[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-657449-105":3,"detail-sidebar-cat-0-en-105":80,"doc-detail-657449-en":130},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":73,"head_meta":75,"extra_data":77,"updated_unix":79},105,"en","chapter-3-structure-analysis-chapter-3-structure-function","Chapter 3 Structure Analysis - Chapter 3 - Structure function","","Chapter 3 introduces the structure function in system reliability theory, using binary component states to model whether a system is functioning or failed. It defines the order-n system, the state variable for each component, and the structure function ϕ(x) that maps the component state vector to system state. The chapter then derives structure functions for series and parallel configurations, and extends the approach to k-out-of-n and specific 2-out-of-3 (2oo3) structures, explaining how component counts determine system success.",{"@graph":14,"@context":72},[15,34,55],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & Report",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/chapter-3-structure-analysis-chapter-3-structure-function/657449/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/chapter-3-structure-analysis-chapter-3-structure-function/657449.png","ImageObject",300,407,{"name":42,"@type":43},"jayni","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-30","2026-10-26",true,{"@type":52,"interactionType":53,"userInteractionCount":33},"InteractionCounter",{"@type":54},"ViewAction",{"@type":56,"mainEntity":57},"FAQPage",[58,64,68],{"name":59,"@type":60,"acceptedAnswer":61},"What is a state variable in this reliability model?","Question",{"text":62,"@type":63},"Each component i is assigned a binary state variable xi, where xi = 1 indicates functioning and xi = 0 indicates failed.","Answer",{"name":65,"@type":60,"acceptedAnswer":66},"How does the structure function ϕ(x) determine system status?",{"text":67,"@type":63},"The structure function ϕ(x) is a binary function of the state vector x; it equals 1 when the system is functioning and 0 when the system is failed.",{"name":69,"@type":60,"acceptedAnswer":70},"What is the difference between series and parallel structures?",{"text":71,"@type":63},"In a series structure, the system functions only if all n components function. 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This system is said to have order n. The components are referred to by the numbers 1; 2; : : : ; n.  \nEach component can have two possible states: functioning and failed. The state of component i is given by the binary variable xi ,  \nxi = ( 10 f componentf component ii s fus innactfaioilneindgstate xi is called the state variable of component i.  \nThe state vector for the system is  \nx = (x1 ; x2 ; : : : ; xn)  \n Marvin Rausand (RAMS Group)  System Reliability Theory  (Version 0.1) 3 / 37   \nStructure function Cut and path sets Pivotal decomposition Critical component Modules  \nStructure function  \nThe state of the system (of order n) can be described by the binary function  \nϕ (x) = ϕ (x1 ; x2 ; : : : ; xn)  \nwhere ϕ (x) = ( 10 if the system is functioning  \nif the system is in a failed state  \nϕ (x) is called the structure function of the system or just the structure.  \nWe o􀀜en talk about structures of order n instead of systems of order n.  \n Marvin Rausand (RAMS Group)  System Reliability Theory  (Version 0.1) 4 / 37   \nStructure function Cut and path sets Pivotal decomposition Critical component Modules  \nSeries structure  \nA system that is functioning if and only if all of its n components are functioning, is called a series structure.  \nA series structure of order n is illustrated by the reliability block diagram  \n1  \n2  \n3  \n b  \n~~ ~~ n   \na  \nThe structure function of the series structure is  \nn  \nϕ (x) = x1 􀀁 x2 􀀁 􀀁 􀀁 xn = Y xi  \ni= 1  \nThe state, ϕ(x) , of the series structure is equal to 1 (i.e., functioning) if and only if all the n components have state 1 (i.e., functioning) .  \nThe series structure fails (i.e., has state 0) as soon as at least one of its components fails (i.e., has state 0) .  \n Marvin Rausand (RAMS Group)  System Reliability Theory  (Version 0.1) 5 / 37   \nStructure function Cut and path sets Pivotal decomposition Critical component Modules  \nParallel structure-1  \nA system that is functioning whenever at least one of its n components is functioning, is called a parallel structure.  \nA parallel structure of order n is illustrated by the reliability block diagram  \nThe structure function of the parallel structure is  \nn ϕ (x) = 1 􀀀 Y( 1 􀀀 xi)  \ni= 1  \nThe state, ϕ(x) , of the parallel structure is equal to 1 (i.e., functioning) if at least one of the n components have state 1 (i.e., functioning) .  \n Marvin Rausand (RAMS Group)  System Reliability Theory  (Version 0.1) 6 / 37   \nStructure function Cut and path sets Pivotal decomposition Critical component Modules  \nParallel structure-2  \nThe structure function of a parallel structure of order n = 2 is hence:ϕ (x) = 1 􀀀 ( 1 􀀀 x1 )( 1 􀀀 x2 ) = x1 + x2 􀀀 x1 x2  \nSometimes, we use the symbol q and write  \nϕ (x) = x1 q x2 = 1 􀀀 ( 1 􀀀 x1 )( 1 􀀀 x2 ) = x1 + x2 􀀀 x1 x2 x1 q x2 may be read as “component 1 in parallel with component 2 .”We may also use the same symbol for a parallel structure of order n  \nn  \nϕ (x) = x1 q x2 q 􀀁 􀀁 􀀁 q x n = a xi  \ni= 1  \n Marvin Rausand (RAMS Group)  System Reliability Theory  (Version 0.1) 7 / 37   \nStructur","cbCailrWeyZSFi5F","https://ap.wps.com/l/cbCailrWeyZSFi5F","pdf",884179,37,"English","# Structure function\n## State variable\n## Structure function definition\n## Series structure\n## Parallel structure\n## k-out-of-n structure\n## 2-out-of-3 structure","[{\"question\":\"What is a state variable in this reliability model?\",\"answer\":\"Each component i is assigned a binary state variable xi, where xi = 1 indicates functioning and xi = 0 indicates failed.\"},{\"question\":\"How does the structure function ϕ(x) determine system status?\",\"answer\":\"The structure function ϕ(x) is a binary function of the state vector x; it equals 1 when the system is functioning and 0 when the system is failed.\"},{\"question\":\"What is the difference between series and parallel structures?\",\"answer\":\"In a series structure, the system functions only if all n components function. In a parallel structure, the system functions when at least one component functions.\"}]","Chapter 3 Structure Analysis - Chapter 3 - Structure function | PDF",1792979281,93]