Module 4. Analysis of Statically Indeterminate Structures by the Direct Stiffness Method. Version 2 CE IIT, Kharagpur
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1 Mdle Analysis f Saically Indeerminae Srcres by he Direc Siffness Mehd Versin CE IIT, Kharagr
2 Lessn The Direc Siffness Mehd: Temerare Changes and Fabricain Errrs in Trss Analysis Versin CE IIT, Kharagr
3 Insrcinal Objecives Afer reading his chaer he sden will be able. Cme sresses develed in he rss members de emerare changes.. Cme sresses develed in rss members de fabricain members.. Cme reacins in lane rss de emerare changes and fabricain errrs.. Inrdcin In he las fr lessns, he direc siffness mehd as alied he rss analysis was discssed. Assembly f member siffness marices, imsiin f bndary cndiins, and he rblem f inclined srs were discssed. De he change in emerare he rss members eiher exand r shrink. Hwever, in he case f saically indeerminae rsses, he lengh f he members is revened frm eiher exansin r cnracin. Ths, he sresses are develed in he members de changes in emerare. Similarly he errr in fabricaing rss members als rdces addiinal sresses in he rsses. Bh hese effecs can be easily accned fr in he siffness analysis.. Temerare Effecs and Fabricain Errrs Versin CE IIT, Kharagr
4 Cnsider rss member f lengh L, area f crss secin A as shwn in Fig...The change in lengh Δl is given by Δl α LΔT (. where α is he cefficien f hermal exansin f he maerial cnsidered. If he member is n allwed change is lengh (as in he case f saically indeerminae rss he change in emerare will indce addiinal frces in he member. As he rss elemen is a ne dimensinal elemen in he lcal crdinae sysem, he hermal lad can be easily calclaed in glbal crdinae sysem by r AEΔ L (.a ( AEΔ L (.b ( ' {( } + AEΔL (. The eqain (. can als be sed calclae frces develed in he rss member in he lcal crdinae sysem de fabricain errr. ΔL will be cnsidered siive if he member is lng. The frces in he lcal crdinae sysem can be ransfrmed glbal crdinae sysem by sing he eqain, ( ( ( ( csθ ' ( ( csθ ' (.a where (, ( and ( (, are he frces in he glbal crdinae sysem a ndes and f he rss member resecively Using eqain (., he eqain (.a may be wrien as, ( csθ ( AEΔL ( csθ ( (.b Versin CE IIT, Kharagr
5 The frce dislacemen eqain fr he enire rss may be wrien as, {} []{} k + {( } (. where, {} is he vecr f exernal jin lads alied n he rss and {( } is he vecr f jin lads develed in he rss de change in emerare/fabricain errr f ne r mre members. As ined earlier. in he rss analysis, sme jin dislacemens are knwn de bndary cndiins and sme jin lads are knwn as hey are alied exernally.ths,ne cld ariin he abve eqain as, k [ k] [ k ] [ k ] [ k ] { } { } k + ( k ( (. where sbscri is sed dene nknwn qaniies and sbscri k is sed dene knwn qaniies f frces and dislacemens. Exanding eqain (., { } [ k ]{ } + [ k ]{ } + {( } k (.a { } [ k ]{ } [ k ]{ } ( k k { } + + (.b k If he knwn dislacemen vecr { } { } k hen sing eqain (.a he nknwn dislacemens can be calclaed as { } [ k ] ({ k } {( k } If { k } hen { } [ k ] ({ } [ k ]{ } {( } (.a k k k (.b Afer evalaing nknwn dislacemens, he nknwn frce vecrs are calclaed sing eqain (.b.afer evalaing dislacemens, he member frces in he lcal crdinae sysem fr each member are evalaed by, r { } [ k ][ T ]{} + { } (.9a ' ' AE L csθ csθ v v ' ( + ' ( Versin CE IIT, Kharagr
6 Exanding he abve eqain, yields v AE { } { csθ csθ } + AEΔL L (.a v And, AE { } { csθ csθ } v Δ L AE L (.b v Few rblems are slved illsrae he alicain f he abve rcedre calclae hermal effecs /fabricain errrs in he rss analysis:- Examle. Analyze he rss shwn in Fig..a, if he emerare f he member ( is raised by C.The secinal areas f members in sqare cenimeers are shwn in he figre. Assme E N / mm and α /, er C. Versin CE IIT, Kharagr
7 The nmbering f jins and members are shwn in Fig..b. The ssible glbal dislacemen degrees f freedm are als shwn in he figre. Ne ha lwer nmbers are sed indicae ncnsrained degrees f freedm. Frm he figre i is bvis ha he dislacemens de bndary cndiins. The emerare f he member ( has been raised by C. Ths, ΔL α LΔT ( (. ΔL m ( The frces in member ( de rise in emerare in glbal crdinae sysem can be calclaed sing eqain (.b.ths, ( csθ ( AEΔL ( ( csθ ( Fr member (, A cm m and θ Versin CE IIT, Kharagr
8 ( (. / ( ( ( ( ( (. kn ( ( In he nex se, wrie siffness marix f each member in glbal crdinae sysem and assemble hem bain glbal siffness marix Elemen (: θ, L m, A m,ndal ins - ' k ( Member (: θ, L m, A m, ndal ins k.... (.... [ ] Member (: θ 9, A m, L m, ndal ins - Versin CE IIT, Kharagr
9 Versin CE IIT, Kharagr x k ( The glbal siffness marix is f he rder,assembling he hree member siffness marices, ne ges [] k ( Wriing he lad dislacemen eqain fr he rss + (9 In he resen case, he dislacemens and are n knwn. All her dislacemens are zer. Als (as n jin lads are alied.ths,
10 Versin CE IIT, Kharagr + ( Ths nknwn dislacemens are (. (.. m m Nw reacins are calclaed as kn (
11 The sr reacins are shwn in Fig..c.The member frces can be easily calclaed frm reacins. The member end frces can als be calclaed by sing eqain (.a and (.b. Fr examle, fr member (, θ ' [ ]. (.. kn. Ths he member ( is in ensin. Member ( θ ' 9.[ ] kn. Ths member ( is in cmressin Versin CE IIT, Kharagr
12 Examle. Analyze he rss shwn in Fig..a, if he member BC is made.m shr befre lacing i in he rss. Assme AE kn fr all members. Versin CE IIT, Kharagr
13 Slin A similar rss wih differen bndary cndiins has already been slved in examle.. Fr he sake f cmleeness he member f ndes and members are shwn in Fig..b.The dislacemens,,,, and are zer de bndary cndiins. Fr he resen rblem he ncnsrained degrees f freedm are and.the assembled siffness marix is f he rder and is available in examle.. In he given rblem he member ( is shr by.m.the frces develed in member ( in he glbal crdinae sysem de fabricain errr is ( ( ( ( csθ AE(. csθ. kn (. Nw frce-dislacemen relains fr he rss are Versin CE IIT, Kharagr
14 Versin CE IIT, Kharagr AE ( Ne ha Ths, slving... AE ( and, m. ( Reacins are calclaed as, AE (..... ( The reacins and member frces are shwn in Fig..c. The member frces can als be calclaed by eqain (.a and (.b. Fr examle, fr member (, 9 θ
15 ' AEΔL [ - ] L ( (...9. kn ( Examle. Evalae he member frces f rss shwn in Fig..a.The emerare f he member BC is raised by C and member BD is raised by C.Assme AEKN fr all members and α er C. Versin CE IIT, Kharagr
16 Slin Fr his rblem assembled siffness marix is available in Fig..b.The jins and members are nmbered as shwn in Fig..b. In he given rblem,,, and reresen ncnsrained degrees f freedm. De sr cndiins,. The emerare f he member ( is raised by C.Ths, ΔL α LΔT. m ( The frces are develed in member (, as i was revened frm exansin. ( f csθ ( f. ( ( f csθ f Versin CE IIT, Kharagr
17 Versin CE IIT, Kharagr ( The dislacemen f he member ( was raised by C. Ths, m T L L., Δ Δ α The frces develed in member ( as i was n allwed exand is ( ( ( ( ( The glbal frce vecr de hermal lad is ( ( ( ( ( ( ( (.... ( Wriing he lad-dislacemen relain fr he enire rss is given belw.
18 Versin CE IIT, Kharagr AE ( In he abve rblem and. Ths slving fr nknwn dislacemens, AE ( Slving eqain (, he nknwn dislacemens are calclaed as m m m m.,.,.,. ( Nw, reacins are cmed as, ( All reacins are zer as rss is exernally deerminae and hence change in emerare des n indce any reacin. Nw member frces are calclaed by sing eqain (.b Member (: Lm, θ
19 ' AE [- ] ( '. Kn Member : Lm, θ 9,ndal ins - ' AE [ - ]. (9. kn Member (: Lm, θ,ndal ins - ' [- ] (.kn Member (: θ 9, L m, ndal ins - ' [ - ] ( Member (: θ, L,ndal ins - ' [ ]. ( -.9 kn Member ( : θ, L,ndal ins - Versin CE IIT, Kharagr
20 ' [ ]. kn. ( Smmary In he las fr lessns, he direc siffness mehd as alied he rss analysis was discssed. Assembly f member siffness marices, imsiin f bndary cndiins, and he rblem f inclined srs were discssed. De he change in emerare he rss members eiher exand r shrink. Hwever, in he case f saically indeerminae rsses, he lengh f he members is revened frm eiher exansin r cnracin. Ths, he sresses are develed in he members de changes in emerare. Similarly he errrs in fabricaing rss members als rdce addiinal sresses in he rsses. In his lessn, hese effecs are accned fr in he siffness analysis. A cle f rblems are slved. Versin CE IIT, Kharagr
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Module 4. Analysis of Statically Indeterminate Structures by the Direct Stiffness Method. Version 2 CE IIT, Kharagpur
Modle Analysis of Statically Indeterminate Strctres by the Direct Stiffness Method Version CE IIT, Kharagr Lesson The Direct Stiffness Method: Trss Analysis (Contined) Version CE IIT, Kharagr Instrctional
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