\udc46\ud835\udc45\ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64\u221a1 + 16\ud835\udc46\ud835\udc452 (\ud835\udf14\ud835\udc50 + \ud835\udf14\ud835\udc60\ud835\udc3f\ud835\udc45\ud835\udf14\ud835\udc50 \u2212 \ud835\udf14\ud835\udc60\ud835\udc3f\ud835\udc452 + 2\ud835\udf14\ud835\udc60\ud835\udc3f\ud835\udc45)2\ud835\udc46\ud835\udc45\ud835\udc53 \ud835\udc3f\ud835\udc5a \ud835\udc3f\ud835\udc45 \ud835\udc3f\u210e \ud835\udc3f\ud835\udc5a\ud835\udf14\ud835\udc50\ud835\udc3f\ud835\udc45\ud835\udf14\ud835\udc50\ud835\udf14\ud835\udc50\ud835\udf14\ud835\udc50 =\ud835\udefe\ud835\udc50\ud835\udc3f\ud835\udc5a[\ud835\udf14\ud835\udc50 + \ud835\udf14\ud835\udc60\ud835\udc3f\ud835\udc45 2 \u2212 \ud835\udc3f\ud835\udc45 ]8 \ud835\udc46\ud835\udc45\ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64\u221a1 + 16(\ud835\udc46\ud835\udc452 \u2212 \ud835\udc3f\ud835\udc45)2\ud835\udf14\ud835\udc60 (\ud835\udf14\ud835\udc60 + \ud835\udf14\ud835\udc5d)\ud835\udf14\ud835\udc50\ud835\udf14\ud835\udc50 =(\ud835\udf14\ud835\udc60 + \ud835\udf14\ud835\udc5d) \ud835\udf09 \ud835\udc3f\ud835\udc45 2 \u2212 \ud835\udc3f\ud835\udc45 \u221a1 + 16(\ud835\udc46\ud835\udc452 \u2212 \ud835\udc3f\ud835\udc45)21 \u2212 \ud835\udf09 \u221a1 + 16 (\ud835\udc46\ud835\udc452 \u2212 \ud835\udc3f\ud835\udc45)2\u03be\ud835\udf09 =\ud835\udefe\ud835\udc50 \ud835\udc3f\ud835\udc5a8 \ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64 \ud835\udc46\ud835\udc45\ud835\udc3f\ud835\udc45\ud835\udc3f\ud835\udc45\ud835\udc46\ud835\udc45 \ud835\udc3f\ud835\udc5aSuspension Ratio = 0.2 Suspension Ratio = 0.4 Suspension Ratio = 0.6 \ud835\udefe\ud835\udc50 \ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc640 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 100000.030.060.090.120.15\ud835\udc73\ud835\udc8e \ud835\udc8e \ud835\udc73\ud835\udc8e \ud835\udc8e \ud835\udc73\ud835\udc8e \ud835\udc8e \ud835\udf4e\ud835\udc84\ud835\udf4e\ud835\udc94 +\ud835\udf4e\ud835\udc91 \ud835\udf4e\ud835\udc84\ud835\udf4e\ud835\udc94 +\ud835\udf4e\ud835\udc91 \ud835\udf4e\ud835\udc84\ud835\udf4e\ud835\udc94 +\ud835\udf4e\ud835\udc91 \ud835\udc46\ud835\udc45 = 0.3 \ud835\udc46\ud835\udc45 = 0.2 \ud835\udc46\ud835\udc45 = 0.1 \ud835\udc53\u210e \ud835\udc3f\u210e \ud835\udeff\ud835\udc5a\ud835\udc4e\ud835\udc65 \ud835\udc39\ud835\udc3f \ud835\udeff\ud835\udc5a\ud835\udc4e\ud835\udc65 \ud835\udc43\ud835\udc3f(\ud835\udeff\ud835\udc5d\ud835\udc52\ud835\udc4e\ud835\udc58)\ud835\udc39\ud835\udc3f\ud835\udc58\ud835\udc38\ud835\udc58\ud835\udc3a \ud835\udc3f\ud835\udc5a \ud835\udefe\ud835\udc50 \ud835\udc38\ud835\udc50 \ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60\ud835\udc65\ud835\udc5d \ud835\udc3f\ud835\udc5d \ud835\udc3f\u210e = 0.2 \u00b7 \ud835\udc3f\ud835\udc5a 0.1 \u00b7 \ud835\udc3f\u210e \ud835\udc65\ud835\udc5d \ud835\udc3f\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udf14\ud835\udc5d \ud835\udc3f\ud835\udc5d \ud835\udc65\ud835\udc5d \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udf14\ud835\udc60 \ud835\udf14\ud835\udc5d \ud835\udf39\ud835\udc8e\ud835\udc82\ud835\udc99\ud835\udc73\ud835\udc8e \ud835\udf39\ud835\udc8e\ud835\udc82\ud835\udc99\ud835\udc73\ud835\udc8e \ud835\udf39\ud835\udc8e\ud835\udc82\ud835\udc99\ud835\udc73\ud835\udc8e \ud835\udc73\ud835\udc91 \ud835\udc73\ud835\udc89\u2044 \ud835\udc73\ud835\udc91 \ud835\udc73\ud835\udc89\u2044 \ud835\udc73\ud835\udc91 \ud835\udc73\ud835\udc89\u2044 \ud835\udc99\ud835\udc91\ud835\udc73\ud835\udc89\u2044 \ud835\udc99\ud835\udc91\ud835\udc73\ud835\udc89\u2044 \ud835\udc99\ud835\udc91\ud835\udc73\ud835\udc89\u2044 0.1 \u00b7 \ud835\udc3f\u210e \ud835\udc3f\u210e = 0.6 \u00b7 \ud835\udc3f\ud835\udc5a 0.1 \u00b7 \ud835\udc3f\u210e \ud835\udc3f\u210e = 1.0 \u00b7 \ud835\udc3f\ud835\udc5a Suspension Ratio = 0.2 Suspension Ratio = 0.6 Suspension Ratio = 1.0 \ud835\udc3f\ud835\udc5a \ud835\udefe\ud835\udc50 \ud835\udc38\ud835\udc50 \ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udf39\ud835\udc8e\ud835\udc82\ud835\udc99 \ud835\udc77\ud835\udc73 \u2212 \ud835\udf39\ud835\udc8e\ud835\udc82\ud835\udc99 \ud835\udc6d\ud835\udc73(\ud835\udf39\ud835\udc91\ud835\udc86\ud835\udc82\ud835\udc8c)\ud835\udc6d\ud835\udc73 \ud835\udf39\ud835\udc8e\ud835\udc82\ud835\udc99 \ud835\udc77\ud835\udc73 \u2212 \ud835\udf39\ud835\udc8e\ud835\udc82\ud835\udc99 \ud835\udc6d\ud835\udc73(\ud835\udf39\ud835\udc91\ud835\udc86\ud835\udc82\ud835\udc8c)\ud835\udc6d\ud835\udc73 0.1 \u00b7 \ud835\udc3f\u210e 0.1 \u00b7 \ud835\udc3f\u210e 0.1 \u00b7 \ud835\udc3f\u210e 0.1 \u00b7 \ud835\udc3f\u210e \ud835\udc3f\u210e = 0.6 \u00b7 \ud835\udc3f\ud835\udc5a \ud835\udc3f\ud835\udc5a \ud835\udc3f\u210e \ud835\udc73\ud835\udc91 \ud835\udc73\ud835\udc89\u2044 \ud835\udc73\ud835\udc91 \ud835\udc73\ud835\udc89\u2044 \ud835\udc99\ud835\udc91\ud835\udc73\ud835\udc89\u2044 \ud835\udc99\ud835\udc91\ud835\udc73\ud835\udc89\u2044 \ud835\udc3f\u210e = 0.2 \u00b7 \ud835\udc3f\ud835\udc5a \ud835\udc3f\ud835\udc5a \ud835\udc3f\u210e \ud835\udc65\ud835\udc5d \ud835\udc3f\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udf14\ud835\udc5d \ud835\udc3f\ud835\udc5d \ud835\udc65\ud835\udc5d \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 Suspension Ratio = 0.6 Suspension Ratio = 0.2 \ud835\udc3f\u210e \ud835\udc53\u210e \ud835\udc58\ud835\udc60\ud835\udc5d\ud835\udc5f\ud835\udc56\ud835\udc5b\ud835\udc54 = \ud835\udc58\ud835\udc38 + \ud835\udc58\ud835\udc3a \ud835\udc3f\ud835\udc5a \ud835\udefe\ud835\udc50 \ud835\udc38\ud835\udc50 \ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60\ud835\udc3f\u210e \ud835\udc3f\ud835\udc5a \ud835\udefe\ud835\udc50 \ud835\udc38\ud835\udc50 \ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc600.2 0.4 0.6 0.8 100.0050.010.0150.020.2 0.4 0.6 0.8 100.010.020.030.04\ud835\udc73\ud835\udc79 = \ud835\udc73\ud835\udc89 \ud835\udc73\ud835\udc8e\u2044 \ud835\udc46\ud835\udc45 = 0.3 \ud835\udc46\ud835\udc45 = 0.2 \ud835\udc46\ud835\udc45 = 0.1 \ud835\udc73\ud835\udc79 = \ud835\udc73\ud835\udc89 \ud835\udc73\ud835\udc8e\u2044 \ud835\udf39\ud835\udc91\ud835\udc86\ud835\udc82\ud835\udc8c\ud835\udc73\ud835\udc8e \ud835\udf39\ud835\udc91\ud835\udc86\ud835\udc82\ud835\udc8c\ud835\udc73\ud835\udc89 \ud835\udc46\ud835\udc45 = 0.3 \ud835\udc46\ud835\udc45 = 0.2 \ud835\udc46\ud835\udc45 = 0.1 \ud835\udc3f\ud835\udc5a 500\u2044 \ud835\udc3f\ud835\udc5a 50\u2044 \ud835\udc47\ud835\udc4c\ud835\udc43 \ud835\udc3f\ud835\udc5a \ud835\udc38\ud835\udc34\ud835\udc54, \ud835\udc38\ud835\udc3c\ud835\udc54 \ud835\udc53 \ud835\udc45\ud835\udc56\ud835\udc54\ud835\udc56\ud835\udc51 \ud835\udc47\ud835\udc5f\ud835\udc62\ud835\udc60\ud835\udc60 \ud835\udc38\ud835\udc59\ud835\udc52\ud835\udc5a\ud835\udc52\ud835\udc5b\ud835\udc61\ud835\udc60 \ud835\udc45\ud835\udc56\ud835\udc54\ud835\udc56\ud835\udc51 \ud835\udc47\ud835\udc5f\ud835\udc62\ud835\udc60\ud835\udc60 \ud835\udc38\ud835\udc59\ud835\udc52\ud835\udc5a\ud835\udc52\ud835\udc5b\ud835\udc61\ud835\udc60 \ud835\udc43\ud835\udc56\ud835\udc5b \ud835\udc46\ud835\udc62\ud835\udc5d\ud835\udc5d\ud835\udc5c\ud835\udc5f\ud835\udc61 \ud835\udc45\ud835\udc5c\ud835\udc59\ud835\udc59\ud835\udc52\ud835\udc5f \ud835\udc46\ud835\udc62\ud835\udc5d\ud835\udc5d\ud835\udc5c\ud835\udc5f\ud835\udc61 (TYP) \ud835\udc3f\u210e \ud835\udc3f\ud835\udc5a \u2212 \ud835\udc3f\u210e 2\u2044 \ud835\udc3f\ud835\udc5a \u2212 \ud835\udc3f\u210e 2\u2044 \ud835\udc3f\ud835\udc5a 500\u2044 \ud835\udf14\ud835\udc5d \ud835\udc49\ud835\udc4e\ud835\udc5f\ud835\udc56\ud835\udc4e\ud835\udc4f\ud835\udc59\ud835\udc52 \ud835\udf14\ud835\udc60 \ud835\udc49\ud835\udc4e\ud835\udc5f\ud835\udc56\ud835\udc4e\ud835\udc4f\ud835\udc59\ud835\udc52 \ud835\udc3f\ud835\udc5a 100\u2044\u1d07\ud835\udc46\ud835\udc45 \ud835\udc53 \ud835\udc3f\ud835\udc5a\u2044 \ud835\udc3f\ud835\udc5a \ud835\udc38\ud835\udc3c\ud835\udc54 \u1d07 \ud835\udc38\ud835\udc34\ud835\udc54 \u1d07 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udf14\ud835\udc5dSuspension Ratio = 0.6 Suspension Ratio = 0.4 Suspension Ratio = 0.2 Deflection Envelope (m) Bending Moment Envelope (MN\u00b7m) Axial Force Envelope (MN) \ud835\udc46\ud835\udc45 = 0.1 \ud835\udc46\ud835\udc45 = 0.2 \ud835\udc46\ud835\udc45 = 0.3 \ud835\udc3f\ud835\udc5a \ud835\udc38\ud835\udc3c\ud835\udc54 \u1d076 \ud835\udc38\ud835\udc34\ud835\udc54 \u1d07 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udf14\ud835\udc5d0.2 0.3 0.4 0.5 0.69\uf02d6\uf02d3\uf02d3690.1 0.15 0.2 0.25 0.39\uf02d6\uf02d3\uf02d3690.2 0.3 0.4 0.5 0.6525\uf02d350\uf02d175\uf02d1753505250.1 0.15 0.2 0.25 0.3525\uf02d350\uf02d175\uf02d1753505250.2 0.3 0.4 0.5 0.6300\uf02d200\uf02d100\uf02d1002003000.1 0.15 0.2 0.25 0.3300\uf02d200\uf02d100\uf02d100200300 \ud835\udc46\ud835\udc45 = 0.1 \ud835\udc46\ud835\udc45 = 0.2 \ud835\udc46\ud835\udc45 = 0.3 \ud835\udc3f\ud835\udc45 = 0.2 \ud835\udc3f\ud835\udc45 = 0.4 \ud835\udc3f\ud835\udc45 = 0.6\ud835\udc3f\ud835\udc45 \ud835\udc46\ud835\udc45Peak Deflections (m) Peak Bending Moments (MN\u00b7m) Peak Axial Force (MN) \ud835\udc39\ud835\udc5d\ud835\udc52\ud835\udc4e\ud835\udc58 \u2245 \u2211\ud835\udc3b\ud835\udc60\ud835\udc61\ud835\udc60 + \ud835\udc3b\ud835\udc50\ud835\udc60 =(\ud835\udf14\ud835\udc60 + \ud835\udf14\ud835\udc5d)[\ud835\udc3f\ud835\udc5a 1 \u2212 \ud835\udc3f\ud835\udc45 ]28 \u210e\ud835\udc47+ \ud835\udc3b\ud835\udc50\ud835\udc5a\ud835\udc3b\ud835\udc50\ud835\udc5a\ud835\udf14\ud835\udc50\ud835\udc5d \u2248 \ud835\udf14\ud835\udc50\u210e\ud835\udc47 \u2248 \ud835\udc53\ud835\udc39\ud835\udc5d\ud835\udc52\ud835\udc4e\ud835\udc58 \u2245(\ud835\udefe\ud835\udc50\ud835\udc5a\ud835\udc34\ud835\udc50\ud835\udc5a + \ud835\udf14\ud835\udc60 + \ud835\udf14\ud835\udc5d)\ud835\udc3f\ud835\udc5a8 \ud835\udc46\ud835\udc45\ud835\udefe\ud835\udc50\ud835\udc5a \ud835\udc34\ud835\udc50\ud835\udc5a\ud835\udc3b\ud835\udc50\ud835\udc5a\/ \ud835\udc49\ud835\udc50\ud835\udc5a = Horizontal\/Vertical reaction from suspension cable \ud835\udc3b\ud835\udc50\ud835\udc60\/ \ud835\udc49\ud835\udc50\ud835\udc60 = Horizontal\/Vertical reaction from anchor cable \u2211\ud835\udc3b\ud835\udc60\ud835\udc61\ud835\udc60\/ \u2211\ud835\udc49\ud835\udc60\ud835\udc61\ud835\udc60= Cumulative horizontal\/vertical reaction from stay cables in side span \u210e\ud835\udc47 = Tower height \ud835\udc49\ud835\udc47\ud835\udc4f\ud835\udc4e\ud835\udc60\ud835\udc52= Reaction at tower base \ud835\udf14\ud835\udc60\/ \ud835\udf14\ud835\udc5d = Dead\/Live loading \ud835\udc3f\ud835\udc45 = Suspension ratio \ud835\udc3f\ud835\udc5a = Main span length \ud835\udc3f\ud835\udc5a 1 \u2212 \ud835\udc3f\ud835\udc45 2\u2044 \u210e\ud835\udc47 \ud835\udc3b\ud835\udc50\ud835\udc5a (\ud835\udf14\ud835\udc60 + \ud835\udf14\ud835\udc5d) \ud835\udc3b\ud835\udc50\ud835\udc5a \u2211\ud835\udc3b\ud835\udc60\ud835\udc61\ud835\udc60 + \ud835\udc3b\ud835\udc50\ud835\udc60 \u2211\ud835\udc49\ud835\udc60\ud835\udc61\ud835\udc60 +\ud835\udc49\ud835\udc50\ud835\udc60 + \ud835\udc49\ud835\udc50\ud835\udc5a \ud835\udc49\ud835\udc47\ud835\udc4f\ud835\udc4e\ud835\udc60\ud835\udc52 Stay Cables \u2211\ud835\udc3b\ud835\udc60\ud835\udc61\ud835\udc60 + \ud835\udc3b\ud835\udc50\ud835\udc60 Main Span Side Span C Tower L Suspension Ratio = 0.2 Suspension Ratio = 0.4 Suspension Ratio = 0.6 \ud835\udc3f\ud835\udc5a \ud835\udc38\ud835\udc34\ud835\udc54 \u1d075 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udf14\ud835\udc5d0 5 10 15 20 25 309\uf02d6\uf02d3\uf02d03690 5 10 15 20 25 309\uf02d6\uf02d3\uf02d03690 5 10 15 20 25 309\uf02d6\uf02d3\uf02d03690 5 10 15 20 25 301800\uf02d1200\uf02d600\uf02d0600120018000 5 10 15 20 25 301800\uf02d1200\uf02d600\uf02d0600120018000 5 10 15 20 25 301800\uf02d1200\uf02d600\uf02d060012001800 \ud835\udc46\ud835\udc45 = 0.1 \ud835\udc46\ud835\udc45 = 0.2 \ud835\udc46\ud835\udc45 = 0.3Peak Bending Moments (MN\u00b7m) Peak Deflections (m) \ud835\udc38\ud835\udc3c\ud835\udc54 \ud835\udc40\ud835\udc41\ud835\udc5a \ud835\udc65 106 \ud835\udc38\ud835\udc3c\ud835\udc54 \ud835\udc40\ud835\udc41\ud835\udc5a \ud835\udc65 106 \ud835\udc38\ud835\udc3c\ud835\udc54 \ud835\udc40\ud835\udc41\ud835\udc5a \ud835\udc65 106 Suspension Ratio = 0.2 Suspension Ratio = 0.4 Suspension Ratio = 0.6 \ud835\udc3f\ud835\udc5a \ud835\udc38\ud835\udc34\ud835\udc54 \u1d075 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udf14\ud835\udc5d0 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 \ud835\udc46\ud835\udc45 = 0.1 \ud835\udc46\ud835\udc45 = 0.2 \ud835\udc46\ud835\udc45 = 0.3\ud835\udf39\ud835\udc91\ud835\udc86\ud835\udc82\ud835\udc8c\ud835\udc7a\ud835\udc76\ud835\udf39\ud835\udc91\ud835\udc86\ud835\udc82\ud835\udc8c\u2212\u2212 \ud835\udc74\ud835\udc91\ud835\udc86\ud835\udc82\ud835\udc8c\ud835\udc7a\ud835\udc76\ud835\udc74\ud835\udc91\ud835\udc86\ud835\udc82\ud835\udc8c++ \ud835\udc74\ud835\udc91\ud835\udc86\ud835\udc82\ud835\udc8c\ud835\udc7a\ud835\udc76\ud835\udc74\ud835\udc91\ud835\udc86\ud835\udc82\ud835\udc8c\u2212\u2212 \ud835\udc38\ud835\udc3c\ud835\udc54 \ud835\udc40\ud835\udc41\ud835\udc5a \ud835\udc65 106 \ud835\udc38\ud835\udc3c\ud835\udc54 \ud835\udc40\ud835\udc41\ud835\udc5a \ud835\udc65 106 \ud835\udc38\ud835\udc3c\ud835\udc54 \ud835\udc40\ud835\udc41\ud835\udc5a \ud835\udc65 106 \ud835\udeff\ud835\udc631 =\ud835\udf0e\ud835\udc50\u2212\ud835\udc53\ud835\udc38\ud835\udc52\ud835\udc53\ud835\udc53\u2212\ud835\udc53(1\ud835\udc60\ud835\udc56\ud835\udc5b\ud835\udf03\ud835\udc50\u2212\ud835\udc531\ud835\udc50\ud835\udc5c\ud835\udc60\ud835\udf03\ud835\udc50\u2212\ud835\udc53)\ud835\udc3f\ud835\udc5a2\ud835\udeff\u210e =\ud835\udf0e\ud835\udc50\u2212\ud835\udc4e\ud835\udc38\ud835\udc52\ud835\udc53\ud835\udc53\u2212\ud835\udc4e(1\ud835\udc50\ud835\udc5c\ud835\udc60\ud835\udf03\ud835\udc50\u2212\ud835\udc4e1\ud835\udc60\ud835\udc56\ud835\udc5b\ud835\udf03\ud835\udc50\u2212\ud835\udc4e) \u210e\ud835\udc47\ud835\udeff\ud835\udc632 =\ud835\udf0e\ud835\udc50\u2212\ud835\udc4e\ud835\udc38\ud835\udc52\ud835\udc53\ud835\udc53\u2212\ud835\udc4e(1\ud835\udc50\ud835\udc5c\ud835\udc60\ud835\udf03\ud835\udc50\u2212\ud835\udc4e1\ud835\udc60\ud835\udc56\ud835\udc5b\ud835\udf03\ud835\udc50\u2212\ud835\udc4e)\ud835\udc3f\ud835\udc5a2\ud835\udf0e\ud835\udc50\u2212\ud835\udc53 \ud835\udc38\ud835\udc52\ud835\udc53\ud835\udc53\u2212\ud835\udc53 \ud835\udf0e\ud835\udc50\u2212\ud835\udc4e \ud835\udc38\ud835\udc52\ud835\udc53\ud835\udc53\u2212\ud835\udc4e\ud835\udf0e\ud835\udc50\u2212\ud835\udc53 = (\ud835\udf14\ud835\udc451 + \ud835\udf14\ud835\udc45) \ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64\ud835\udf0e\ud835\udc50\u2212\ud835\udc4e = (\ud835\udf14\ud835\udc451 + \ud835\udf14\ud835\udc45 \u2212 4\ud835\udc3f\ud835\udc46\ud835\udc452)\ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc64\ud835\udf14\ud835\udc45 \ud835\udf14\ud835\udc5d: \ud835\udf14\ud835\udc60 \ud835\udc3f\ud835\udc46\ud835\udc45\ud835\udc3f\ud835\udc60: \ud835\udc3f\ud835\udc5a \ud835\udeff\u210e \ud835\udc3f\ud835\udc60 \ud835\udc3f\ud835\udc5a 2\u2044 \u210e\ud835\udc47 \ud835\udf03\ud835\udc50\u2212\ud835\udc53 \ud835\udf03\ud835\udc50\u2212\ud835\udc4e \ud835\udf14\ud835\udc60 \ud835\udf14\ud835\udc5d \ud835\udeff\ud835\udc631 \ud835\udeff\ud835\udc632 (From Elongation of Forestay) (From Elongation of Anchor Cable) Forestay Anchor Cable \ud835\udeff\ud835\udc631 + \ud835\udeff\ud835\udc632\ud835\udeff\ud835\udc631= 1 + (1 + \ud835\udf14\ud835\udc451 + \ud835\udf14\ud835\udc45 \u2212 4\ud835\udc3f\ud835\udc46\ud835\udc452) (\ud835\udc60\ud835\udc56\ud835\udc5b\ud835\udf03\ud835\udc50\u2212\ud835\udc53\ud835\udc60\ud835\udc56\ud835\udc5b\ud835\udf03\ud835\udc50\u2212\ud835\udc4e\ud835\udc50\ud835\udc5c\ud835\udc60\ud835\udf03\ud835\udc50\u2212\ud835\udc53\ud835\udc50\ud835\udc5c\ud835\udc60\ud835\udf03\ud835\udc50\u2212\ud835\udc4e)\ud835\udf03\ud835\udc50\u2212\ud835\udc53 \ud835\udf03\ud835\udc50\u2212\ud835\udc4e\ud835\udeff\ud835\udc631 + \ud835\udeff\ud835\udc632\ud835\udeff\ud835\udc631= 1 + (1 + \ud835\udf14\ud835\udc451 + \ud835\udf14\ud835\udc45 \u2212 4\ud835\udc3f\ud835\udc46\ud835\udc452) (2\ud835\udc3f\ud835\udc46\ud835\udc45) (\u210e\ud835\udc47\ud835\udc45 + \ud835\udc3f\ud835\udc46\ud835\udc4524\u210e\ud835\udc47\ud835\udc452 + 1)\u210e\ud835\udc47\ud835\udc45 \u210e\ud835\udc47: \ud835\udc3f\ud835\udc5a \ud835\udf14\ud835\udc450.1 0.2 0.3 0.4 0.512340.1 0.15 0.2 0.25 0.31234 \u210e\ud835\udc47\ud835\udc45 = 0.1 \u210e\ud835\udc47\ud835\udc45 = 0.2 \u210e\ud835\udc47\ud835\udc45 = 0.3 \ud835\udc3f\ud835\udc46\ud835\udc45 = 0.35 \ud835\udc3f\ud835\udc46\ud835\udc45 = 0.40 \ud835\udc3f\ud835\udc46\ud835\udc45 = 0.45\ud835\udc3f\ud835\udc46\ud835\udc45 \u210e\ud835\udc47\ud835\udc45\ud835\udeff\ud835\udc631 + \ud835\udeff\ud835\udc632\ud835\udeff\ud835\udc631Typical Range |\ud835\udeff\ud835\udc52|\ud835\udc5d\ud835\udc52\ud835\udc4e\ud835\udc58|\ud835\udeff\ud835\udc5f|\ud835\udc5d\ud835\udc52\ud835\udc4e\ud835\udc58 \ud835\udc3f\ud835\udc5a \ud835\udc3f\ud835\udc46\ud835\udc45 \ud835\udc38\ud835\udc3c\ud835\udc54 \u1d07 \ud835\udc38\ud835\udc34\ud835\udc54 \u1d07 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udf14\ud835\udc5dSuspension Ratio = 0.6 Suspension Ratio = 0.4 Suspension Ratio = 0.2 Deflection Envelope (m) Bending Moment Envelope (MN\u00b7m) Axial Force Envelope (MN) \ud835\udc46\ud835\udc45 = 0.1 \ud835\udc46\ud835\udc45 = 0.2 \ud835\udc46\ud835\udc45 = 0.3 \ud835\udc3f\ud835\udc5a \ud835\udc3f\ud835\udc46\ud835\udc45 \ud835\udc38\ud835\udc3c\ud835\udc54 \u1d076 \ud835\udc38\ud835\udc34\ud835\udc54 \u1d07 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udf14\ud835\udc5d \ud835\udc3f\ud835\udc5a \ud835\udc3f\ud835\udc46\ud835\udc45 \ud835\udc38\ud835\udc3c\ud835\udc54 \u1d076 \ud835\udc38\ud835\udc34\ud835\udc54 \u1d07 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udf14\ud835\udc5d0.1 0.15 0.2 0.25 0.312340.2 0.3 0.4 0.5 0.69\uf02d6\uf02d3\uf02d3690.1 0.15 0.2 0.25 0.39\uf02d6\uf02d3\uf02d3690.2 0.3 0.4 0.5 0.6525\uf02d350\uf02d175\uf02d1753505250.1 0.15 0.2 0.25 0.3525\uf02d350\uf02d175\uf02d175350525\ud835\udc46\ud835\udc45|\ud835\udeff\ud835\udc52|\ud835\udc5d\ud835\udc52\ud835\udc4e\ud835\udc58|\ud835\udeff\ud835\udc5f|\ud835\udc5d\ud835\udc52\ud835\udc4e\ud835\udc58\ud835\udc3f\ud835\udc45 = 0.6 \ud835\udc3f\ud835\udc45 = 0.4 \ud835\udc3f\ud835\udc45 = 0.2 \ud835\udc46\ud835\udc45 = 0.1 \ud835\udc46\ud835\udc45 = 0.2 \ud835\udc46\ud835\udc45 = 0.3 \ud835\udc3f\ud835\udc45 = 0.2 \ud835\udc3f\ud835\udc45 = 0.4 \ud835\udc3f\ud835\udc45 = 0.6\ud835\udc3f\ud835\udc45 \ud835\udc46\ud835\udc45Peak Deflections (m) Peak Bending Moments (MN\u00b7m) Suspension Ratio = 0.2 Suspension Ratio = 0.4 Suspension Ratio = 0.6 \ud835\udc3f\ud835\udc5a \ud835\udc3f\ud835\udc46\ud835\udc45 \ud835\udc38\ud835\udc34\ud835\udc54 \u1d075 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udf14\ud835\udc5d0 5 10 15 20 25 309\uf02d6\uf02d3\uf02d03690 5 10 15 20 25 309\uf02d6\uf02d3\uf02d03690 5 10 15 20 25 309\uf02d6\uf02d3\uf02d03690 5 10 15 20 25 301800\uf02d1200\uf02d600\uf02d0600120018000 5 10 15 20 25 301800\uf02d1200\uf02d600\uf02d0600120018000 5 10 15 20 25 301800\uf02d1200\uf02d600\uf02d060012001800 \ud835\udc46\ud835\udc45 = 0.1 \ud835\udc46\ud835\udc45 = 0.2 \ud835\udc46\ud835\udc45 = 0.3Peak Bending Moments (MN\u00b7m) Peak Deflections (m) \ud835\udc38\ud835\udc3c\ud835\udc54 \ud835\udc40\ud835\udc41\ud835\udc5a \ud835\udc65 106 \ud835\udc38\ud835\udc3c\ud835\udc54 \ud835\udc40\ud835\udc41\ud835\udc5a \ud835\udc65 106 \ud835\udc38\ud835\udc3c\ud835\udc54 \ud835\udc40\ud835\udc41\ud835\udc5a \ud835\udc65 106 \ud835\udc51 \u221d \ud835\udc3f\ud835\udc5a\ud835\udeff\ud835\udc5d\ud835\udc52\ud835\udc4e\ud835\udc58 =5\ud835\udf14\ud835\udc5d\ud835\udc3f\ud835\udc5a4384\ud835\udc38\ud835\udc3c\ud835\udc54\ud835\udc3c\ud835\udc54 \u221d \ud835\udc3f\ud835\udc5a3Suspension Ratio = 0.2 Suspension Ratio = 0.4 Suspension Ratio = 0.6 \ud835\udc38\ud835\udc3c\ud835\udc54 \ud835\udc3f\ud835\udc5a3 \ud835\udc38\ud835\udc34\ud835\udc54 \u1d075 \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 \ud835\udf14\ud835\udc5d250 500 750 10009\uf02d6\uf02d3\uf02d369250 500 750 10009\uf02d6\uf02d3\uf02d369250 500 750 10009\uf02d6\uf02d3\uf02d369250 500 750 10000.6\uf02d0.4\uf02d0.2\uf02d0.20.40.6250 500 750 10000.6\uf02d0.4\uf02d0.2\uf02d0.20.40.6250 500 750 10000.6\uf02d0.4\uf02d0.2\uf02d0.20.40.6 \ud835\udc46\ud835\udc45 = 0.1 \ud835\udc46\ud835\udc45 = 0.2 \ud835\udc46\ud835\udc45 = 0.3\ud835\udc3f\ud835\udc5a \ud835\udc5a \ud835\udc3f\ud835\udc5a \ud835\udc5a \ud835\udc3f\ud835\udc5a \ud835\udc5a \ud835\udf39\ud835\udc91\ud835\udc86\ud835\udc82\ud835\udc8c\ud835\udc73\ud835\udc8e \ud835\udc74\ud835\udc91\ud835\udc86\ud835\udc82\ud835\udc8c\ud835\udc73\ud835\udc8e\ud835\udfd0 [\ud835\udc74\ud835\udc75\ud835\udc8e] (\ud835\udf0e\ud835\udc4f,\ud835\udc3f\ud835\udc3f)\ud835\udc5d\ud835\udc52\ud835\udc4e\ud835\udc58 =\ud835\udc40\ud835\udc5d\ud835\udc52\ud835\udc4e\ud835\udc58\ud835\udc50\ud835\udc3c\ud835\udc54\ud835\udc50\ud835\udc50 \ud835\udc51\ud835\udc50 = \ud835\udc58\ud835\udc51\ud835\udc58 \ud835\udc51\ud835\udc3f\ud835\udc5a \ud835\udc3c\ud835\udc54 \ud835\udc3f\ud835\udc5a3(\ud835\udf0e\ud835\udc4f,\ud835\udc3f\ud835\udc3f)\ud835\udc5d\ud835\udc52\ud835\udc4e\ud835\udc58 \u221d\ud835\udc40\ud835\udc5d\ud835\udc52\ud835\udc4e\ud835\udc58\ud835\udc3f\ud835\udc5a2\ud835\udc51 \u221d \ud835\udc3f\ud835\udc5a \ud835\udc3c\ud835\udc54 \u221d \ud835\udc3f\ud835\udc5a3(\ud835\udf0e\ud835\udc4e,\ud835\udc3f\ud835\udc3f)\ud835\udc5d\ud835\udc52\ud835\udc4e\ud835\udc58 \u2245\ud835\udf14\ud835\udc5d\ud835\udc3f\ud835\udc5a8 \ud835\udc46\ud835\udc45\ud835\udc34\ud835\udc54\ud835\udc34\ud835\udc54\ud835\udc3c\ud835\udc54 \ud835\udc51\ud835\udc51\ud835\udc3c\ud835\udc54 \u2245\ud835\udc34\ud835\udc54\ud835\udc5124\ud835\udc51 \u221d \ud835\udc3f\ud835\udc5a \ud835\udc3c\ud835\udc54 \u221d \ud835\udc3f\ud835\udc5a3\ud835\udc34\ud835\udc54 \u221d \ud835\udc3f\ud835\udc5a\ud835\udc51 \u221d \ud835\udc3f\ud835\udc5a\ud835\udc3c\ud835\udc54 \u221d \ud835\udc3f\ud835\udc5a3(\ud835\udf0e\ud835\udc4e,\ud835\udc37\ud835\udc3f)\ud835\udc5d\ud835\udc52\ud835\udc4e\ud835\udc58 \u2245 \ud835\udefe\ud835\udc50\ud835\udc5a\ud835\udc34\ud835\udc50\ud835\udc5a + \ud835\udf14\ud835\udc60 \ud835\udc3f\ud835\udc5a8 \ud835\udc46\ud835\udc45\ud835\udc34\ud835\udc54(\ud835\udf0e\ud835\udc4e,\ud835\udc37\ud835\udc3f)\ud835\udc5d\ud835\udc52\ud835\udc4e\ud835\udc58 \u2245(\ud835\udefe\ud835\udc60\ud835\udc34\ud835\udc54\ud835\udefc\ud835\udc60\ud835\udc51\ud835\udc59)\ud835\udc3f\ud835\udc5a8 \ud835\udc46\ud835\udc45\ud835\udc34\ud835\udc54 \ud835\udc51 \ud835\udc34\ud835\udc54 2\u2044 \ud835\udc34\ud835\udc54 2\u2044 \ud835\udc34\ud835\udc54 Transverse Stiffener Longitudinal Stiffener (TYP) Top Plate Bottom Plate Idealized Top Plate Idealized Bottom Plate \ud835\udefc\ud835\udc60\ud835\udc51\ud835\udc59\ud835\udc34\ud835\udc54\ud835\udf14\ud835\udc45 = \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60\u2044\ud835\udc38\ud835\udc3c\ud835\udc54 \u1d07 \ud835\udc3f\ud835\udc5a \ud835\udc3f\ud835\udc46\ud835\udc45 \ud835\udc46\ud835\udc45 \ud835\udc3f\ud835\udc45 \ud835\udc38\ud835\udc3c\ud835\udc54 \u1d07 \ud835\udc38\ud835\udc34\ud835\udc54 \ud835\udf14\ud835\udc5d3\uf02d2\uf02d1\uf02d012375\uf02d50\uf02d25\uf02d0255075Deflection Envelope (m) Bending Moment Envelope (MN\u00b7m) \ud835\udf14\ud835\udc45 = 0.6\ud835\udf14\ud835\udc45 = 0.4\ud835\udf14\ud835\udc45 = 0.2\ud835\udc46\ud835\udc45 = 0.2\ud835\udc3f\ud835\udc45 = 0.4\ud835\udeff\ud835\udc5d\ud835\udc52\ud835\udc4e\ud835\udc582\ud835\udeff\ud835\udc5d\ud835\udc52\ud835\udc4e\ud835\udc581\u2245(\ud835\udf14\ud835\udc4521 + \ud835\udf14\ud835\udc452)(\ud835\udf14\ud835\udc4511 + \ud835\udf14\ud835\udc451)\ud835\udeff\ud835\udc5d\ud835\udc52\ud835\udc4e\ud835\udc582\ud835\udeff\ud835\udc5d\ud835\udc52\ud835\udc4e\ud835\udc581\u2245(\ud835\udf14\ud835\udc4521 + \ud835\udf14\ud835\udc452) + (\ud835\udf14\ud835\udc4521 + \ud835\udf14\ud835\udc452 \u2212 4\ud835\udc3f\ud835\udc46\ud835\udc452)\ud835\udc3e\ud835\udc5f(\ud835\udf14\ud835\udc4511 + \ud835\udf14\ud835\udc451) + (\ud835\udf14\ud835\udc4511 + \ud835\udf14\ud835\udc451 \u2212 4\ud835\udc3f\ud835\udc46\ud835\udc452)\ud835\udc3e\ud835\udc5f\ud835\udc3e\ud835\udc5f\ud835\udc3e\ud835\udc5f\ud835\udc3e\ud835\udc5f = (2\ud835\udc3f\ud835\udc46\ud835\udc45)(\u210e\ud835\udc47\ud835\udc45 + \ud835\udc3f\ud835\udc46\ud835\udc4524\u210e\ud835\udc47\ud835\udc452 + 1) \ud835\udc3f\ud835\udc5a \ud835\udc3f\ud835\udc46\ud835\udc45 \ud835\udc46\ud835\udc45 \ud835\udc3f\ud835\udc45 \ud835\udc38\ud835\udc3c\ud835\udc54 \u1d07 \ud835\udc38\ud835\udc34\ud835\udc54 \ud835\udf14\ud835\udc5d0.2 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 0.60.40.50.60.70.80.91\ud835\udf14\ud835\udc45 \ud835\udf14\ud835\udc45Peak 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 \u0733\u0bbf\u0bd4\u0be1 \u2245\u07e9\u0be6\u0be7\u07ea\u0be6\u0be7 \u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f\u072e\u0be0\u0b36 \u0d642 \u0d6c\u210e\u0bcd\u072e\u0be0\u0d70 +16 \u0d6c\u072e\u0be0\u210e\u0bcd\u0d70\u0d68 (5.1) \u0733\u0bc1\u0bd4\u0be5\u0be3 \u2245\u07e9\u0be6\u0be7\u07ea\u0be6\u0be7 \u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f\u072e\u0be0\u0b36 \u0d64\u0d6c\u210e\u0bcd\u072e\u0be0\u0d70 +14 \u0d6c\u072e\u0be0\u210e\u0bcd\u0d70\u0d68 (5.2) \u07e9\u0be6\u0be7 and \u07ea\u0be6\u0be7 are the density and design stress of the stay cable material; \u07f1 \u0be6 and \u07f1 \u0be3 are the magnitude of the uniformly distributed dead and live load; \u072e\u0be0 is the main span length; and \u210e\u0bcd 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 (\u210e\u0bcd: \u072e\u0be0). 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 0.500.01.02.03.04.0\u072a\u073d\u074e\u074c\u0728\u073d\u074a\u088e\u0880 \u0878\u0893\u2044\u087d\u08cb\u0899\u089a\u08cc\u0899\u089a \u0d6b\u08d3\u0899 + \u08d3\u0896\u0d6f\u0878\u0893\u0adb \t5.1 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 (Sch\u00fcller, 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 \u2018semi-fan\u2019 configuration which is picture in Figure 5.4. The only difference relative to the fan configuration is that the anchorage zone at the towers is extended downwards. Figure 5.4: The Semi-Fan Arrangement 5.1.2 Hangers In addition to the conventional vertical arrangement of hangers, a diagonal arrangement of hangers (Figure 5.5) has also been employed. Notwithstanding, the diagonal arrangement has only been applied in three vehicular bridges: the Severn Bridge (1966), the Bosporus Bridge (1973), and the Humber Bridge (1981). All three were designed by the same engineering firm, each serving as a model for the latter. (Kawada, 2010) 5.1 Cable Arrangement 159 The primary advantage of diagonal hangers is that they form a truss-like structure between the suspension cables and the superstructure which significantly suppresses strain-free deformations. Consequently, the stiffness of the cable system, and thus the bridge as a whole, is greatly increased. Nevertheless, diagonal hangers were first employed in the Severn Bridge for a different reason, as explained below. The suppression of strain-free deformations from live, wind, and other forms of loading causes a cyclic variation of forces in the hangers. Prior to the Severn Bridge, heavy space trusses were used exclusively in suspension bridges due to ongoing concerns regarding aerodynamic stability following the Tacoma Narrows Disaster in 1940. The Severn Bridge was the first bridge to use to a lightweight, streamlined box girder for the superstructure. Because of its reduced gravity stiffness, attention was directed towards enhancing structural damping. (Kawada, 2010) This was achieved by using diagonal hangers in conjunction with helical cables. The helical cables have a unique hysteresis which is activated by the cyclic variation of wind forces in the hangers. Unfortunately, it was not until near the opening of the Humber Bridge, that severe structural problems started to emerge with the hangers in the Severn Bridge. There were many contributing factors. Poor penetration of the hanger socketing material was observed, in addition to a lack of axial and angular alignment along the hanger pin centre axes (Flint & Smith, 1992). It also became evident that live load demands had reached levels close to three times the original design estimate (Bradley, 2010). Nevertheless, ironically, the foremost cause of the problems can be linked to the rationale behind the employment of the diagonal arrangement. Although advantageous from a structural damping perspective, the cyclic variation of forces in the hangers from live load severely reduced their fatigue life. Moreover, the slackening of the hangers made them particularly susceptible to wind-induced vibration which further exacerbated their fatigue. Prior to the Severn Bridge, wind-(Vertical Arrangement) (Diagonal Arrangement) Figure 5.5: Types of Longitudinal Arrangements for Hangers 5.2 Derivation of Bridge Quantities 160 induced vibration of hangers had never been observed. (Kawada, 2010) When studies were undertaken to replace the hanger system, it was found that by using a vertical arrangement the combined bending and axial stress range in the hangers could be reduced by roughly 50% at mid-span, and 85% at the quarter span. However, replacing the original hangers with a vertical arrangement would have required repositioning of the hanger clamps or, alternatively, the installation of new deck attachment stools. Furthermore, the deck and tower would have had to be substantially strengthened. For these reasons, the diagonal arrangement was retained. (Flint & Smith, 1992) Notwithstanding, following the Humber Bridge, the diagonal arrangement was completely abandoned in vehicular bridges. The reasons that led to the abandonment of the diagonal arrangement in suspension bridges are no less valid in regard to hybrid cable bridges. Consequently, the diagonal arrangement will not be considered when deriving the optimum proportions for a self-anchored discontinuous hybrid cable bridge. 5.2 Derivation of Bridge Quantities As previously mentioned, the accurate calculation of quantities required in principal bridge components is an exceedingly complex task. Numerous assumptions are required in order to generalize and simplify the calculations. In addition, extreme caution must be exercised in differentiating parameters which are arbitrary from those which have a prominent effect. Bearing this in mind, the estimation approach utilized in Gimsing & Georgakis (2012) for conventional cable bridges is adapted herein for the hybrid system. Inherent in the adopted approach are several general assumptions: 1. The governing loading scenarios for the principal bridge components are assumed to occur when either the entire bridge, or the entire main span, is loaded with dead and live load. Under these loading scenarios the majority of the applied load is transferred through the cable system and, on that account, the bending stiffness of the superstructure can be neglected. This assumption is particularly valid in modern cable bridges where the superstructure is often of slender construction. It allows principal bridge components to be considered isolated from the rest of the bridge when deriving their internal forces which greatly simplifies the calculations involved. 2. Secondary forms of loading (i.e. wind, temperature, earthquake, etc\u2026) 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. \u210e\u0bcd is the height of the towers above deck; \u073e is the height above deck of the first tower connection; and \u073d is the length of the array. Figure 5.6: Idealized Array of Stay Cables \u073d\u210e\u0bcd\u073e\t\u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f\u0754C\t\tTowerL\t\u0740\u0754(\u210e\u0bcd \u2212 \u073e)\u073d \u0754\t5.2 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 (\u07f1\u0be6 + \u07f1\u0be3) acting over a tributary length of \u0740\u0754, is given by the following expression: \u0740\u0736 = \u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f\u0da7\u1242\u1240\u210e\u0bcd \u2212 \u073e\u073d \u1241 \u0754 + \u073e\u1243\u0b36+ \u0754\u0b36\u1242\u1240\u210e\u0bcd \u2212 \u073e\u073d \u1241 \u0754 + \u073e\u1243\u0740\u0754 + 12\u0740\u0736\u07ea\u0be6\u0be7 \u07db\u0be6\u0be7\u0da8\u0d64\u0d6c\u210e\u0bcd \u2212 \u073e\u073d \u0d70\u0754 + \u073e\u0d68\u0b36+ \u0754\u0b36 (5.3) where \u07ea\u0be6\u0be7 and \u07db\u0be6\u0be7 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 \u0740\u0736 is isolated, the following equation is obtained, \u0740\u0736 =\u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f\u0da7\u1242\u1240\u210e\u0bcd \u2212 \u073e\u073d \u1241 \u0754 + \u073e\u1243\u0b36+ \u0754\u0b36\u1242\u1240\u210e\u0bcd \u2212 \u073e\u073d \u1241 \u0754 + \u073e\u1243 \u12481 \u221212\u07db\u0be6\u0be7\u07ea\u0be6\u0be7 \u0da7\u1242\u1240\u210e\u0bcd \u2212 \u073e\u073d \u1241 \u0754 + \u073e\u1243\u0b36+ \u0754\u0b36\u1249\u0740\u0754 (5.4) The quantity of cable steel in the infinitesimal segment, \u0740\u0733, will then be given by the density of the stay cable material, \u07e9\u0be6\u0be7, multiplied by the cable area required, multiplied by the length of the segment: \u0740\u0733 = \u07e9\u0be6\u0be7\u0740\u0736\u07ea\u0be6\u0be7\u0da8\u0d64\u0d6c\u210e\u0bcd \u2212 \u073e\u073d \u0d70\u0754 + \u073e\u0d68\u0b36+ \u0754\u0b36 (5.5) When Equation (5.4) is substituted in for \u0740\u0736, the quantity becomes, \u0740\u0733 = \u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f\u07e9\u0be6\u0be7\u07ea\u0be6\u0be7\u1242\u1240\u210e\u0bcd \u2212 \u073e\u073d \u1241 \u0754 + \u073e\u1243\u0b36+ \u0754\u0b36\u1242\u1240\u210e\u0bcd \u2212 \u073e\u073d \u1241 \u0754 + \u073e\u1243 \u12481 \u221212\u07db\u0be6\u0be7\u07ea\u0be6\u0be7 \u0da7\u1242\u1240\u210e\u0bcd \u2212 \u073e\u073d \u1241 \u0754 + \u073e\u1243\u0b36+ \u0754\u0b36\u1249\u0740\u0754 (5.6) Thereupon, the total quantity of cable steel in the array can be obtained by integrating \u0740\u0733 over the length of the array, \u0733 = \u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f\u07e9\u0be6\u0be7\u07ea\u0be6\u0be7 \u0db1\u1242\u1240\u210e\u0bcd \u2212 \u073e\u073d \u1241 \u0754 + \u073e\u1243\u0b36+ \u0754\u0b36\u1242\u1240\u210e\u0bcd \u2212 \u073e\u073d \u1241 \u0754 + \u073e\u1243 \u12481 \u221212\u07db\u0be6\u0be7\u07ea\u0be6\u0be7 \u0da7\u1242\u1240\u210e\u0bcd \u2212 \u073e\u073d \u1241 \u0754 + \u073e\u1243\u0b36+ \u0754\u0b36\u1249\u0bd4\u0b34\u0740\u0754 (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 (\u073e = \u210e\u0bcd). 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 (\u073e = 3\/4\u210e\u0bcd). 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: \u073d = 500m, \u07ea\u0bd4\u0bdf\u0bdf\u0be2\u0bea = 800MPa, \u07db\u0be6\u0be7\u0bd4\u0bec = 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 \u07db\u0be6\u0be7\u0bd4\u0bec equal to zero, at which point, the expression becomes, \u0733\u0bd4\u0be3\u0be3\u0be5\u0be2\u0beb =\u07e9\u0be6\u0be7\u07ea\u0be6\u0be7 \u073d\u0b36\u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f \u0d6c\u073d3\u210e\u0bcd\u0d70 \u12481 + 3 \u0d6c\u210e\u0bcd\u073d \u0d70\u0b36\u1249 (5.8) Note that if \u073d is replaced by \u072e\u0be0\/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 \u0735\u0741\u0749\u0745 \u2212 \u0728\u073d\u074a\u0728\u073d\u074a\u088e\u0880 (\u0adb\u0887)\u2044\u087d\u08cb\u0899\u089a\u08cc\u0899\u089a \u0d6b\u08d3\u0899 + \u08d3\u0896\u0d6f\u0887\u0adb\t\t5.2 Derivation of Bridge Quantities 164 Figure 5.8: The Influence of Self-Weight on Cable Steel Quantity in a Fan Arrangement Parameters: \u07ea\u0be6\u0be7 = 800MPa, \u07db\u0be6\u0be7 = 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 \u072e\u0bcb as the suspension ratio, \u210e\u0bcd\u0bcb as the tower height-to-main span ratio, \u072e\u0bcc\u0bcb as the side-to-main span length ratio, and \u072e\u0be0 as the main span length, the final expression may be written as, \u0733\u0be6\u0be7 =\u07e9\u0be6\u0be7\u07ea\u0be6\u0be7 \u072e\u0be0\u0b36\u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f \u1248(1 \u2212 \u072e\u0bcb)\u0b3712\u210e\u0bcd\u0bcb + (1 \u2212 \u072e\u0bcb)\u210e\u0bcd\u0bcb +23\u072e\u0bcc\u0bcb\u0b37\u210e\u0bcd\u0bcb + 2\u210e\u0bcd\u0bcb\u072e\u0bcc\u0bcb\u1249 (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, \u0723\u0bd6\u0be0 =1\u07db\u0bd6\u0be0\u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f \u07e6 \u072e\u0bcb (2 \u2212 \u072e\u0bcb)\u0da81 + 16 \u1240 \u210e\u0bcd\u0bcb2 \u2212 \u072e\u0bcb\u1241\u0b361 \u2212 \u07e6 \u0da81 + 16 \u1240 \u210e\u0bcd\u0bcb2 \u2212 \u072e\u0bcb\u1241\u0b36 (5.10) where as before, \u07e6 = \u07db\u0bd6\u0be0 \u072e\u0be08 \u07ea\u0bd6\u0be0\u210e\u0bcd\u0bcb Using Equation (5.10), the quantity of cable steel can then be expressed as, 0.00 0.10 0.20 0.30 0.40 0.500.960.970.980.991.00\u088e\u0880 (\u0adb\u0887)\u2044\u087d\u0887\u0896\u0896\u0898\u0895\u089e\u087d \t\t\u073d = 500\u0749\t\u073d = 400\u0749\t\u073d = 300\u0749\t\u073d = 200\u0749\t\u073d = 100\u0749\t5.2 Derivation of Bridge Quantities 165 \u0733 = \u07e9\u0bd6\u0be0\u0723\u0bd6\u0be0\u0d632\u072e\u0bd6\u0be0_\u0b35 + \u072e\u0bd6\u0be0_\u0b36\u0d67 (5.11) where \u07e9\u0bd6\u0be0 and \u0723\u0bd6\u0be0 are the density and area of the cable material. In addition, \u072e\u0bd6\u0be0_\u0b35 is the length of the cable in the stayed region (the value is multiplied by two because there are two stayed regions) and \u072e\u0bd6\u0be0_\u0b36 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 \u072e\u0bd6\u0be0_\u0b35 and \u072e\u0bd6\u0be0_\u0b36. The expression for the cable quantity then becomes, \u0733 = \u07e9\u0bd6\u0be0\u0723\u0bd6\u0be0 \u124e2\u0db1 \u0da81 + \u0d6c\u0740\u0755(\u0754)\u0740\u0754 \u0d70\u0b36\u0740\u0754 + \u0db1 \u0da81 + \u0d6c\u0740\u0755(\u0754)\u0740\u0754 \u0d70\u0b36\u0740\u0754\u0bc5\u0cd8(\u0b35\u0b3e\u0bc5\u0cc3)\u0b36\u0bc5\u0cd8(\u0b35\u0b3f\u0bc5\u0cc3)\u0b36\u0bc5\u0cd8(\u0b35\u0b3f\u0bc5\u0cc3)\u0b36\u0b34\u124f (5.12) The ordinates of the cable curve, represented by \u0755(\u0754), 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, \u072e\u0bd6\u0be0_\u0b35_\u0bd4\u0be3\u0be3\u0be5\u0be2\u0beb = \u0da8\u1246\u072e\u0be0(1 \u2212 \u072e\u0bcb)2 \u1247\u0b36+ \u0755\u0bbb\u0b36 (5.13) where \u0755\u0bbb 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 \u0755\u0bbb, then after substitution of Equations (4.9) and (4.11), Equation (5.13) reduces to, \u072e\u0bd6\u0be0_\u0b35_\u0bd4\u0be3\u0be3\u0be5\u0be2\u0beb =\u072e\u0be0(1 \u2212 \u072e\u0bcb)2 \u0da81 + \u210e\u0bcd\u0bcb\u0b36 \u0d6c 42 \u2212 \u072e\u0bcb\u0d70\u0b36 (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, \u072e\u0bd6\u0be0_\u0b36_\u0bd4\u0be3\u0be3\u0be5\u0be2\u0beb = \u0db1 \u12481 +12 \u0d6c\u0740\u0755(\u0754)\u0740\u0754 \u0d70\u0b36\u1249 \u0740\u0754\u0bc5\u0cd8(\u0b35\u0b3e\u0bc5\u0cc3)\u0b36\u0bc5\u0cd8(\u0b35\u0b3f\u0bc5\u0cc3)\u0b36 (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 \u072e\u0bd6\u0be0_\u0b36_\u0bd4\u0be3\u0be3\u0be5\u0be2\u0beb = \u072e\u0be0\u072e\u0bcb \u12481 +83 \u210e\u0bcd\u0bcb\u0b36 \u0d6c 12 \u2212 \u072e\u0bcb\u0d70\u0b36\u1249 (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), \u0733\u0bd4\u0be3\u0be3\u0be5\u0be2\u0beb = \u07e9\u0bd6\u0be0\u0723\u0bd6\u0be0\u072e\u0be0 \u0d66(1 \u2212 \u072e\u0bcb)\u0da81 + \u0d6c4\u210e\u0bcd\u0bcb2 \u2212 \u072e\u0bcb\u0d70\u0b36+ \u072e\u0bcb \u12481 +83 \u0d6c\u210e\u0bcd\u0bcb2 \u2212 \u072e\u0bcb\u0d70\u0b36\u1249\u0d6a (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: \u072e\u0be0 = 1000m, \u07ea\u0bd6\u0be0 = 800MPa, \u07db\u0bd6\u0be0 = 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, \u0733\u0bd6\u0be0 =1\u0743 \u072e\u0be0\u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f\u072e\u0bcb(2 \u2212 \u072e\u0bcb)\u07e6\u07df1 \u2212 \u07e6\u07df \u1248(1 \u2212 \u072e\u0bcb)\u07df + \u072e\u0bcb(\u07df\u0b36 + 5)6 \u1249 (5.18) where \u0743 is the standard acceleration due to gravity, and the additional dimensionless parameter, \u07df, is defined as, 0.00 0.10 0.20 0.30 0.40 0.500.980.991.001.011.02\u088e\u0880\u087e\u087d\u0887\u0896\u0896\u0898\u0895\u089e\u087d \t\t \u072e\u0bcb = 0.2 \u072e\u0bcb = 0.4 \u072e\u0bcb = 0.65.2 Derivation of Bridge Quantities 167 \u07df = \u0da81 + 16 \u0d6c \u210e\u0bcd\u0bcb2 \u2212 \u072e\u0bcb\u0d70\u0b36 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 \u2018cut\u2019 at the end of the stayed region, and the suspension cable has been \u2018cut\u2019 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 \u072a\u0bd6\u0be0\/ \u0bd6\u0738\u0be0 =\tHorizontal\/Vertical\treaction\tfrom\tsuspension\tcable\tin\tmain\tspan\t\t\u0733\u0bd6\u0be0\t=\tSuspension\tcable\tquantity\t\t\u0733\u0bd6\u0be6\t=\tAnchor\tcable\tquantity\t\t\u0733\u0be6\u0be7\u0be0\t=\tMain\tspan\tstay\tcable\tquantity\t\t\u0733\u0be6\u0be7\u0be6\t=\tSide\tspan\tstay\tcable\tquantity\t\t\u0743=\tStandard\tacceleration\tdue\tto\tgravity\t\t\u210e\u0bcd\t=\tTower\theight\t\t\u0734\u0be3=\tReaction\tat\tanchor\tpier\t\t\u07f1\u0be6\/\t\u07f1\u0be3\t=\tDead\/Live\tloading\t\t\u072e\u0bcb\t=\tSuspension\tratio\t\t\u072e\u0be0\t=\tMain\tspan\tlength\t\t\u072e\u0be6 =\tSide\tspan\tlength\u072e\u0be0(1 \u2212 \u072e\u0bcb) 6\u2044\u072e\u0be0(1 \u2212 \u072e\u0bcb) 2\u2044\u072e\u0be6\t\u072e\u0be6 3\u2044 \t\u210e\u0bcd\u072a\u0bd6\u0be0\t\u0733\u0bd6\u0be6\u07434 + \u0bd6\u0738\u0be0\t\u07f1\u0be6\u07f1\u0be3\u0734\u0be3\t\u0733\u0be6\u0be7\u0be0\u0743 2\u2044\u0733\u0be6\u0be7\u0be6\u0743 2\u2044 \t\u0733\u0bd6\u0be6\u07434 \tMain\tSpanSide\tSpan\tC\t\tTower\tL\t\t\u072a\u0bd6\u0be05.2 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, \u0734\u0be3 =\u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f\u072e\u0be0(1 \u2212 \u072e\u0bcb)\u0b368\u072e\u0bcc\u0bcb + \u072a\u0bd6\u0be0\u210e\u0bcd\u0bcb\u072e\u0bcc\u0bcb +\u0733\u0be6\u0be7\u0be0\u074312\u072e\u0bcc\u0bcb (1 \u2212 \u072e\u0bcb) \u2212\u0733\u0be6\u0be7\u0be6\u07436\u2212 12\u07f1\u0be6\u072e\u0be0\u072e\u0bcc\u0bcb \u2212\u0733\u0bd6\u0be6\u07434 (5.19) Vertical equilibrium at the anchor pier then gives the vertical component of the anchor cable chord tension, \u0bd6\u0738\u0be6 =\u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f\u072e\u0be0(1 \u2212 \u072e\u0bcb)\u0b368\u072e\u0bcc\u0bcb + \u072a\u0bd6\u0be0\u210e\u0bcd\u0bcb\u072e\u0bcc\u0bcb +\u0733\u0be6\u0be7\u0be0\u074312\u072e\u0bcc\u0bcb (1 \u2212 \u072e\u0bcb) \u2212\u0733\u0be6\u0be7\u0be6\u07436 \u221212\u07f1\u0be6\u072e\u0be0\u072e\u0bcc\u0bcb (5.20) Equation (4.9) can be substituted in for \u072a\u0bd6\u0be0 (during the substitution \u07f1\u0be6 must be replaced by \u07f1\u0be6 + \u07f1\u0be3 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, \u0bd6\u0736\u0be6 =124\u072e\u0bcc\u0bcb \u0d633\u072e\u0be0\u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f + 3\u0733\u0bd6\u0be0\u0743 + 2\u0733\u0be6\u0be7\u0be0\u0743(1 \u2212 \u072e\u0bcb) \u2212 4\u0733\u0be6\u0be7\u0be6\u0743(\u072e\u0bcc\u0bcb)\u2212 12\u072e\u0be0\u072e\u0bcc\u0bcb\u0b36\u07f1\u0be6\u0d67\u0da7\u072e\u0bcc\u0bcb\u0b36 + \u210e\u0bcd\u0bcb\u0b36\u210e\u0bcd\u0bcb (5.21) It then follows that the total quantity of cable steel in both anchor cables is, \u0733\u0bd6\u0be6 = 2\u07e9\u0bd6\u0be6 \u0bd6\u0736\u0be6\u07ea\u0bd6\u0be6 \u072e\u0be0\u0da7\u072e\u0bcc\u0bcb\u0b36 + \u210e\u0bcd\u0bcb\u0b36 (5.22) where \u07e9\u0bd6\u0be6 and \u07ea\u0bd6\u0be6 are the density and design stress of the anchor cable material. When expanded, Equation (5.22) becomes, \u0733\u0bd6\u0be6 =112\u07e9\u0bd6\u0be6\u07ea\u0bd6\u0be6 \u072e\u0be0\u0d633\u072e\u0be0\u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f + 3\u0733\u0bd6\u0be0\u0743 + 2\u0733\u0be6\u0be7\u0be0\u0743(1 \u2212 \u072e\u0bcb) \u2212 4\u0733\u0be6\u0be7\u0be6\u0743(\u072e\u0bcc\u0bcb)\u2212 12\u072e\u0be0\u072e\u0bcc\u0bcb\u0b36\u07f1\u0be6\u0d67 \u0d6c\u072e\u0bcc\u0bcb\u210e\u0bcd\u0bcb +\u210e\u0bcd\u0bcb\u072e\u0bcc\u0bcb\u0d70 (5.23)5.2 Derivation of Bridge Quantities 169 where from Equation (5.9), \u0733\u0be6\u0be7\u0be0 =\u07e9\u0be6\u0be7\u07ea\u0be6\u0be7 \u072e\u0be0\u0b36\u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f(1 \u2212 \u072e\u0bcb)\u210e\u0bcd\u0bcb \u1248(1 \u2212 \u072e\u0bcb)\u0b3612\u210e\u0bcd\u0bcb\u0b36+ 1\u1249 (5.24)and, \u0733\u0be6\u0be7\u0be6 =\u07e9\u0be6\u0be7\u07ea\u0be6\u0be7 \u072e\u0be0\u0b36\u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f\u210e\u0bcd\u0bcb\u072e\u0bcc\u0bcb \u124823 \u0d6c\u072e\u0bcc\u0bcb\u210e\u0bcd\u0bcb\u0d70\u0b36+ 2\u1249 (5.25) Note from Equation (5.23) that the contribution from \u0733 \u0be6\u0be7\u0be0 and \u0733 \u0be6\u0be7\u0be6 amount to zero when the stay cables are symmetrically arranged about the centre of the towers (i.e. when \u072e\u0bcc\u0bcb = (1 \u2212 \u072e\u0bcb)\/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 \u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f\u0755\u0bbb \u0742\u072e\u0be0(1 \u2212 \u072e\u0bcb) 2\u2044\u072e\u0be0(1 \u2212 \u072e\u0bcb) 2\u2044 \u072e\u0bdb\u072e\u0be0\u0755(\u0754)\t\u0754\t5.2 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, \u0733 = \u07e9\u0bdb\u0d6b\u0723\u0bdb_\u0be7\u0be2\u0be7\u0bd4\u0bdf\u0d6f\u0d6b\u072e\u0bdb_\u0bd4\u0be9\u0bda\u0d6f (5.26) When expanded, the expression becomes, \u0733 = \u07e9\u0bdb\u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f\u072e\u0bdb\u07ea\u0bdb \u0d6c\u210e\u0bcd \u2212 \u0755\u0bbb3 \u0d70 (5.27) where \u07ea\u0bdb is the design stress of the hangers. Substituting in Equations (4.9) and (4.11) then gives, \u0733 = 13\u07e9\u0bdb\u07ea\u0bdb \u072e\u0be0\u0b36\u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f\u210e\u0bcd\u0bcb\u072e\u0bcb\u0b37 \u0d66\u07f1\u0be6 + \u1240\u0733\u0bd6\u0be0\u0743\u072e\u0be0 \u1241\u07f1\u0be6\u072e\u0bcb(2 \u2212 \u072e\u0bcb) + \u1240\u0733\u0bd6\u0be0\u0743\u072e\u0be0 \u1241\u0d6a (5.28) However, a simpler expression can be obtained by neglecting the contribution from the self-weight of the suspension cable, \u0733\u0bd4\u0be3\u0be3\u0be5\u0be2\u0beb =13\u07e9\u0bdb\u07ea\u0bdb \u072e\u0be0\u0b36\u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f\u210e\u0bcd\u0bcb\u072e\u0bcb\u0b36 \u0d6c12 \u2212 \u072e\u0bcb\u0d70 (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 \u0733\u0bdb (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: \u072e\u0be0 = 1000m, \u07ea\u0bd6\u0be0 = 800MPa, \u07db\u0bd6\u0be0 = 0.09MN\/m3, \u07f1\u0bcb = 0.6 Figure 5.13: Hanger Cable Quantity in Relation to Total Cable Steel Quantity Parameters: \u072e\u0be0 = 1000m, \u07f1\u0bcb = 0.6 5.2.5 Tower Quantity The derivation of the tower quantity is particularly challenging. The required quantity strongly depends on both in-plane, and out-of-plane loading. For the in-plane loading, the towers must be capable of sustaining considerable axial and bending demands; each governed by different loading scenarios. Furthermore, the magnitude of the bending demands will depend on the geometry of the tower section as well as the articulation scheme for the superstructure; two parameters which are very difficult to generalize. Likewise, for out-of-plane loading, bending demands are no less difficult to quantify in general terms. 0.00 0.10 0.20 0.30 0.40 0.501.001.051.101.150.00 0.10 0.20 0.30 0.40 0.500.0000.0200.040\u088e\u0880\u087e\u087d\u0887\u0896\u0896\u0898\u0895\u089e\u087d \t\t\u072e\u0bcb = 0.2 \u072e\u0bcb = 0.4 \u072e\u0bcb = 0.6\t\u088e\u0880\u087e\u087d\u088e\u087d\u088e + \u087d\u0889\u0893 + \u087d\u0889\u0899 + \u087d\u0899\u089a\t\t\u072e\u0bcb = 0.2 \u072e\u0bcb = 0.4 \u072e\u0bcb = 0.6\t5.2 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, \u0be9\u0730 =\u0733\u0be6\u0be7\u0be0\u07432 \u0d6c1 +1 \u2212 \u072e\u0bcb6\u072e\u0bcc\u0bcb \u0d70 +\u0733\u0be6\u0be7\u0be6\u07433 + \u0bd6\u0738\u0be0 + \u072a\u0bd6\u0be0 \u0d6c\u210e\u0bcd\u0bcb\u072e\u0bcc\u0bcb\u0d70 + \u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f\u072e\u0be0\u072e\u0bcc\u0bcb2+ \u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f(1 \u2212 \u072e\u0bcb)\u072e\u0be02 \u0d6c1 +1 \u2212 \u072e\u0bcb4\u072e\u0bcc\u0bcb \u0d70 +\u0733\u0bd6\u0be6\u07434 (5.30) Substituting in the horizontal and vertical components of the suspension cable force then gives, \u0be9\u0730= 124\u072e\u0bcc\u0bcb \u0d633\u072e\u0be0\u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f(2\u072e\u0bcc\u0bcb + 1)\u0b36 + 2\u0733\u0be6\u0be7\u0be0\u0743(6\u072e\u0bcc\u0bcb \u2212 \u072e\u0bcb + 1) + 8\u0733\u0be6\u0be7\u0be6\u0743(\u072e\u0bcc\u0bcb)+ 3\u0733\u0bd6\u0be0\u0743(4\u072e\u0bcc\u0bcb + 1) + 6\u0733\u0bd6\u0be6\u0743(\u072e\u0bcc\u0bcb)\u0d67 (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, \u0738\u0bcd = \u0be9\u0730 + \u0be7\u0739(\u0756) (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, \u0756. The area required in each tower can be obtained as, \u0723\u0be7 = \u0be9\u0730+ \u0be7\u0739(\u0756)\u07da\u0be7\u07ea\u0be7 (5.33) where \u07ea\u0be7 is the design stress of the tower and \u07da\u0be7 is the reduction coefficient to account for the effect of out-of-plane loading (\u07da\u0be7 = 0.7). Notwithstanding, the self-weight of the tower is also a function of the tower area, \u0be7\u0739(\u0756) = \u0723\u0be7\u07db\u0be7\u0756 (5.34) where \u07db\u0be7 is the unit weight of the tower material. Substituting Equation (5.34) into Equation (5.33) results in, \u0723\u0be7 = \u0be9\u0730\u07da\u0be7\u07ea\u0be7 \u2212 \u07db\u0be7\u0756 (5.35) Accordingly, the total quantity of material required in both towers can be calculated as, \u0733\u0be7 = 2\u07e9\u0be7 \u0be9\u0730 \u0db1 \u0d6c1\u07da\u0be7\u07ea\u0be7 \u2212 \u07db\u0be7\u0756\u0d70\u0bdb\u0cc5\u0b3e\u0bdb\u0cb3\u0b34\u0740\u0756 (5.36)which then simplifies to, \u0733\u0be7 =2 \u0be9\u0730\u0743 \u0748\u074a \u0d6611 \u2212 (\u210e\u0bcd + \u210e\u0bbb)\u07db\u0be7\u07da\u0be7\u07ea\u0be7\u0d6a (5.37) The only parameter left to assign is \u210e\u0bbb 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: \uf0a7 The base of the towers is at water level; \uf0a7 The vertical profile of the roadway is parabolic; 5.2 Derivation of Bridge Quantities 174 \uf0a7 The vertical clearance required at mid-span is 50 metres; and \uf0a7 The vertical slope of the roadway cannot exceed 5%. Then, the minimum tower height below deck can be calculated as, \u210e\u0bbb = 50 \u0d6c1 \u2212\u072e\u0be04000\u0d70 [\u0745\u074a \u0749\u0741\u0750\u074e\u0741\u074f] (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,\t\u07f1\u0be6, an expression for the superstructure quantity can be readily obtained, \u0733\u0be6 =\u07f1\u0be6\u072e\u0be0\u0743\u07d9\u0be6\u0bd7\u0bdf (1 + 2\u072e\u0bcc\u0bcb) (5.39) However,\t\u07f1\u0be6 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, \u07d9\u0be6\u0bd7\u0bdf \u2265 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 (\u07f1\u0be6) and the magnitude of the live load (\u07f1\u0be3). 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 \u07f1\u0be6\u0be2 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, \u0394\u07f1\u0be6 = \u07db\u0be6[(\u0723\u0be6\u0be0 \u2212 \u0723\u0be6\u0be0\u0be2) + (\u0723\u0be6\u0bd4 \u2212 \u0723\u0be6\u0bd4\u0be2) + (\u0723\u0be6\u0bd5 \u2212 \u0723\u0be6\u0bd5\u0be2)] (5.40) where \u07db\u0be6 is the unit weight of the superstructure material. In addition, \u0723\u0be6\u0be0\/\u0723\u0be6\u0be0\u0be2 is the area required for miscellaneous transverse support members (i.e. floor beams, diaphragms, etc.), \u0723\u0be6\u0bd4\/\u0723\u0be6\u0bd4\u0be2 is the area required for longitudinal axial demands, and \u0723\u0be6\u0bd5\/\u0723\u0be6\u0bd5\u0be2 is the area required for longitudinal bending demands. Accordingly, when the longitudinal bridge proportions are varied (\u0723\u0be6\u0be0 \u2212 \u0723\u0be6\u0be0\u0be2) 5.2 Derivation of Bridge Quantities 175 can be assumed equal to zero, upon which, \u0394\u07f1\u0be6 = \u07db\u0be6[(\u0723\u0be6\u0bd4 \u2212 \u0723\u0be6\u0bd4\u0be2) + (\u0723\u0be6\u0bd5 \u2212 \u0723\u0be6\u0bd5\u0be2)] (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 \u0728\u0be6\u0b35 and \u0728\u0be3\u0b35 are, together, equal to the horizontal force in the suspension cable. Thus, \u0728\u0be6\u0b35 + \u0728\u0be3\u0b35 = \u072a\u0bd6\u0be0 (5.42) Substituting in the appropriate expression for \u072a\u0bd6\u0be0 then yields, \u0728\u0be6\u0b35 + \u0728\u0be3\u0b35 =18\u210e\u0bcd\u0bcb \u0d63\u0733\u0bd6\u0be0\u0743 + \u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f\u072e\u0bcb(2 \u2212 \u072e\u0bcb)\u0d67 (5.43) The uniform axial force components in the side span (\u0728\u0be6\u0b36\u0bd4 and \u0728\u0be3\u0b36) are a result of the anchorage force required to equilibrate the unbalanced portion of the dead and live load in the stayed regions of \u07f1\u0be3\u07f1\u0be6\t\u0728\u0be6\u0b37\u0728\u0be3\u0b37\u0728\u0be3\u0b36\t\u0728\u0be3\u0b35\t\u0728\u0be6\u0b35\t\u0728\u0be6\u0b36\u0bd4\t\u0728\u0be6\u0b36\u0bd5\tDead\tLoad\tAxial\tForcesLive\tLoad\tAxial\tForces5.2 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, \u0728\u0be6\u0b36\u0bd4 + \u0728\u0be3\u0b36 = \u0bd6\u0738\u0be6\u0d24\u0d24\u0d24\u0d24\u072e\u0bcc\u0bcb\u210e\u0bcd\u0bcb (5.44) where \u0bd6\u0738\u0be6\u0d24\u0d24\u0d24\u0d24 is the vertical component of the anchorage force. \u0bd6\u0738\u0be6\u0d24\u0d24\u0d24\u0d24 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, \u0728\u0be6\u0b36\u0bd4 + \u0728\u0be3\u0b36 =124\u210e\u0bcd\u0bcb \u0d633\u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f\u072e\u0be0(1 \u2212 \u072e\u0bcb)\u0b36 + 2\u0733\u0be6\u0be7\u0be0\u0743(1 \u2212 \u072e\u0bcb) \u2212 4\u0733\u0be6\u0be7\u0be6\u0743\u072e\u0bcc\u0bcb\u2212 12\u07f1\u0be6\u072e\u0be0\u072e\u0bcc\u0bcb\u0b36\u0d67 (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 \u0728\u0be6\u0b36\u0bd5. 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, \u0728\u0be6\u0b36\u0bd5\u0bd4 = \u0db1\u07f1\u0be6\u0754\u210e\u0bcd \u0740\u0754\u0beb\u0b34= 12\u07f1\u0be6\u0754\u0b36\u210e\u0bcd (5.46) where the origin for \u0754 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, \u0728\u0be6\u0b37 + \u0728\u0be3\u0b37 = \u0db1\u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f\u0754\u210e\u0bcd \u0740\u0754\u0beb\u0b34= 12\u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f\u0754\u0b36\u210e\u0bcd (5.47) where in this case, the origin for \u0754 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, \u0738\u0731\u072e\u0be6\u0bd4 =1\u07ea\u0be6 \u124e\u0d6b\u0728\u0be6\u0b35 + \u0728\u0be3\u0b35\u0d6f\u072e\u0be0(1 + 2\u072e\u0bcc\u0bcb) + 2\u0d6b\u0728\u0be6\u0b36\u0bd4 + \u0728\u0be3\u0b36\u0d6f(\u072e\u0bcc\u0bcb\u072e\u0be0)+ 2\u0db1 (\u0728\u0be6\u0b36\u0bd5\u0bd4)\u0740\u0754\u0bc5\u0cc4\u0cc3\u0bc5\u0cd8\u0b34+ 2\u0db1 \u0d6b\u0728\u0be6\u0b37 + \u0728\u0be3\u0b37\u0d6f\u0740\u0754\u0bc5\u0cd8(\u0b35\u0b3f\u0bc5\u0cc3)\u0b36\u0b34\u124f (5.48) which simplifies to, 5.2 Derivation of Bridge Quantities 177 \u0738\u0731\u072e\u0be6\u0bd4 =\u072e\u0be024\u210e\u0bcd\u0bcb\u07ea\u0be6 \u1242\u072e\u0749\u0d6b\u07f1\u074f + \u07f1\u074c\u0d6f\u0d6b1 \u2212 \u072e\u07343 + 3\u072e\u0734 + 6\u072e\u0735\u0734\u0d6f \u2212 16\u072e\u0749\u072e\u0735\u07343\u07f1\u074f+ 4\u0733\u074f\u0750\u0749\u0743\u072e\u0735\u0734(1 \u2212 \u072e\u0734) \u2212 8\u0733\u074f\u0750\u074f\u0743(\u072e\u0735\u0734)2 + 3\u0733\u073f\u0749\u0743(2\u072e\u0735\u0734 + 1)\u1243 (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), \u0723\u0be6\u0bd4 =\u0738\u0731\u072e\u0be6\u0bd4\u072e\u0be0(1 + 2\u072e\u0bcc\u0bcb) (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, \u0394\u07f1\u0be6 = \u07db\u074f(\u0723\u073d\u074f \u2212 \u0723\u074f\u073d\u074b) (5.51) and the dead load of the superstructure can be written as, \u07f1\u0be6 = \u07f1\u0be6\u0be2 + \u07db\u074f(\u0723\u073d\u074f \u2212 \u0723\u074f\u073d\u074b) (5.52) However, because \u07f1 \u0be6 depends on \u0723\u0bd4\u0be6, which is also a function of \u07f1 \u0be6, 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 \u07f1\u0be6, 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 \u07f1 \u0be6. 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, \u0733\u0be6 =\u072e\u0be0(1 + 2\u072e\u0bcc\u0bcb)\u0743 [\u07f1\u0be6 \u2212 (\u07d9\u0be6\u0bd7\u0bdf \u2212 1)\u07f1\u0be6\u0be2] (5.53) 5.2 Derivation of Bridge Quantities 178 5.2.8 Summary of Equations \u072e\u0be0\t=\tMain\tspan\tlength \u07f1\u0be6 =\tSuperstructure\tdead\tload\t \u072e\u0bcb\t=\tSuspension\tratio \u07f1\u0be6\u0be2 =\tReference\tsuperstructure\tdead\tload \u072e\u0bcc\u0bcb\t=\tSide-to-main\tspan\tlength\tratio \u07f1\u0be3 =\tLive\tload \u210e\u0bcd\u0bcb\t=\tTower\theight-to-span\tratio \u07e9\u0be0 \/\t\u07db\u0be0 =\tDensity\/Unit\tweight\tof\tmaterial\t\u2018m\u2019 \u07da\u0be7\t=\tFactor\tfor\tout-of-plane\tloading\ton\ttowers \u07ea\u0be0 =\tDesign\tstress\tof\tmaterial\t\u2018m\u2019\t \u07d9\u0be6\u0bd7\u0bdf\t=\tFactor\tfor\tsuperimposed\tdead\tload \u0743 =\tStandard\tacceleration\tdue\tto\tgravity\t Stay Cable Steel Quantity Main\tSpan\t \t \u0733\u0be6\u0be7\u0be0 =\u07e9\u0be6\u0be7\u07ea\u0be6\u0be7 \u072e\u0be0\u0b36\u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f(1 \u2212 \u072e\u0bcb)\u210e\u0bcd\u0bcb \u1248(1 \u2212 \u072e\u0bcb)\u0b3612\u210e\u0bcd\u0bcb\u0b36+ 1\u1249 (5.54) Side\tSpan\t \t \u0733\u0be6\u0be7\u0be6 =\u07e9\u0be6\u0be7\u07ea\u0be6\u0be7 \u072e\u0be0\u0b36\u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f\u210e\u0bcd\u0bcb\u072e\u0bcc\u0bcb \u124823 \u0d6c\u072e\u0bcc\u0bcb\u210e\u0bcd\u0bcb\u0d70\u0b36+ 2\u1249 (5.55) Suspension Cable Quantity \u0733\u0bd6\u0be0 =1\u0743 \u072e\u0749\u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f\u072e\u0bcb(2 \u2212 \u072e\u0bcb)\u07e6\u07df1 \u2212 \u07e6\u07df \u1248(1 \u2212 \u072e\u0bcb)\u07df + \u072e\u0bcb(\u07df\u0b36 + 5)6 \u1249 (5.56) where,\t \u07df = \u0da81 + 16 \u0d6c \u210e\u0736\u07342 \u2212 \u072e\u0bcb\u0d70\u0b36 and \u07e6 =\u07db\u0bd6\u0be0 \u072e\u0be08 \u07ea\u0bd6\u0be0\u210e\u0bcd\u0bcb Anchor Cable Quantity \u0733\u0bd6\u0be6 =112\u07e9\u0bd6\u0be6\u07ea\u0bd6\u0be6 \u072e\u0be0\u0d633\u072e\u0be0\u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f + 3\u0733\u0bd6\u0be0\u0743 + 2\u0733\u0be6\u0be7\u0be0\u0743(1 \u2212 \u072e\u0bcb) \u2212 4\u0733\u0be6\u0be7\u0be6\u0743(\u072e\u0bcc\u0bcb)\u2212 12\u072e\u0be0\u072e\u0bcc\u0bcb\u0b36\u07f1\u0be6\u0d67 \u0d6c\u072e\u0bcc\u0bcb\u210e\u0bcd\u0bcb +\u210e\u0bcd\u0bcb\u072e\u0bcc\u0bcb\u0d70 (5.57) Hanger Cable Quantity \u0733\u0bdb =13\u07e9\u0bdb\u07ea\u0bdb \u072e\u0be0\u0b36\u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f\u210e\u0bcd\u0bcb\u072e\u0bcb\u0b36 \u0d6c12 \u2212 \u072e\u0bcb\u0d70 (5.58)5.2 Derivation of Bridge Quantities 179 Tower Quantity \u0733\u0be7 =2 \u0be9\u0730\u0743 \u0748\u074a \u0d6611 \u2212 (\u210e\u0bcd\u0bcb\u072e\u0be0 + \u210e\u0bbb)\u07db\u0be7\u07da\u0be7\u07ea\u0be7\u0d6a (5.59) where,\t \u0be9\u0730 =124\u072e\u0bcc\u0bcb \u0d633\u072e\u0be0\u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f(2\u072e\u0bcc\u0bcb + 1)\u0b36 + 2\u0733\u0be6\u0be7\u0be0\u0743(6\u072e\u0bcc\u0bcb \u2212 \u072e\u0bcb + 1) + 8\u0733\u0be6\u0be7\u0be6\u0743(\u072e\u0bcc\u0bcb)+ 3\u0733\u0bd6\u0be0\u0743(4\u072e\u0bcc\u0bcb + 1) + 6\u0733\u0bd6\u0be6\u0743(\u072e\u0bcc\u0bcb)\u0d67 and,\t \u210e\u0bbb = 50 \u0d6c1 \u2212\u072e\u0be04000\u0d70 [\u0745\u074a\t\u0749\u0741\u0750\u074e\u0741\u074f] Superstructure Quantity \u0733\u0be6 =\u072e\u0be0(1 + 2\u072e\u0bcc\u0bcb)\u0743 [\u07f1\u0be6 \u2212 (\u07d9\u0be6\u0bd7\u0bdf \u2212 1)\u07f1\u0be6\u0be2] (5.60) Load Correction Equation \u07f1\u0be6 = \u07f1\u0be6\u0be2 + \u07db\u0be6(\u0723\u0be6\u0bd4 \u2212 \u0723\u0be6\u0bd4\u0be2) (5.61) where,\t \u0723\u0be6\u0bd4 =124\u210e\u0bcd\u0bcb\u07ea\u0be6(2\u072e\u0bcc\u0bcb + 1) \u0d63\u072e\u0be0\u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f\u0d6b1 \u2212 \u072e\u0bcb\u0b37 + 3\u072e\u0bcb + 6\u072e\u0bcc\u0bcb\u0d6f \u2212 16\u072e\u0be0\u072e\u0bcc\u0bcb\u0b37\u07f1\u0be6+ 4\u0733\u0be6\u0be7\u0be0\u0743\u072e\u0bcc\u0bcb(1 \u2212 \u072e\u0bcb) \u2212 8\u0733\u0be6\u0be7\u0be6\u0743(\u072e\u0bcc\u0bcb)\u0b36 + 3\u0733\u0bd6\u0be0\u0743(2\u072e\u0bcc\u0bcb + 1)\u0d67 and,\t \u0723\u0be6\u0bd4\u0be2 is computed using the same expression as \u0723\u0be6\u0bd4 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,\t\u0725\u0bc1, can be expressed as, \u0725\u0bc1 = \u073f\u0be6\u0be7(\u0733\u0be6\u0be7\u0be0 + \u0733\u0be6\u0be7\u0be6) + \u073f\u0bd6\u0be0(\u0733\u0bd6\u0be0 + \u0733\u0bd6\u0be6) + \u073f\u0bdb\u0733\u0bdb + \u073f\u0be7\u0733\u0be7 + \u073f\u0be6\u0733\u0be6 + \u0bd9\u073f\u0733\u0bd9 (5.62)where \u073f\u0be0 represents the unit cost of component \u2018m\u2019 and \u0733\u0bd9 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, \u0733 \u0bd9 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, \u0740\u0725\u0bc1\u0740\u210e\u0bcd\u0bcb = 0 (5.63)Neglecting the change in the foundation quantity, Equation (5.63) may also be written as, \u0740(\u0733\u0be6\u0be7\u0be0 + \u0733\u0be6\u0be7\u0be6)\u0740\u210e\u0bcd\u0bcb +\u073f\u0bd6\u0be0\u073f\u0be6\u0be7\u0740(\u0733\u0bd6\u0be0 + \u0733\u0bd6\u0be6)\u0740\u210e\u0bcd\u0bcb +\u073f\u0bdb\u073f\u0be6\u0be7\u0740\u0733\u0bdb\u0740\u210e\u0bcd\u0bcb +\u073f\u0be7\u073f\u0be6\u0be7\u0740\u0733\u0be7\u0740\u210e\u0bcd\u0bcb +\u073f\u0be6\u073f\u0be6\u0be7\u0740\u0733\u0be6\u0740\u210e\u0bcd\u0bcb = 0 (5.64)whereupon it also becomes clear that the optimal solution does not depend on the specific values assigned for the unit costs \u2013 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 (\u07f1\u0be3) and superstructure dead load (\u07f1\u0be6). 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, \u07f1\u0bcb\u0be2, which is defined as, 5.3 Span Proportions 181 \u07f1\u0bcb\u0be2 =\u07f1\u074c\u07f1\u074f\u074b (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)\u07db (kN\/m3) 24 77 24 77 90 90 90 \u07ea (MPa) 25 250 25 250 800 800 800 \u07da\u0be7 n\/a n\/a 0.7 0.7 n\/a n\/a n\/a \u07d9\u0be6\u0bd7\u0bdf 1.1 1.1 n\/a n\/a n\/a n\/a n\/a \u073f \u073f\u0be6\u0be7\u2044 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 (\u072e\u0bcb\u0be2 = 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 (\u210e\u0bcd\u0bcb\u0be2) and the live load ratio (\u07f1\u0bcb\u0be2). The tower height-to-span ratio is set at a conventional value for cable-stayed bridges, \u210e\u0bcd\u0bcb\u0be2 = 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 \u07f1\u0bcb\u0be2 = 0.6 and \u07f1\u0bcb\u0be2 = 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 (\u072e\u0bcb) and multiple side-to-main span 5.3 Span Proportions 182 length ratios (\u072e\u0bcc\u0bcb). 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: \u072e\u0be0 = 500m,\t\u072e\u0be0\u0be2 = \u072e\u0be0, \u072e\u0bcc\u0bcb\u0be2 = \u072e\u0bcc\u0bcb, \u210e\u0bcd\u0bcb\u0be2 = 0.25, also refer to Table 5.1 Based on Table 5.2, the optimum tower height-to-span ratio, \uf0a7 Decreases when steel is used in place of concrete for the towers; \uf0a7 Increases when steel is used in place of concrete for the superstructure; \uf0a7 Increases with increasing suspension ratio; and \uf0a7 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 \u0878\u087f\u087e \u0878\u087e 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: \u072e\u0be0 = 500m,\t\u072e\u0bcb = 0.4, \u072e\u0bcc\u0bcb = 0.3, \u072e\u0be0\u0be2 = \u072e\u0be0, \u072e\u0bcc\u0bcb\u0be2 = \u072e\u0bcc\u0bcb, \u210e\u0bcd\u0bcb\u0be2 = 0.25, also refer to Table 5.1 *Excludes foundation cost Another important observation from Figure 5.15 is that the hanger cost is inconsequential relative to the cost of the other components. This is also true when the value of the suspension ratio is increased. Thus, the cost of the hangers can be effectively negated in the calculation of the total cost. It is also of note that the optimum tower height-to-span ratio is fairly impervious to changes in the 0.1 0.2 0.3 0.4 0.500.20.40.60.811.21.40.1 0.2 0.3 0.4 0.500.20.40.60.811.21.40.1 0.2 0.3 0.4 0.500.20.40.60.811.21.40.1 0.2 0.3 0.4 0.500.20.40.60.811.21.4\u088e\u0880\u087e \u088e\u0880\u087e\t\u086f\u0874\u086f\u0874\u0895\t\u086f\u0874\u086f\u0874\u0895\t5.3 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:\t\u072e\u0bcb = 0.4, \u072e\u0bcc\u0bcb = 0.3, \u072e\u0be0\u0be2 = \u072e\u0be0, \u072e\u0bcc\u0bcb\u0be2 = \u072e\u0bcc\u0bcb, \u210e\u0bcd\u0bcb\u0be2 = 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 (\u07f1\u0bcb\u0be2, \u210e\u0bcd\u0bcb\u0be2), and the input factor which accounts for superimposed dead load (\u07d9\u0be6\u0bd7\u0bdf). The only exception is the design stress and unit cost of the suspension\/anchor cable which have a notable 200 400 600 800 10000.200.220.240.260.280.30\u088e\u0880\u087e_\u0895\u0896\u089a\t\u0878\u0893 (\u0893)5.3 Span Proportions 185 influence when the superstructure is composed of concrete. Tower Material (Concrete) (Steel) Superstructure Material (Concrete) Percent\tChange\tin\tOptimum\tTower\tHeight-to-Span\tRatio\t(Steel) Percent\tChange\tin\tOptimum\tTower\tHeight-to-Span\tRatio\t Pecent\tChange\tin\tInput\tParameters Pecent\tChange\tin\tInput\tParameters\t Input\tParameters: \u073f\u0bd6\u0be0 \u073f\u0be6\u0be7\u2044 \u073f\u0bdb \u073f\u0be6\u0be7\u2044 \u073f\u0be6 \u073f\u0be6\u0be7\u2044 \u073f\u0be7 \u073f\u0be6\u0be7\u2044 \u07f1\u0bcb\u0be2 \u07ea\u0be6\u0be7 \u07ea\u0bd6\u0be0 \u07ea\u0bdb \u07ea\u0be6 \u07da\u0be7\u07ea\u0be7 \u210e\u0bcd\u0bcb\u0be2 \u07e9\u0be6\u0be7 \u07e9\u0bd6\u0be0 \u07e9\u0bdb \u07d9\u0be6\u0bd7\u0bdf Figure 5.17: Sensitivity of Optimum Tower Height-to-Span Ratio Parameters: \u072e\u0be0 = 500m,\t\u072e\u0bcb = 0.4, \u072e\u0bcc\u0bcb = 0.3, \u072e\u0be0\u0be2 = \u072e\u0be0, \u072e\u0bcc\u0bcb\u0be2 = \u072e\u0bcc\u0bcb, \u210e\u0bcd\u0bcb\u0be2 = 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\u2212 10\u2212 0 10 2010\u22125\u2212051020\u2212 10\u2212 0 10 2010\u22125\u2212051020\u2212 10\u2212 0 10 2010\u22125\u2212051020\u2212 10\u2212 0 10 2010\u22125\u221205105.3 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, \u0bd9\u0730 = \u0be9\u0730 +12\u0733\u0be7\u0743 (5.66)where \u0730 \u0be9 is the vertical force at the top of the towers from the cable system (Equation (5.31)), and \u0733 \u0be7 is the tower quantity. The relationship between \u0bd9\u0730 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: \u072e\u0be0 = 500m,\t\u072e\u0bcb = 0.4, \u072e\u0bcc\u0bcb = 0.3, \u072e\u0be0\u0be2 = \u072e\u0be0, \u072e\u0bcc\u0bcb\u0be2 = \u072e\u0bcc\u0bcb, \u210e\u0bcd\u0bcb\u0be2 = 0.25, also refer to Table 5.1 It can be seen from Figure 5.18 that when the tower height-to-span ratio is varied within the optimal range the vertical force on the foundation varies by less than 5%. Still, the tower foundation cost generally constitutes a large portion of the overall bridge cost. Therefore, for cases when the tower is composed of concrete, it would be prudent to slightly reduce the value of the optimum tower height-to-span ratio reported in Table 5.2. Moreover, there are other justifications for reducing the tower height-to-span ratio. Aesthetically, lofty towers can be overly striking on most landscapes and, structurally, nonlinear effects and tower bending moments will increase with increasing tower height. Ultimately, for the aforementioned reasons, a tower height-to-span ratio in the range of 0.20-0.25 is 0.20 0.25 0.300.951.001.05\u087a\u0de1\u088c\t\u088e\u0880\u087e5.3 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, \u2018Optimizing 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\u2019 (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, \u0734\u0be3 \u2245\u07f1\u0be6\u072e\u0be08\u072e\u0bcc\u0bcb \u0d631 + \u07f1\u0bcb \u2212 4\u072e\u0bcc\u0bcb\u0b36\u0d67 (5.67) The relationship between the side-to-main span ratio (\u072e\u0bcc\u0bcb) and the uplift force (\u0734\u0be3) is plotted in Figure 5.19. The ordinates of the plot are normalized so that they are independent of the superstructure dead load (\u07f1\u0be6), and the main span length (\u072e\u0be0). 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 \u07f1 \u0be6\u0be6 and \u07f1 \u0be6\u0be0 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 0.500.00.51.01.52.0\u087e\u0896\u08d3\u0899\u0878\u0893\t\u0878\u087f\u087e \u07f1\u0bcb = 0.2 \u07f1\u0bcb = 0.4 \u07f1\u0bcb = 0.6 \u07f1\u0bcb = 0 5.3 Span Proportions 189 \u0734\u0be3 \u2245\u07f1\u0be6\u0be0\u072e\u0be08\u072e\u0bcc\u0bcb \u0d641 + \u07f1\u0bcb\u0be0 \u2212 4\u072e\u0bcc\u0bcb\u0b36 \u0d6c\u07f1\u0be6\u0be6\u07f1\u0be6\u0be0\u0d70\u0d68 (5.68) where \u07f1 \u0bcb\u0be0 is the unfactored live load ratio for the main span only (\u07f1\u0bcb\u0be0 = \u07f1 \u0be3 \u07f1\u0be6\u0be0\u2044 ). Solving for the superstructure dead load ratio (\u07f1\u0be6\u0be6 \u07f1\u0be6\u0be0\u2044 ) when \u0734 \u0be3 equals zero then gives the balancing condition as, \u07f1\u0be6\u0be6\u07f1\u0be6\u0be0 \u22451 + \u07f1\u0bcb\u0be04\u072e\u0bcc\u0bcb\u0b36 (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 0.501.02.03.04.05.06.07.08.09.010.0\u08d3\u0899\u0899\u08d3\u0899\u0893\t\u0878\u087f\u087e \u07f1\u0bcb\u0be0 = 0.2\u07f1\u0bcb\u0be0 = 0.4\u07f1\u0bcb\u0be0 = 0.6\u07f1\u0bcb\u0be0 = 0 \u07f1\u0be6\u0be0\u07f1\u0be3 \u072e\u0be0\u072e\u0bcc\u0bcb\u072e\u0be0 \u072e\u0bcc\u0bcb\u072e\u0be00 0 \u07f1\u0be6\u0be6 \u07f1\u0be6\u0be6\u07f1\u0bcb\u0be0 = \u07f1\u0be3 \u07f1\u0be6\u0be0\u2044 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, \u0bd9\u0730 \u2245\u072e\u0be08\u072e\u0bcc\u0bcb \u0d6b\u07f1\u0be6 + \u07f1\u0be3\u0d6f(2\u072e\u0bcc\u0bcb + 1)\u0b36\u039b (5.70)where, \u039b = 1 + \u0748\u074a \u0d66 11 \u2212 (\u210e\u0bcd\u0bcb\u072e\u0be0 + \u210e\u0bbb)\u07db\u0be7\u07da\u0be7\u07ea\u0be7\u0d6a Since all of the parameters can be assumed independent of \u072e\u0bcc\u0bcb, 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 0.501.01.21.31.51.61.8\u087a\u088c\u0878\u0893\u0d6b\u08d3\u0899 + \u08d3\u0896\u0d6f\u08ab\t\u0878\u087f\u087e5.3 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 \u2013 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, \u072e\u0bcc\u0bcb\u072e\u0be0\t\u072e\u0bcc\u0bcb\u072e\u0be0\t\u072e\u0be0(1 \u2212 \u072e\u0bcb\u0b35) 2\u2044\u072e\u0be0(1 \u2212 \u072e\u0bcb\u0b36) 2\u2044=\t>\tBridge 1 Bridge 2 5.3 Span Proportions 193 \u072e\u0bcb \u2264 1 \u2212 2\u072e\u0bcc\u0bcb (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 \u2264 \u072e\u0bcb \u2264 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 \u2264 \u072e\u0bcc\u0bcb_\u0be2\u0be3\u0be7 \u2264 0.4 0.2 \u2264 \u072e\u0bcb_\u0be2\u0be3\u0be7 \u2264 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, (\u072a\u0bd6\u0be0)\u0bbd\u0bc5 \u2245\u072e\u0be0\u07f1\u0be68\u0735\u0bcb \u0d64\u07f1\u0bd6\u0be0\u07f1\u0be6 + \u072e\u0bcb(2 \u2212 \u072e\u0bcb)\u0d68 (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 0.301.001.101.201.301.401.50\u0d6b\u0874\u0de1\u0889\u0893\u0d6f\u0870\u0878\t\u0878\u087e5.4 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: \u210e\u0bcd\u0bcb = 0.25,\t\u072e\u0bcb = 0.3, \u072e\u0bcc\u0bcb = 0.35, \u072e\u0be0\u0be2 = \u072e\u0be0, \u072e\u0bcc\u0bcb\u0be2 = \u072e\u0bcc\u0bcb, \u210e\u0bcd\u0bcb\u0be2 = \u210e\u0bcd\u0bcb, also refer to Table 5.1 *Difference in foundation cost assumed negligible (refer to Equation (5.70)) Based on Equation (5.62), it can be observed from Figure 5.24 that the total cost of a self-anchored discontinuous hybrid cable bridge is slightly greater than the cost which would be incurred by a conventional cable-stayed bridge. Notwithstanding, considering that it is generally uneconomic to employ concrete superstructures for relatively long spans, the percent change in cost between the two systems can be expected to be less than 5% for any practical span length. This result should not be interpreted to underrate the potential economic advantages of self-anchored discontinuous hybrid cable bridges. There are many factors which are not accounted for in Equation (5.62). Primarily, in the hybrid system, Equation (5.62) treats the uniform compression force transferred to the superstructure by the suspension cable as a disadvantage because it assumes additional200 400 600 800 10000246810200 400 600 800 10000246810200 400 600 800 10000246810200 400 600 800 10000246810\u0878\u0893(\u0893) \u0878\u0893(\u0893)\t%\t\u086f\u088e\u0887\u0894\u088d\u088b\t%\t\u086f\u088e\u0887\u0894\u088d\u088b\t5.5 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. \ud835\udc38\ud835\udc34\ud835\udc54\ud835\udc38\ud835\udc3c\ud835\udc54\ud835\udc38\ud835\udc34\ud835\udc61\ud835\udc38\ud835\udc3c\ud835\udc61 \ud835\udc39\ud835\udc56\ud835\udc65\ud835\udc52\ud835\udc51 \ud835\udc46\ud835\udc62\ud835\udc5d\ud835\udc5d\ud835\udc5c\ud835\udc5f\ud835\udc61 (TYP) \ud835\udc45\ud835\udc5c\ud835\udc59\ud835\udc59\ud835\udc52\ud835\udc5f \ud835\udc46\ud835\udc62\ud835\udc5d\ud835\udc5d\ud835\udc5c\ud835\udc5f\ud835\udc61 (TYP) \ud835\udc3f\ud835\udc5a 50 (\ud835\udc47\ud835\udc4c\ud835\udc43) \ud835\udc38\ud835\udc34\ud835\udc54,\ud835\udc38\ud835\udc3c\ud835\udc54 \ud835\udc3f\ud835\udc5a 500 \ud835\udf14\ud835\udc5d = 75\ud835\udc58\ud835\udc41\/\ud835\udc5a \ud835\udc49\ud835\udc4e\ud835\udc5f\ud835\udc56\ud835\udc4f\ud835\udc59\ud835\udc52 37.5\ud835\udc5a \ud835\udc49\ud835\udc4e\ud835\udc5f\ud835\udc56\ud835\udc4f\ud835\udc59\ud835\udc52 0.36(\ud835\udc3f\ud835\udc5a) 0.36(\ud835\udc3f\ud835\udc5a) 0.28(\ud835\udc3f\ud835\udc5a) 0.225(\ud835\udc3f\ud835\udc5a) \ud835\udf14\ud835\udc60 = \ud835\udf14\ud835\udc5d 0.6 \ud835\udc38\ud835\udc34\ud835\udc61,\ud835\udc38\ud835\udc3c\ud835\udc61 \ud835\udc3f\ud835\udc5a = 1000\ud835\udc5a 0.36(\ud835\udc3f\ud835\udc5a) 0.36(\ud835\udc3f\ud835\udc5a) \ud835\udc47\ud835\udc4e\ud835\udc4f\ud835\udc59\ud835\udc52 6.2 \ud835\udc65 \ud835\udc66 Hanger (TYP) Stay Cable (TYP) Main Span Suspension Cable Anchor Cable \ud835\udf14\ud835\udc5d \ud835\udf14\ud835\udc60 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\u00b7m) 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\u00b7m) AS1 AS2 AS3 \u0197AS2 Deflections (m) Axial Forces (MN) Shear Forces (MN) Moments (MN\u00b7m) Cable Stress Range (MN) Cable Stress Range \u0197 (MN) \ud835\udc3b\ud835\udc66\ud835\udc4f\ud835\udc5f\ud835\udc56\ud835\udc51 \ud835\udc35\ud835\udc5f\ud835\udc56\ud835\udc51\ud835\udc54\ud835\udc52 \ud835\udc36\ud835\udc5c\ud835\udc5b\ud835\udc63\ud835\udc52\ud835\udc5b\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\ud835\udc4e\ud835\udc59 \ud835\udc36\ud835\udc4e\ud835\udc4f\ud835\udc59\ud835\udc52 \ud835\udc46\ud835\udc61\ud835\udc4e\ud835\udc66\ud835\udc52\ud835\udc51 \ud835\udc35\ud835\udc5f\ud835\udc56\ud835\udc51\ud835\udc54\ud835\udc52 Main Span Side Span Deflections (m) Shear Forces (MN) Moments (MN\u00b7m) AS1 AS2 AS3 \ud835\udc36\ud835\udc5c\ud835\udc5b\ud835\udc63\ud835\udc52\ud835\udc5b\ud835\udc61\ud835\udc56\ud835\udc5c\ud835\udc5b\ud835\udc4e\ud835\udc59 \ud835\udc36\ud835\udc4e\ud835\udc4f\ud835\udc59\ud835\udc52 \ud835\udc46\ud835\udc61\ud835\udc4e\ud835\udc66\ud835\udc52\ud835\udc51 \ud835\udc35\ud835\udc5f\ud835\udc56\ud835\udc51\ud835\udc54\ud835\udc52 \ud835\udc3b\ud835\udc66\ud835\udc4f\ud835\udc5f\ud835\udc56\ud835\udc51 \ud835\udc35\ud835\udc5f\ud835\udc56\ud835\udc51\ud835\udc54\ud835\udc52 \ud835\udc38\ud835\udc3c\ud835\udc54 \ud835\udc38\ud835\udc3c\ud835\udc61\ud835\udc38\ud835\udc34\ud835\udc50\uf0a7 \uf0a7 \uf0a7 \u0197 With Cable Clamp Without Cable Clamp Span Cable Stress Range (MN) Cable Stress Range \u0197 (MN) Cross Stay (TYP) Anchor Cable Stay Cable (TYP) Hanger (TYP) \ud835\udc65\ud835\udc5d\ud835\udc56\ud835\udc3f\ud835\udc60\ud835\udc65\ud835\udc5d\ud835\udc56 \ud835\udc3f\ud835\udc60 \ud835\udc65\ud835\udc5d\ud835\udc56 \ud835\udc3f\ud835\udc60 \ud835\udc65\ud835\udc5d\ud835\udc56 \ud835\udc3f\ud835\udc60 \ud835\udc65\ud835\udc5d\ud835\udc56 \ud835\udc3f\ud835\udc60 \ud835\udc65\ud835\udc5d\ud835\udc56 \ud835\udc3f\ud835\udc60 \ud835\udc65\ud835\udc5d\ud835\udc56 \ud835\udc3f\ud835\udc60 \ud835\udc65\ud835\udc5d\ud835\udc56 \ud835\udc3f\ud835\udc60 \u0197AS2 Deflections (m) Axial Forces (MN) Shear Forces (MN) Moments (MN\u00b7m) Cable Stress Range (MN) Cable Stress Range \u0197 (MN) \ud835\udc65\ud835\udc5d\ud835\udc56 \ud835\udc3f\ud835\udc60 = 0 \ud835\udc65\ud835\udc5d\ud835\udc56 \ud835\udc3f\ud835\udc60 = 0.5 \ud835\udc65\ud835\udc5d\ud835\udc56 \ud835\udc3f\ud835\udc60 = 0.75 \ud835\udc65\ud835\udc5d\ud835\udc56 \ud835\udc3f\ud835\udc60 = 0.25 \ud835\udc73\ud835\udc94 \ud835\udc99\ud835\udc91\ud835\udc8a Main Span Side Span Deflections (m) Shear Forces (MN) Moments (MN\u00b7m) AS1 AS2 AS3 \ud835\udc65\ud835\udc5d\ud835\udc56 \ud835\udc3f\ud835\udc60 = 0.5 \ud835\udc65\ud835\udc5d\ud835\udc56 \ud835\udc3f\ud835\udc60 = 0.75 \ud835\udc65\ud835\udc5d\ud835\udc56 \ud835\udc3f\ud835\udc60 = 0.25 \ud835\udc65\ud835\udc5d\ud835\udc56 \ud835\udc3f\ud835\udc60 = 0 3 Equal Spaces 5 Equal Spaces 4 Equal Spaces 3 Equal Spaces 5 Equal Spaces 4 Equal Spaces \u0197AS2 Deflections (m) Axial Forces (MN) Shear Forces (MN) Moments (MN\u00b7m) Cable Stress Range (MN) Cable Stress Range \u0197 (MN) \ud835\udc41\ud835\udc5c \ud835\udc3c\ud835\udc5b\ud835\udc61.\ud835\udc43\ud835\udc56\ud835\udc52\ud835\udc5f 3 \ud835\udc3c\ud835\udc5b\ud835\udc61.\ud835\udc43\ud835\udc56\ud835\udc52\ud835\udc5f\ud835\udc60 4 \ud835\udc3c\ud835\udc5b\ud835\udc61.\ud835\udc43\ud835\udc56\ud835\udc52\ud835\udc5f\ud835\udc60 2 \ud835\udc3c\ud835\udc5b\ud835\udc61.\ud835\udc43\ud835\udc56\ud835\udc52\ud835\udc5f\ud835\udc60 Main Span Side Span Deflections (m) Shear Forces (MN) Moments (MN\u00b7m) AS1 AS2 AS3 \ud835\udc41\ud835\udc5c \ud835\udc3c\ud835\udc5b\ud835\udc61.\ud835\udc43\ud835\udc56\ud835\udc52\ud835\udc5f 3 \ud835\udc3c\ud835\udc5b\ud835\udc61.\ud835\udc43\ud835\udc56\ud835\udc52\ud835\udc5f\ud835\udc60 4 \ud835\udc3c\ud835\udc5b\ud835\udc61.\ud835\udc43\ud835\udc56\ud835\udc52\ud835\udc5f\ud835\udc60 2 \ud835\udc3c\ud835\udc5b\ud835\udc61.\ud835\udc43\ud835\udc56\ud835\udc52\ud835\udc5f\ud835\udc60 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 (\ud835\udc3b\ud835\udc50\ud835\udc5a)\ud835\udc37\ud835\udc3f \u2245\ud835\udc3f\ud835\udc5a\ud835\udf14\ud835\udc60\ud835\udc5a8\ud835\udc46\ud835\udc45[\ud835\udf14\ud835\udc50\ud835\udc5a\ud835\udf14\ud835\udc60\ud835\udc5a+ \ud835\udc3f\ud835\udc45(2 \u2212 \ud835\udc3f\ud835\udc45)]\ud835\udf14\ud835\udc60\ud835\udc5a \ud835\udf14\ud835\udc60\ud835\udf14\ud835\udc60\ud835\udc5a \ud835\udf14\ud835\udc60\ud835\udc60\ud835\udc3f\ud835\udc5a \ud835\udc46\ud835\udc45 \ud835\udc3f\ud835\udc45 \ud835\udf14\ud835\udc50\ud835\udc3b\ud835\udc46\ud835\udc45 \u2245 \ud835\udf07\ud835\udc49\ud835\udc5d\ud835\udc53\ud835\udf07 \ud835\udc49\ud835\udc5d\ud835\udc53\ud835\udc49\ud835\udc5d\ud835\udc53\ud835\udc3f\ud835\udc45 1 \u2212 2\ud835\udc3f\ud835\udc46\ud835\udc45\ud835\udc49\ud835\udc5d\ud835\udc53 \u2245 (\ud835\udf14\ud835\udc60\ud835\udc60 \u2212 \ud835\udf14\ud835\udc60\ud835\udc5a)\ud835\udc3f\ud835\udc46\ud835\udc45\ud835\udc3f\ud835\udc5a \u2212 (\ud835\udc3b\ud835\udc50\ud835\udc5a)\ud835\udc37\ud835\udc3f (\ud835\udc46\ud835\udc45\ud835\udc3f\ud835\udc46\ud835\udc45)\ud835\udc3f\ud835\udc46\ud835\udc45\ud835\udc3b\ud835\udc46\ud835\udc45(\ud835\udc3b\ud835\udc50\ud835\udc5a)\ud835\udc37\ud835\udc3f= \ud835\udf072\ud835\udc46\ud835\udc45(1 \u2212 \ud835\udc3f\ud835\udc45)[\ud835\udf14\ud835\udc50\ud835\udc5a\ud835\udf14\ud835\udc60\ud835\udc5a+ (1 \u2212 \ud835\udc3f\ud835\udc45)2 (1 \u2212 2\ud835\udf14\ud835\udc60\ud835\udc60\ud835\udf14\ud835\udc60\ud835\udc5a) + 1\ud835\udf14\ud835\udc50\ud835\udc5a\ud835\udf14\ud835\udc60\ud835\udc5a\u2212 (1 \u2212 \ud835\udc3f\ud835\udc45)2 + 1]\ud835\udf14\ud835\udc50Vertical Component of Suspension Cable Force Surplus Dead Load from Side Span Superstructure \ud835\udc3f\ud835\udc5a \ud835\udc3f\ud835\udc46\ud835\udc45 \ud835\udc3f\ud835\udc45 \ud835\udf14\ud835\udc45 \ud835\udefe\ud835\udc50 \ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc640.20 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 0.300.01.02.03.04.0\ud835\udc46\ud835\udc45 = 0.25 \ud835\udc46\ud835\udc45 = 0.225 \ud835\udc46\ud835\udc45 = 0.20 \ud835\udc73\ud835\udc79 \ud835\udc73\ud835\udc79 \ud835\udc73\ud835\udc79 \ud835\udc6f\ud835\udc7a\ud835\udc79(\ud835\udc6f\ud835\udc84\ud835\udc8e)\ud835\udc6b\ud835\udc73 \ud835\udc6f\ud835\udc7a\ud835\udc79(\ud835\udc6f\ud835\udc84\ud835\udc8e)\ud835\udc6b\ud835\udc73 \ud835\udc6f\ud835\udc7a\ud835\udc79(\ud835\udc6f\ud835\udc84\ud835\udc8e)\ud835\udc6b\ud835\udc73 \ud835\udf41 = \ud835\udfce.\ud835\udfd3 \ud835\udf41 = \ud835\udfce.\ud835\udfd1 \ud835\udf41 = \ud835\udfce.\ud835\udfd5 \ud835\udc3f\ud835\udc5a \ud835\udc3f\ud835\udc46\ud835\udc45 \ud835\udc3f\ud835\udc45 \ud835\udf14\ud835\udc45 \ud835\udefe\ud835\udc50 \ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc640.20 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 0.301.02.03.04.05.0\ud835\udc46\ud835\udc45 = 0.25 \ud835\udc46\ud835\udc45 = 0.225 \ud835\udc46\ud835\udc45 = 0.20 \ud835\udc73\ud835\udc79 \ud835\udc73\ud835\udc79 \ud835\udc73\ud835\udc79 \ud835\udf4e\ud835\udc94\ud835\udc94\ud835\udf4e\ud835\udc94\ud835\udc8e \ud835\udf4e\ud835\udc94\ud835\udc94\ud835\udf4e\ud835\udc94\ud835\udc8e \ud835\udf4e\ud835\udc94\ud835\udc94\ud835\udf4e\ud835\udc94\ud835\udc8e \ud835\udf41 = \ud835\udfce.\ud835\udfd3 \ud835\udf41 = \ud835\udfce.\ud835\udfd1 \ud835\udf41 = \ud835\udfce.\ud835\udfd5 \uf0a7 \uf0a7 \uf0a7 \ud835\udc3f\ud835\udc45 \ud835\udc3f\ud835\udc5a \ud835\udefe\ud835\udc50 \ud835\udf0e\ud835\udc4e\ud835\udc59\ud835\udc59\ud835\udc5c\ud835\udc640.20 0.22 0.24 0.26 0.28 0.3024578\ud835\udc46\ud835\udc45 = 0.25 \ud835\udc46\ud835\udc45 = 0.225 \ud835\udc46\ud835\udc45 = 0.20 \ud835\udc73\ud835\udc79 \ud835\udc68\ud835\udc84 (\ud835\udc94\ud835\udc96\ud835\udc94\ud835\udc91\ud835\udc86\ud835\udc8f\ud835\udc94\ud835\udc8a\ud835\udc90\ud835\udc8f \ud835\udc83\ud835\udc93\ud835\udc8a\ud835\udc85\ud835\udc88\ud835\udc86)\ud835\udc68\ud835\udc84 (\ud835\udc89\ud835\udc9a\ud835\udc83\ud835\udc93\ud835\udc8a\ud835\udc85 \ud835\udc83\ud835\udc93\ud835\udc8a\ud835\udc85\ud835\udc88\ud835\udc86) 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 \u073a (longitudinal), \u073b (vertical), and \u073c (transverse) directions by the vectors \u0394\u073a\u0bcd, \u0394\u073b\u0bcd , and \u0394\u073c\u0bcd 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 \u0745, 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 \u0754\u2217 and \u0755 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. \u0b35\u0730\u2026\u0be1 Nodal Numbering Scheme \u0394X\u0b58 Projected Length of Cable in \u2018x\u2019 Direction\u0b35\u0735\u2026\u0be1 Segment Numbering Scheme \u0394Y\u0b58 Projected Length of Cable in \u2018y\u2019 Direction\u07f1\u0bd6 \u0197Self-Weight of Suspension Cable \u0394Z\u0b58 Projected Length of Cable in \u2018z\u2019 Direction\u0728\u210e\u0bdc Hanger Force Acting at Node i \u074a Number of Cable Segments \u0197Not Shown for Clarity Figure A1: 3D View of a Suspension Cable \u073b\u073c\u0728\u210e\u0b36\u0b35\u0730\u07c2\u073a\u0bcd \u0b35\u0735\u07c2\u073c\u0bcd\u07c2\u073b\u0bcd \u0728\u210e\u0b37\u0728\u210e\u0bdc\u0b3f\u0b35\u0728\u210e\u0bdc\u0728\u210e\u0bdc\u0b3e\u0b35\u0728\u210e\u0be1\u0b37\u0730\u0bdc\u0730\u0b3f\u0b35\u0bdc\u0730\u0bdc\u0730\u0b3e\u0b35\u0be1\u0730 \u0735\u0b36\u0735\u0b37\u0bdc\u0735\u0b3f\u0b35\u0bdc\u0735\u0bdc\u0735\u0b3e\u0b35\u0735\u0be1 \u0b36\u0730 \u073aAppendix 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 \u0745 may thereby be expressed as (from Figure 3.1 and Equation 3.3), \u0755\u0bdc =\u072a\u0bdc\u07f1\u0bd6 \u073f\u074b\u074f\u210e \u0d6c\u07f1\u0bd6\u0754\u0bdc\u2217\u072a\u0bdc + \u0723\u0bdc\u0d70 + \u0724\u0bdc (A1)where, \u0723\u0bdc = \u073d\u074f\u0745\u074a\u210e \u124e\u07f1\u0bd6\u210e\u0bdc2\u072a\u0bdc\u074f\u0745\u074a\u210e \u1240\u07f1\u0bd6\u073d\u0bdc2\u072a\u0bdc \u1241\u124f \u2212 \u07f1\u0bd6\u073d\u0bdc2\u072a\u0bdc and \u0724\u0bdc = \u2212\u072a\u0bdc\u07f1\u0bd6 \u073f\u074b\u074f\u210e\u123a\u0723\u0bdc\u123b \u07f1\u0bd6 \u0197Self-Weight of Suspension Cable \u072a\u0bdc \u2018Horizontal\u2019 Cable Force in Segment i\u123a\u0734\u0754\u0bdc, \u0734\u0755\u0bdc, \u0734\u0756\u0bdc\u123b Cable Force Components at Beginning of Segment i\u123a\u07c2\u0754\u0bdc, \u07c2\u0755\u0bdc, \u07c2\u0756\u0bdc\u123b Projected Dimensions of Segment i \u0197Not Shown for Clarity Figure A2: Free Body Diagram of Segment i \u0755\u0756\u07c2\u0754\u0bdc \u07c2\u0755\u0bdc \u0754\u0754\u2217\u07c2\u0756\u0bdc\u0734\u0754\u0bdc\u0734\u0755\u0bdc = \u2212\u072a\u0bdc\u0740\u0755\u0740\u0754\u2217\u0e2c\u0beb\u2217\u0b40\u0b34 = \u2212\u072a\u0bdc\u074f\u0745\u074a\u210e\u123a\u0723\u0bdc\u123b \u0734\u0756\u0bdc\u072a\u0bdc\u0734\u0754\u0bdc\u0734\u0756\u0bdc \u072a\u0bdc\u0740\u0755\u0740\u0754\u2217\u0e2c\u0beb\u2217\u0b40\u0bd4\u0cd4= \u072a\u0bdc\u074f\u0745\u074a\u210e \u0d6c\u07f1\u0bd6\u073d\u0bdc\u072a\u0bdc + \u0723\u0bdc\u0d70\u210e\u0bdc Appendix A: Procedure to Determine Cable Shape in Three Dimensions 269 The direction of the \u0754\u2217 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), \u072a\u0bdc = \u0da5\u123a\u0734\u0754\u0bdc\u123b\u0b36 + \u123a\u0734\u0756\u0bdc\u123b\u0b36 (A2) \u073d\u0bdc = \u0da5\u123a\u0394\u0754\u0bdc\u123b\u0b36 + \u123a\u0394\u0756\u0bdc\u123b\u0b36 (A3) \u210e\u0bdc = \u0394y\u0bdc (A4) \u0754\u0bdc\u2217 = \u0754\u0bdc\u0da81 + \u0d6c\u0394\u0756\u0bdc\u0394\u0754\u0bdc\u0d70\u0b36 (A5) where, the local \u0756 and \u0754 axes are related through the following relationship, \u0756\u0bdc = \u0754\u0bdc \u0d6c\u0394\u0756\u0bdc\u0394\u0754\u0bdc\u0d70 (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 \u0754 direction are known parameters, there remain 5 \u00d7 \u074a unknowns in the form of \u0734\u0754\u0b35\u2026\u0be1, \u0734\u0755\u0b35\u2026\u0be1, \u0734\u0756\u0b35\u2026\u0be1, \u0394\u0755\u0b35\u2026\u0be1, and \u0394\u0756\u0b35\u2026\u0be1. However, given the support reactions at Node 1 \u123a\u0734\u0754\u0b35, \u0734\u0755\u0b35, \u0734\u0756\u0b35\u123b, the transverse and vertical projected dimensions of Segment 1 can be determined from the geometrical conditions upon which, \u0394\u0756\u0bdc = \u0394\u0754\u0bdc \u0d6c\u0734\u0756\u0bdc\u0734\u0754\u0bdc\u0d70 (A7)and \u0734\u0755\u0bdc = \u2212\u072a\u0bdc\u074f\u0745\u074a\u210e\u123a\u0723\u0bdc\u123b (A8)where, \u0723\u0bdc is a function of \u0394\u0755\u0bdc. 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 \u0734\u0754\u0bdc\u0b3e\u0b35 = \u0734\u0754\u0bdc + \u0728\u210e\u0beb\u0bdc\u0b3e\u0b35 (A9) \u0734\u0755\u0bdc\u0b3e\u0b35 = \u0728\u210e\u0bec\u0bdc\u0b3e\u0b35 \u2212 \u072a\u0bdc\u074f\u0745\u074a\u210e \u0d6c\u07f1\u0bd6\u073d\u0bdc\u072a\u0bdc + \u0723\u0bdc\u0d70 (A10) \u0734\u0756\u0bdc\u0b3e\u0b35 = \u0734\u0756\u0bdc + \u0728\u210e\u0bed\u0bdc\u0b3e\u0b35 (A11) where, \u0728\u210e\u0beb\u0cd4, \u0728\u210e\u0bec\u0cd4, and \u0728\u210e\u0bed\u0cd4 denote the respective longitudinal, vertical, and transverse components of the hanger force acting at node \u0745. 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 |\u0728\u210e|, the hanger force components at a given node \u0745 can be computed as (neglecting the sag effect of the hangers), \u0728\u210e\u0beb\u0bdc = |\u0728\u210e\u0bdc|\u06c9\u06c7 \u07dc\u210e\u0beb\u0da7\u07dc\u210e\u0beb\u0b36 + \u07dc\u210e\u0bec\u0b36 + \u07dc\u210e\u0bed\u0b36\u06cc\u06ca\u0bdc(A12) \u0728\u210e\u0bec\u0bdc = |\u0728\u210e\u0bdc|\u06c9\u06c7 \u07dc\u210e\u0bec\u0da7\u07dc\u210e\u0beb\u0b36 + \u07dc\u210e\u0bec\u0b36 + \u07dc\u210e\u0bed\u0b36\u06cc\u06ca\u0bdc(A13) \u0728\u210e\u0bed\u0bdc = |\u0728\u210e\u0bdc|\u06c9\u06c7 \u07dc\u210e\u0bed\u0da7\u07dc\u210e\u0beb\u0b36 + \u07dc\u210e\u0bec\u0b36 + \u07dc\u210e\u0bed\u0b36\u06cc\u06ca\u0bdc(A14) where, \u07dc\u210e\u0beb\u0bdc = \u210e\u0beb\u0bdc \u2212 \u0dcd \u0394\u0754\u0be0\u0bdc\u0b3f\u0b35\u0be0\u0b40\u0b35 \u07dc\u210e\u0bec\u0bdc = \u210e\u0bec\u0bdc \u2212 \u0dcd \u0394y\u0be0\u0bdc\u0b3f\u0b35\u0be0\u0b40\u0b35 JOINT EQUILIBRIUM EQUATIONS Appendix A: Procedure to Determine Cable Shape in Three Dimensions 271 \u07dc\u210e\u0bed\u0bdc = \u210e\u0bed\u0bdc \u2212 \u0dcd \u0394z\u0be0\u0bdc\u0b3f\u0b35\u0be0\u0b40\u0b35 As shown in Figure A3, \u07dc\u210e\u0beb, \u07dc\u210e\u0bec, and \u07dc\u210e\u0bed represent the projected dimensions of the hanger in the \u0754, \u0755, and \u0756 directions, respectively. And, \u0d6b\u210e\u0beb, \u210e\u0bec, \u210e\u0bed\u0d6f\u0bdc denote a set of specified coordinates for the end node of hanger \u0745 , 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, \u0728\u210e\u0beb\u0bdc = \u0728\u210e\u0bec\u0bdc \u1246\u07dc\u210e\u0beb\u07dc\u210e\u0bec\u1247\u0bdc (A15) \u0728\u210e\u0bed\u0bdc = \u0728\u210e\u0bec\u0bdc \u1246\u07dc\u210e\u0bed\u07dc\u210e\u0bec\u1247\u0bdc (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 \u0734 \u0754\u0b35, \u0734 \u0755\u0b35, and \u0734 \u0756\u0b35. However, since these Figure A3: YZ Section at Node i *Nodes (2\u2026 i-1) not shown for clarity\u210e\u0bed\u0bdcBridge Superstructure \u073b\u073c\u0b35\u0730\u0bdc\u0730\u210e\u0bec\u0bdc\u07dc\u210e\u0bed\u0bdc\u07dc\u210e\u0bec\u0bdc 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 \u0394\u073a\u0bcd, which yield the specified end offsets of the cable (see Figure A1). \u0394\u073b\u0bcd = \u0dcd \u0394\u0755\u0bdc\u0be1\u0bdc\u0b40\u0b35= \u0735\u074c\u0741\u073f\u0745\u0742\u0745\u0741\u0740 \u0738\u073d\u0748\u0751\u0741 (A17) \u0394\u073c\u0bcd = \u0dcd \u0394z\u0bdc\u0be1\u0bdc\u0b40\u0b35= \u0735\u074c\u0741\u073f\u0745\u0742\u0745\u0741\u0740 \u0738\u073d\u0748\u0751\u0741 (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 \u2018Y\u2019 direction) from the origin to the cable at, \u073a = \u0394\u073a\u0bcd2 is chosen. This value, referred to hereon as the vertical cable sag, \u0bec\u0742, is computed for given set of support reactions as. \u0dcd \u0394\u0755\u0bdc\u0be4\u0bdc\u0b40\u0b35 if there is a hanger at midspan \u0bec\u0742 = (A19) \u0dcd \u0394\u0755\u0bdc\u0be3\u0b3f\u0b35\u0bdc\u0b40\u0b35+ \u0755\u0be3 \u0d6c\u0754\u0be3 =\u0394\u0754\u0be32 \u0d70 if there is no hanger at midspan where, \u074d = \u074a2 and \u074c =\u074a + 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\u2019s 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\u2019s 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, \u07f1\u0bd6 = \u07db\u0bd6\u0723\u0bd6. 2. An array containing \u0394\u0754\u0b35\u2026\u0be1 3. An array containing \u0728\u210e\u0b35\u2026\u0be1, or alternatively \u0728\u210e\u0bec\u0b35\u2026\u0be1 4. Target values for \u0bec\u0742, \u0394\u073b\u0bcd , and \u0394Z\u0bcd. Also, the tolerance accepted in achieving the target values, denoted as \u0736\u0731\u072e. 5. An initial guess for the support reactions at Node 1, denoted as \u0734\u0754\u073d\u0b35, \u0734\u0755\u073d\u0b35, and \u0734\u0756\u073d\u0b35. Steps: 1. Set the support reactions at Node 1 equal to \u123a\u0734\u0754\u073d\u0b35, \u0734\u0755\u073d\u0b35, \u0734\u0756\u073d\u0b35\u123b. 2. Compute \u0394z\u0bdc using Equation (A7). 3. Solve for \u0394\u0755\u0bdc using Equation (A8). 4. Decompose \u0728\u210e\u0bdc using Equations (A12)-(A14). 5. Compute \u0734\u0754\u0bdc\u0b3e\u0b35, \u0734\u0755\u0bdc\u0b3e\u0b35, and \u0734\u0756\u0bdc\u0b3e\u0b35 using the joint equilibrium equations, (A9)-(A11). 6. Repeat Steps 2 through 5 for \u0745 = 1 \u2026 \u074a. 7. Compute the vertical cable end offset (denoted as \u0394\u073b\u0bcd \u0bd4), the transverse cable end offset (denoted as \u0394\u073c\u0bcd\u0bd4), and the vertical cable sag (denoted as \u0bec\u0742\u0bd4) corresponding to \u123a\u0734\u0754\u073d\u0b35, \u0734\u0755\u073d\u0b35, \u0734\u0756\u073d\u0b35\u123b using Equations (A17)-(A19) combined with the geometric parameters obtained in Steps 2 through 6. 8. Determine the error in the target parameters, \u0394\u073b\u0bcd \u0bbe\u0be5\u0be5\u0be2\u0be5 = \u0394\u073b\u0bcd \u2212 \u0394\u073b\u0bcd \u0bd4, \u0394Z\u0bcd\u0bbe\u0be5\u0be5\u0be2\u0be5 = \u0394Z\u0bcd \u2212 \u0394Z\u0bcd\u0bd4, and \u0bec\u0742\u0bbe\u0be5\u0be5\u0be2\u0be5 = \u0bec\u0742 \u2212 \u0bec\u0742\u0bd4 9. Check convergence a. If \u1240\u0e2b\u0394\u073b\u0bcd \u0bbe\u0be5\u0be5\u0be2\u0be5\u0e2b \u2227 \u0e2b\u0394Z\u0bcd\u0bbe\u0be5\u0be5\u0be2\u0be5\u0e2b \u2227 \u125a \u0bec\u0742\u0bbe\u0be5\u0be5\u0be2\u0be5\u125a\u1241 > \u0736\u0731\u072e advance to Step 10. b. If \u1240\u0e2b\u0394\u073b\u0bcd \u0bbe\u0be5\u0be5\u0be2\u0be5\u0e2b \u2227 \u0e2b\u0394Z\u0bcd\u0bbe\u0be5\u0be5\u0be2\u0be5\u0e2b \u2227 \u125a \u0bec\u0742\u0bbe\u0be5\u0be5\u0be2\u0be5\u125a\u1241 \u2264 \u0736\u0731\u072e advance to Step 13. See Figure A1 Appendix A: Procedure to Determine Cable Shape in Three Dimensions 274 10. Numerically compute the Jacobian Matrix, \u123e\u072c\u123f =\u06cf\u06ce\u06ce\u06ce\u06cd \u0bd7\u0bd9\u0ce4\u0ccc\u0ccd\u0bd7\u0bcb\u0beb\u0ccc\u0ccd\u0bd7\u0bd9\u0ce4\u0ccc\u0cce\u0bd7\u0bcb\u0bec\u0ccc\u0cce\u0bd7\u0bd9\u0ce4\u0ccc\u0ccf\u0bd7\u0bcb\u0bed\u0ccc\u0ccf\u0b62\u0b7c\u0b5d\u0cc5\u0ccc\u0ccd\u0bd7\u0bcb\u0beb\u0ccc\u0ccd\u0b62\u0b7c\u0b5d\u0cc5\u0ccc\u0cce\u0bd7\u0bcb\u0bec\u0ccc\u0cce\u0b62\u0b7c\u0b5d\u0cc5\u0ccc\u0ccf\u0bd7\u0bcb\u0bed\u0ccc\u0ccf\u0b62\u0b7c\u0b5e\u0cc5\u0ccc\u0ccd\u0bd7\u0bcb\u0beb\u0ccc\u0ccd\u0b62\u0b7c\u0b5e\u0cc5\u0ccc\u0cce\u0bd7\u0bcb\u0bec\u0ccc\u0cce\u0b62\u0b7c\u0b5e\u0cc5\u0ccc\u0ccf\u0bd7\u0bcb\u0bed\u0ccc\u0ccf \u06d2\u06d1\u06d1\u06d1\u06d0 a. Set, \u0734\u0754\u073e\u0b35 = \u123a1 \u2212 \u0736\u0731\u072e\u123b\u0734\u0754\u073d\u0b35, \u0734\u0755\u073f\u0b35 = \u123a1 \u2212 \u0736\u0731\u072e\u123b\u0734\u0755\u073d\u0b35, and \u0734\u0756\u0740\u0b35 =\u123a1 \u2212 \u0736\u0731\u072e\u123b\u0734\u0756\u073d\u0b35. b. Repeat Steps 2 through 6 except with the support reactions at Node 1 equal to \u123a\u0734\u0754\u073e\u0b35, \u0734\u0755\u073d\u0b35, \u0734\u0756\u073d\u0b35\u123b and label the cable parameters in Step 7 \u0d6b\u0394\u073b\u0bcd \u0bd5, \u0394\u073c\u0bcd\u0bd5, \u0bec\u0742\u0bd5\u0d6f. c. Compute first column of the Jacobian Matrix, \u0bd7\u0bd9\u0ce4\u0ccc\u0ccd\u0bd7\u0bcb\u0beb\u0ccc\u0ccd =\u0bd9\u0ce4\u0ccc\u0b3f\u0bd9\u0ce4\u0ccd\u0bcb\u0beb\u0bd4\u0c2d\u0b3f\u0bcb\u0beb\u0bd5\u0c2d ; \u0b62\u0b7c\u0b5d\u0cc5\u0ccc\u0ccd\u0bd7\u0bcb\u0beb\u0ccc\u0ccd =\u0b7c\u0b5d\u0cc5\u0ccc\u0b3f\u0b7c\u0b5d\u0cc5\u0ccd\u0bcb\u0beb\u0bd4\u0c2d\u0b3f\u0bcb\u0beb\u0bd5\u0c2d ; \u0b62\u0b7c\u0b5e\u0cc5\u0ccc\u0ccd\u0bd7\u0bcb\u0beb\u0ccc\u0ccd =\u0b7c\u0b5e\u0cc5\u0ccc\u0b3f\u0b7c\u0b5e\u0cc5\u0ccd\u0bcb\u0beb\u0bd4\u0c2d\u0b3f\u0bcb\u0beb\u0bd5\u0c2d d. Repeat Steps 2 through 6 except with the support reactions at Node 1 equal to \u123a\u0734\u0754\u073d\u0b35, \u0734\u0755\u073f\u0b35, \u0734\u0756\u073d\u0b35\u123b and label the cable parameters in Step 7 \u0d6b\u0394\u073b\u0bcd \u0bd6, \u0394\u073c\u0bcd\u0bd6, \u0bec\u0742\u0bd6\u0d6f. e. Compute second column of the Jacobian Matrix, \u0bd7\u0bd9\u0ce4\u0ccc\u0cce\u0bd7\u0bcb\u0bec\u0ccc\u0cce =\u0bd9\u0ce4\u0ccc\u0b3f\u0bd9\u0ce4\u0cce\u0bcb\u0bec\u0bd4\u0c2d\u0b3f\u0bcb\u0bec\u0bd6\u0c2d ; \u0b62\u0b7c\u0b5d\u0cc5\u0ccc\u0cce\u0bd7\u0bcb\u0bec\u0ccc\u0cce =\u0b7c\u0b5d\u0cc5\u0ccc\u0b3f\u0b7c\u0b5d\u0cc5\u0cce\u0bcb\u0bec\u0bd4\u0c2d\u0b3f\u0bcb\u0bec\u0bd6\u0c2d ; \u0b62\u0b7c\u0b5e\u0cc5\u0ccc\u0cce\u0bd7\u0bcb\u0bec\u0ccc\u0cce =\u0b7c\u0b5e\u0cc5\u0ccc\u0b3f\u0b7c\u0b5e\u0cc5\u0cce\u0bcb\u0bec\u0bd4\u0c2d\u0b3f\u0bcb\u0bec\u0bd6\u0c2d f. Repeat Steps 2 through 6 except with the support reactions at Node 1 equal to \u123a\u0734\u0754\u073d\u0b35, \u0734\u0755\u073d\u0b35, \u0734\u0756\u0740\u0b35\u123b and label the cable parameters in Step 7 \u0d6b\u0394\u073b\u0bcd \u0bd7, \u0394\u073c\u0bcd\u0bd7, \u0bec\u0742\u0bd7\u0d6f. g. Compute third column of Jacobian Matrix, \u0bd7\u0bd9\u0ce4\u0ccc\u0ccf\u0bd7\u0bcb\u0bed\u0ccc\u0ccf =\u0bd9\u0ce4\u0ccc\u0b3f\u0bd9\u0ce4\u0ccf\u0bcb\u0bed\u0bd4\u0c2d\u0b3f\u0bcb\u0bed\u0bd7\u0c2d ; \u0b62\u0b7c\u0b5d\u0cc5\u0ccc\u0ccf\u0bd7\u0bcb\u0bed\u0ccc\u0ccf =\u0b7c\u0b5d\u0cc5\u0ccc\u0b3f\u0b7c\u0b5d\u0cc5\u0ccf\u0bcb\u0bed\u0bd4\u0c2d\u0b3f\u0bcb\u0bed\u0bd7\u0c2d ; \u0b62\u0b7c\u0b5e\u0cc5\u0ccc\u0ccf\u0bd7\u0bcb\u0bed\u0ccc\u0ccf =\u0b7c\u0b5e\u0cc5\u0ccc\u0b3f\u0b7c\u0b5e\u0cc5\u0ccf\u0bcb\u0bed\u0bd4\u0c2d\u0b3f\u0bcb\u0bed\u0bd7\u0c2d 11. Update the initial guess values for the support reactions at Node 1, \u123e\u0734\u073d\u123f\u0bc7\u0bbe\u0bd0 = \u123e\u0734\u073d\u123f +\u123e\u0394\u0734\u073d\u123f a. Set, \u123e\u0734\u073d\u123f = \u0d65\u0734\u0754\u073d\u0b35\u0734\u0755\u073d\u0b35\u0734\u0756\u073d\u0b35\u0d69 b. Compute the requisite change in the support reactions, \u123e\u0394\u0734\u073d\u123f = \u123e\u072c\u123f\u0b3f\u0b35 \u124e\u0bec\u0742\u0bbe\u0be5\u0be5\u0be2\u0be5\u0394\u073b\u0bcd \u0bbe\u0be5\u0be5\u0be2\u0be5\u0394Z\u0bcd\u0bbe\u0be5\u0be5\u0be2\u0be5\u124f 12. Repeat Steps 1 through 11 until the convergence criterion in Step 9b is met. 13. With \u123a\u0734\u0754\u073d\u0b35, \u0734\u0755\u073d\u0b35, \u0734\u0756\u073d\u0b35\u123b 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 \uf0a7 All initial inputs should be entered as positive or negative values according to the coordinate systems specified in Figure A1 and Figure A2. \uf0a7 As \u0728\u210e\u0b35 acts at a support node, its value should be set equal to zero. \uf0a7 The convergence of Newton\u2019s 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 \u0734\u0754\u073d\u0b35, \u0734\u0755\u073d\u0b35, and \u0734\u0756\u073d\u0b35. \uf0a7 If the transverse force component of all hangers is zero \u0d6b\u0728\u210e\u0bed\u0b35\u2026\u0be1 = 0\u0d6f, then the third row and third column of the Jacobian Matrix must be omitted to prevent the matrix from becoming singular. \uf0a7 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, \u0e2b\u123e\u0394\u0734\u073d\u123f\u0bdd\u0e2b \u2264 0.5\u0e2b\u123e\u0734\u073d\u123f\u0bdd\u0e2b be placed on Step 11 for \u0746 = 1 \u2026 \u074e\u074b\u0753\u074f\u123a\u123e\u0394\u0734\u073d\u123f\u123b. GENERAL COMMENTS \uf0a7 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). \u0b35\u0736 = \u0da7\u0734\u0754\u0b35\u0b36 + \u0734\u0755\u0b35\u0b36 + \u0734\u0755\u0b35\u0b36 (A20) \u0be1\u0736\u0b3e\u0b35 = \u0da7\u0734\u0754\u0be1\u0b3e\u0b35\u0b36 + \u0734\u0755\u0be1\u0b3e\u0b35\u0b36 + \u0734\u0755\u0be1\u0b3e\u0b35\u0b36 (A21)Also, the angles formed between the ends of the cable and the \u0754, \u0755, and \u0756 axes are given by, \u07e0\u0beb\u0b35 = \u073d\u073f\u074b\u074f \u1246|\u0734\u0754\u0b35|\u0b35\u0736\u1247 and \u07e0\u0beb\u0be1\u0b3e\u0b35 = \u073d\u073f\u074b\u074f \u1246|\u0734\u0754\u0be1\u0b3e\u0b35|\u0be1\u0736\u0b3e\u0b35\u1247 (A22) \u07e0\u0bec\u0b35 = \u073d\u073f\u074b\u074f \u1246|\u0734\u0754\u0b35|\u0b35\u0736\u1247 and \u07e0\u0bec\u0be1\u0b3e\u0b35 = \u073d\u073f\u074b\u074f \u1246|\u0734\u0755\u0be1\u0b3e\u0b35|\u0be1\u0736\u0b3e\u0b35\u1247 (A23) \u07e0\u0bed\u0b35 = \u073d\u073f\u074b\u074f \u1246|\u0734\u0756\u0b35|\u0b35\u0736\u1247 and \u07e0\u0bed\u0be1\u0b3e\u0b35 = \u073d\u073f\u074b\u074f \u1246|\u0734\u0756\u0be1\u0b3e\u0b35|\u0be1\u0736\u0b3e\u0b35\u1247 (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)), \u0737\u0735\u072e = \u0d65\u0dcd \u0725\u0bdc \u2212 \u0394\u0bdc\u0be1\u0bdc\u0b40\u0b35\u0d69 (A25)where, \u0725\u0bdc =\u072a\u0bdc\u07f1\u0bd6 \u0d64\u074f\u0745\u074a\u210e \u0d6c\u07f1\u0bd6\u073d\u0bdc\u072a\u0bdc + \u0723\u0bdc\u0d70 \u2212 \u074f\u0745\u074a\u210e\u123a\u0723\u0bdc\u123b\u0d68 (A26)and, \u0394\u0bdc =\u072a\u0bdc\u073d\u0bdc\u0727\u0bd6\u0723\u0bd6 \u1248\u07f1\u0bd6\u0394\u0755\u0bdc\u0b362\u072a\u0bdc\u073d\u0bdc \u073f\u074b\u0750\u210e \u0d6c\u07f1\u0bd6\u073d\u0bdc2\u072a\u0bdc \u0d70 +12 +\u072a\u0bdc2\u07f1\u0bd6\u073d\u0bdc \u074f\u0745\u074a\u210e \u0d6c\u07f1\u0bd6\u073d\u0bdc\u072a\u0bdc \u0d70\u1249 (A27)\uf0a7 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). \uf0a7 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. \uf0a7 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\u00d710-10, in less than ten iterations. ","attrs":{"lang":"en","ns":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","classmap":"oc:AnnotationContainer"},"iri":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","explain":"Simple Knowledge Organisation System; Notes are used to provide information relating to SKOS concepts. 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