Module 4. Analysis of Statically Indeterminate Structures by the Direct Stiffness Method. Version 2 CE IIT, Kharagpur

Transcription

1 Mode 4 Anaysis of Staticay Indeterinate Strctres by the Direct Stiffness Method Version CE IIT, Kharagr

2 esson 4 The Direct Stiffness Method: Trss Anaysis Version CE IIT, Kharagr

3 Instrctiona Objecties After reading this chater the stdent wi be abe to. Derie eber stiffness atrix of a trss eber.. Define oca and goba co-ordinate syste. 3. Transfor disaceents fro oca co-ordinate syste to goba co-ordinate syste. 4. Transfor forces fro oca to goba co-ordinate syste. 5. Transfor eber stiffness atrix fro oca to goba co-ordinate syste. 6. Assebe eber stiffness atrices to obtain the goba stiffness atrix. 7. Anayse ane trss by the direct stiffness atrix. 4. Introdction An introdction to the stiffness ethod was gien in the reios chater. The basic rincies inoed in the anaysis of beas, trsses were discssed. The robes were soed with hand cotation by the direct aication of the basic rincies. The rocedre discssed in the reios chater thogh enightening are not sitabe for coter rograing. It is necessary to kee hand cotation to a ini whie ieenting this rocedre on the coter. In this chater a fora aroach has been discssed which ay be readiy rograed on a coter. In this esson the direct stiffness ethod as aied to anar trss strctre is discssed. Pane trsses are ade of short thin ebers interconnected at hinges to for triangated atterns. A hinge connection can ony transit forces fro one eber to another eber bt not the oent. For anaysis rose, the trss is oaded at the joints. Hence, a trss eber is sbjected to ony axia forces and the forces reain constant aong the ength of the eber. The forces in the eber at its two ends st be of the sae agnitde bt act in the oosite directions for eqiibri as shown in Fig. 4.. Version CE IIT, Kharagr

4 Now consider a trss eber haing cross sectiona area A, Yong s ods of ateria E, and ength of the eber. et the eber be sbjected to axia tensie force F as shown in Fig. 4.. Under the action of constant axia force F, aied at each end, the eber gets eongated by as shown in Fig. 4.. The eongation ay be cacated by (ide esson, ode ). F (4.) Now the force-disaceent reation for the trss eber ay be written as, F (4.) Version CE IIT, Kharagr

5 F k (4.3) where k is the stiffness of the trss eber and is defined as the force reqired for nit deforation of the strctre. The aboe reation (4.3) is tre aong the centroida axis of the trss eber. Bt in reaity there are any ebers in a trss. For exae consider a aner trss shown in Fig For each eber of the trss we cod write one eqation of the tye F kaong its axia direction (which is caed as oca co-ordinate syste). Each eber has different oca co ordinate syste. To anayse the aner trss shown in Fig. 4.3, it is reqired to write force-disaceent reation for the coete trss in a co ordinate syste coon to a ebers. Sch a co-ordinate syste is referred to as goba co ordinate syste. 4. oca and Goba Co-ordinate Syste oads and disaceents are ector qantities and hence a roer coordinate syste is reqired to secify their correct sense of direction. Consider a anar trss as shown in Fig In this trss each node is identified by a nber and each eber is identified by a nber encosed in a circe. The disaceents and oads acting on the trss are defined with resect to goba co-ordinate syste xyz. The sae co ordinate syste is sed to define each of the oads and disaceents of a oads. In a goba co-ordinate syste, each node of a aner trss can hae ony two disaceents: one aong x -axis and another aong y - axis. The trss shown in figre has eight disaceents. Each disaceent Version CE IIT, Kharagr

6 (degree of freedo) in a trss is shown by a nber in the figre at the joint. The direction of the disaceents is shown by an arrow at the node. Howeer ot of eight disaceents, fie are nknown. The disaceents indicated by nbers 6,7 and 8 are zero de to sort conditions. The disaceents denoted by nbers -5 are known as nconstrained degrees of freedo of the trss and disaceents denoted by 6-8 reresent constrained degrees of freedo. In this corse, nknown disaceents are denoted by ower nbers and the known disaceents are denoted by higher code nbers. To anayse the trss shown in Fig. 4.4, the strctra stiffness atrix K need to be eaated for the gien trss. This ay be achieed by sitaby adding a the eber stiffness atrices k', which is sed to exress the force-disaceent reation of the eber in oca co-ordinate syste. Since a ebers are oriented at different directions, it is reqired to transfor eber disaceents and forces fro the oca co-ordinate syste to goba co-ordinate syste so that a goba oad-disaceent reation ay be written for the coete trss. 4.3 Meber Stiffness Matrix Consider a eber of the trss as shown in Fig. 4.5a in oca co-ordinate syste x' y'. As the oads are aied aong the centroida axis, ony ossibe disaceents wi be aong x' -axis. et the ' and ' be the disaceents of trss ebers in oca co-ordinate syste i.e. aong x' -axis. Here sbscrit refers to node of the trss eber and sbscrit refers to node of the trss eber. Gie disaceent ' at node of the eber in the ositie x' direction, keeing a other disaceents to zero. This disaceent in trn Version CE IIT, Kharagr

7 indces a coressie force of agnitde ' in the eber. Ths, q ' ' and q' ' (4.4a) ( e as it acts in the e direction for eqiibri). Siiary by giing ositie disaceents of ' at end of the eber, tensie force of agnitde ' is indced in the eber. Ths, q" ' and q " ' (4.4b) Now the forces deeoed at the ends of the eber when both the disaceents are iosed at nodes and resectiey ay be obtained by ethod of serosition. Ths (ide Fig. 4.5d) (4.5a) ' ' ' ' ' ' (4.5b) Or we can write Version CE IIT, Kharagr

8 ' ' ' ' (4.6a) { '} [ k' ]{ ' } Ths the eber stiffness atrix is (4.6b) k ' (4.7) This ay aso be obtained by giing nit disaceent at node and hoding disaceent at node to zero and cacating forces deeoed at two ends. This wi generate the first con of stiffness atrix. Siiary the second con of stiffness atrix is obtained by giing nit disaceent at and hoding disaceent at node to zero and cacating the forces deeoed at both ends. 4.4 Transforation fro oca to Goba Co-ordinate Syste. Disaceent Transforation Matrix A trss eber is shown in oca and goba co ordinate syste in Fig et x' y' z' be in oca co ordinate syste and xyz be the goba co ordinate syste. Version CE IIT, Kharagr

9 The nodes of the trss eber be identified by and. et ' and ' be the disaceent of nodes and in oca co ordinate syste. In goba co ordinate syste, each node has two degrees of freedo. Ths,, and, are the noda disaceents at nodes and resectiey aong x - and y - directions. et the trss eber be incined to x axis by θ as shown in figre. It is obsered fro the figre that ' is eqa to the rojection of on x' axis s rojection of on x' -axis. Ths, (ide Fig. 4.7) This ay be written as cosθ + (4.8a) ' cosθ + (4.8b) ' ' ' cosθ cosθ (4.9) Introdcing direction cosines cosθ ; ; the aboe eqation is written as Version CE IIT, Kharagr

10 ' ' (4.a) Or, (4.b) { } [ ]{} T ' In the aboe eqation is the disaceent transforation atrix which transfors the for goba disaceent coonents to two disaceent coonent in oca coordinate syste. [ ] T Version CE IIT, Kharagr

11 et co-ordinates of node be ( x ) and node be ( ), y x, y. Now fro Fig. 4.8, x x cosθ (4.a) y y (4.b) and ( x y (4.c) x) + ( y ) Force transforation atrix et ', ' be the forces in a trss eber at node and resectiey rodcing disaceents ' and ' in the oca co-ordinate syste and,, be the force in goba co-ordinate syste at node and, 3 4 resectiey rodcing disaceents and, (refer Fig. 4.9a-d)., Version CE IIT, Kharagr

12 Referring to fig. 4.9c, the reation between and, ay be written as, ' ' cosθ (4.a) ' (4.b) Siiary referring to Fig. 4.9d, yieds 3 ' cosθ (4.c) 4 ' (4.d) Now the reation between forces in the goba and oca co-ordinate syste ay be written as 3 4 cosθ ' cosθ ' (4.3) T { } [ T ] { ' } (4.4) where atrix { } stands for goba coonents of force and atrix{ ' } are the coonents of forces in the oca co-ordinate syste. The serscrit T stands for the transose of the atrix. The eqation (4.4) transfors the forces in the oca co-ordinate syste to the forces in goba co-ordinate syste. This is accoished by force transforation atrix[ T ] T. Force transforation atrix is the transose of disaceent transforation atrix. Meber Goba Stiffness Matrix Fro eqation (4.6b) we hae, { '} [ k' ]{ ' } Sbstitting for { '} in eqation (4.4), we get T { } [ T ] [ k' ] { ' } (4.5) Version CE IIT, Kharagr

13 Making se of the eqation (4.b), the aboe eqation ay be written as T T k' T (4.6) { } [ ] [ ][ ]{ } { } []{} k (4.7) Eqation (4.7) reresents the eber oad disaceent reation in goba coordinates and ths [ is the eber goba stiffness atrix. Ths, k] T {} k [ T ] [ k' ][ T ] (4.8) [] k cos θ cosθ cos θ cosθ cosθ sin θ cosθ sin θ cos θ cosθ cos θ cosθ cosθ sin θ cosθ sin θ k (4.9) [] Each coonent k of the eber stiffness atrix [ k ] in goba co-ordinates ij reresents the force in x -or y -directions at the end i reqired to case a nit disaceent aong x or y directions at end j. We obtained the eber stiffness atrix in the goba co-ordinates by transforing the eber stiffness atrix in the oca co-ordinates. The eber stiffness atrix in goba co-ordinates can aso be deried fro basic rincies in a direct ethod. Now gie a nit disaceent aong x -direction at node of the trss eber. De to this nit disaceent (see Fig. 4.) the eber ength gets changed in the axia direction by an aont eqa to Δ cos θ. This axia change in ength is reated to the force in the eber in two axia directions by F '' cosθ (4.a) Version CE IIT, Kharagr

14 This force ay be resoed aong and directions. Ths horizonta coonent of force F'' is k cos θ (4.b) Vertica coonent of force F'' is k cosθ (4.c) The forces at the node are readiy fond fro static eqiibri. Ths, k3 k cos θ (4.d) k 4 k cosθ (4.e) The aboe for stiffness coefficients constitte the first con of a stiffness atrix in the goba co-ordinate syste. Siiary, reaining cons of the stiffness atrix ay be obtained. 4.5 Anaysis of ane trss. Nber a the joints and ebers of a ane trss. Aso indicate the degrees of freedo at each node. In a ane trss at each node, we can hae two disaceents. Denote nknown disaceents by ower nbers and known disaceents by higher nbers as shown in Fig In the next ste eaate eber stiffness atrix of a the ebers in the goba co ordinate Version CE IIT, Kharagr

15 syste. Assebe a the stiffness atrices in a articar order, the stiffness atrix K for the entire trss is fond. The assebing rocedre is best exained by considering a sie exae. For this rose consider a two eber trss as shown in Fig. 4.. In the figre, joint nbers, eber nbers and ossibe disaceents of the joints are shown. The area of cross-section of the ebers, its ength and its incination with the x - axis are aso shown. Now the eber stiffness atrix in the goba coordinate syste for both the ebers are gien by Version CE IIT, Kharagr

16 Goba 3 4 Meb 3 4 k (4.a) On the eber stiffness atrix the corresonding eber degrees of freedo and goba degrees of freedo are aso shown. k (4.b) [ ] Note that the eber stiffness atrix in goba co-ordinate syste is deried referring to Fig. 4.b. The node and node reain sae for a the ebers. Howeer in the trss, for eber, the sae node ( i.e. node and in Fig. 4.b) are referred by and resectiey. Siiary for eber, the nodes and are referred by nodes 3 and 4 in the trss. The eber stiffness atrix is of the order 4 4. Howeer the trss has six ossibe disaceents and hence trss stiffness atrix is of the order 6 6. Now it is reqired to t eeents of the eber stiffness atrix of the entire trss. The stiffness atrix of the entire trss is known as assebed stiffness atrix. It is aso known as strctre stiffness atrix; as oera stiffness atrix. Ths, it is cear that by agebraicay adding the aboe two stiffness atrix we get goba stiffness atrix. For exae the eeent k of the eber stiffness atrix of eber st go to ocation ( 3, 3) in the goba stiffness atrix. Siiary k st go to ocation ( 3, 3) in the goba stiffness atrix. The aboe rocedre ay be syboicay written as, K k n i i (4.) + (4.3a) Version CE IIT, Kharagr

17 The assebed stiffness atrix is of the order 6 6. Hence, it is easy to isaize asseby if we exand the eber stiffness atrix to 6 6 size. The issing cons and rows in atrices and are fied with zeroes. Ths, k k + K (4.4) Adding aroriate eeents of first atrix with the aroriate eeents of the second atrix, Version CE IIT, Kharagr

18 K If ore than one eber eet at a joint then the stiffness coefficients of eber stiffness atrix corresonding to that joint are added. After eaating goba stiffness atrix of the trss, the oad disaceent eqation for the trss is written as, { } [ K] { } (4.6) { } { } where is the ector of joint oads acting on the trss, is the ector of joint disaceents and [ K] is the goba stiffness atrix. The aboe eqation is known as the eqiibri eqation. It is obsered that soe joint oads are known and soe are nknown. Aso soe disaceents are known de to sort conditions and soe disaceents are nknown. Hence the aboe eqation ay be artitioned and written as, { { k } } [ k] [ k ] { } [ k ] [ k ] { } k (4.7) where { k }{, k } denote ector of known forces and known disaceents resectiey. And { }, { } denote ector of nknown forces and nknown disaceents resectiey. Exanding eqation 4.7, [ ] [ ] { } k { } + k { } (4.8a) k k Version CE IIT, Kharagr

19 [ ] [ ] { } k { } + k { } (4.8b) k In the resent case (ide Fig. 4.a) the known disaceents are 6. The known disaceents are zero de to bondary conditions. Ths, { } { } k. And fro eqation (4.8a), k Soing { } [ k ] { } [ k ]{ } 3 4,, { k } (4.9) where [ k ] corresonding to stiffness atrix of the trss corresonding to nconstrained degrees of freedo. Now the sort reactions are eaated fro eqation (4.8b). [ k ]{ } { } (4.3) The eber forces are eaated as foows. Sbstitting eqation (4.b) T in eqation (4.6b) ' k' ', one obtains { '} [ ]{} { } [ ]{ } { } [ k' ][ T ]{ } ' (4.3) 5 and Exanding this eqation, ' ' cosθ cosθ (4.3) Version CE IIT, Kharagr

20 Exae 4. Anayse the two eber trss shown in Fig. 4.a. Asse to be constant for a ebers. The ength of each eber is. 5 Version CE IIT, Kharagr

21 The co-ordinate axes, the nber of nodes and ebers are shown in Fig.4.b. The degrees of freedo at each node are aso shown. By insection it is cear that the disaceent. Aso the externa oads are kn ; kn. () Now eber stiffness atrix for each eber in goba co-ordinate syste θ 3. is ( ) k () [ ] k (3) [ ] The goba stiffness atrix of the trss can be obtained by assebing the two eber stiffness atrices. Ths, K (4) [ ] Again stiffness atrix for the nconstrained degrees of freedo is,.5 K (5) 5.5 [ ] Writing the oad disaceent-reation for the trss for the nconstrained degrees of freedo [ ] { } k { } (6) k Version CE IIT, Kharagr

22 (7) ; (8) Sort reactions are eaated sing eqation (4.3). [ k ]{ } { } (9) Sbstitting aroriate aes in eqation (9), { } () () The answer can be erified by eqiibri of joint. Aso, Now force in each eber is cacated as foows, Meber :.866 ;.5 ; 5. { '} [ k' ]{ ' } [ k' ][ T ]{ } Version CE IIT, Kharagr

23 4 3 ' ' { } [ ] 4 3 ' { } [ ] ' kn Meber : 5 ;.5 ; ' ' { } [ ] 4 3 ' { } [ ] ' kn 5 Sary The eber stiffness atrix of a trss eber in oca co-ordinate syste is defined. Sitabe transforation atrices are deried to transfor disaceents and forces fro the oca to goba co-ordinate syste. The eber stiffness atrix of trss eber is obtained in goba co-ordinate syste by sitabe transforation. The syste stiffness atrix of a ane trss is obtained by assebing eber atrices of indiida ebers in goba co-ordinate syste. In the end, a few ane trss robes are soed sing the direct stiffness atrix aroach. Version CE IIT, Kharagr