Structural analysis - displacement method for planar frames and gridworks

Transcription

1 Structura anayss - dspacement method for panar frames and grdworks Petr Řeřcha October, 6 Contents Summary of the sope defecton method notaton and formuas Dspacement method for panar frames. End forces and moments Pn supported beams Coordnate transformaton Equbrum equatons Exampes Frame Grdwork frame structures 8. Beam torsona stffness A smpe rectanguar grdwork Genera grdworks, matrx formuaton Grdwork matrx formuaton, exampes grdexampe

2 Summary of the sope defecton method notaton and formuas V, M, f M, ψ,, V, M, f M, A straght prsmatc beam, n a pane probem can be represented by ts end ont rotatons ϕ, ϕ and beam rotaton ψ, on the dspacement sde and by the end moments M, M, and shear forces V, and V,. Axa deformaton s negected so the beam s rgd n the beam axs drecton. Further, beam bendng stffness k, s ntroduced for convenence: ( ) EJ k, = () The basc formua of the sope defecton method s the one for the end moments: M, = k, (ϕ + ϕ ψ, ) + M f, (), Fg. Notaton and postve sense for end dspacements and forces n the sope defecton method Fxed end moments M f, are defned n fgure. If end s pn supported (M, = ) then a modfed formua s vad: M, = k, (.5ϕ.5ψ, ) + M f,.5m f, () Some fxed end moments formuas are aso repeated to keep the notes reasonaby sef contaned: M M M a a Fxed-end moments oad M f, M f, oad M f, M f, f F b b M M M M F ab Mb a b F a b Ma b a f / f / M M M f a f t= t upper tower h α,ej constant M M M fa 6 8a+a fa a 4 f / f / t α EJ h t α EJ h Dspacement method for panar frames Beam axa deformaton s accounted for n the dspacement method. Basc equatons are obtaned as a smpe extenson of the equatons of the sope defecton method. Notaton and sgn conventons are ntroduced n fgure. Note that postve oca cartesan x axs ponts from ont to ont! Ths defnes postve drectons and senses of a dspacement and force quanttes. Note aso that for any partcuar beam wth end onts 5,4, for nstance, two aternatves of the oca system are possbe: the beam may be ether 5,4 or 4,5! Needess to say, postve rotatons and moments are aways counter-cockwse n the dspacement method. Each beam has ts own oca coordnate system defned above. Snce dfferent beams can have dfferent orentatons n a frame, another coordnate system s necessary, unque n the frame. Ths system s caed the goba system.

3 M X u EJ,, v v u Fg. Notaton and postve sense for end dspacements and forces M X In order to have strcty conugate dspacement and force quanttes, end forces and X assume opposte postve senses than the correspondng nterna forces (reca that end moment M, n the sope defecton method aso has opposte postve sense than the correspondng bendng moment at secton ). The atera and axa deformatons and forces reman decouped n a straght beam. Note that the atera end forces depend on the dfference v v of the atera end dspacements, not on ther absoute vaues and the same appes to the axa end forces and dspacements.. End forces and moments End moments can be expressed n terms of the end dspacements wth the ad of equaton () snce ψ, = (v v )/, : M = k, (, (v v ) + M f In order to smpfy the formuas, the doube ndces of the end moments and forces are repaced by smpe ndces of the onts when confuson s not possbe. The formua for M foows when the subscrpts are swapped. End atera force at ont s = V = M + M + f, = k (, + + ) (v v ) +,,, Owng to the postve sense of, however, = V = M + M + f, = k (, + + ) + (v v ),,, f {}}{ (M f + M f ) + f,, f {}}{ (M f + M f ) + f,, f, f, Fg. Free-end atera forces f, defnton. Free-end atera forces f, are defned n fgure as the reactons of a smpe beam, whereas f are the fxed-end atera forces on the beam wth both ends camped. Axa end force for prsmatc beams s X = EA,, (u u ) = a, (u u )

4 where a, denotes beam axa stffness. Agan, formua for X s obtaned by swappng ndces and. The beam oad components n the beam axs drecton are not accounted for n ths formua for smpcty. If present, they are assgned to onts by some trbutary rue, 5% each for nstance. Orderng the knematc and force quanttes propery, a matrx equaton can be wrtten for the end forces of a beam. Subscrpt, of the beam s omtted for brevty n the equaton and n the rest of the present secton snce a quanttes refer to a snge beam: {S} { }} { X M X M = [K] {u} {}}{{}}{ a a 6 k k 6 k k k k k a 6 k k k u v u v + {F } { }} { f M f f M f (4) where the sub-dagona symmetrc part of matrx [K] s omtted. Matrx [K] aone can represent the beam eastc propertes for the rest of the frame. Computer codes for panar frames are amost excusvey based on ths matrx equaton. The square matrx [K] s the beam stffness matrx... Pn supported beams In anaogy to the sope defecton method, one rotaton and the conugate end moment can be emnated from the matrx equaton for beams wth one end pn supported. The stffness matrx for such a beam s obtaned by a) omttng or puttng to zero the coumn and row assocated wth the rotaton of the pn supported ont (the thrd or sxth coumn and row), and, b) repacng a factors, and 6 by factor / =.5 n the remang eements of matrx (4). The stffness matrx of a beam wth end pn supported reads then: {S} { }} { X M X M = [K] {u} {}}{{}}{ a a.5 k.5 k.5 k.5k.5 k a.5 k u v u v + {F } {}} { f, M f, f, M f, = M f.5m f, f, = M f, + f, = M f.5m f + f,, f, = M f.5m f + f,,,, Ths pre-emnaton of a degree of freedom (DOF) s a partcuar case of a standard matrx operaton denoted condensaton of DOF (or DOFs) n the context of the fnte eement method (FEM). Condensaton can aternatvey be carred out n the goba stffness matrx of the whoe frame (on the goba eve) as n exampe... Condensaton on the goba eve, however, s dffcut for nterna hnges for whch the above modfcaton of the beam stffness matrx s recommended. Matrces are dvded n submatrces x n equaton (4) n order to factate the next step, whch s generazaton to ncned beams. (5) 4

5 .. Coordnate transformaton g (y,v,) (y,v,) α (x,u,n) g (x,u,n) Fg. 4 Goba and oca (beam) coordnate For ont u v = Transatons and forces components n equaton (4) are n the beam oca coordnate system wth x-axs n the beam axs. Beams of varous orentatons occur n a genera frame. Some unque coordnate system needs to be estabshed n each frame n order to have common prmary unknowns and guarantee correct superposton of the end forces at onts. Ths common coordnate system s the goba one. If the beam axs n ncned wth respect to the goba x coordnate axs by ange α then standard transformaton matrx [T ] can be utzed to obtan the correct components of the transaton and force vectors. Nether rotatons nor the end moments M depend on the coordnate system orentaton. cos α sn α sn α cos α The same matrx s appcabe to the force components u v g or {u } = [T ]{u } g (6) {S } = [T ]{S } g The transformaton matrx for a components of a beam can be composed and for the end forces { {u } {u } } = [ [T ] [T ] ] { {u } g {u } g Both transformaton matrces [T ] and [T ] are orthogona, t hods } or {u} = [T ]{u} g (7) {S} = [T ]{S} g (8) [T ] = [T ] T [T ] = [T ] T Equaton (4) s rewrtten, reca that t s derved n the oca beam system: Substtutons and orthogonaty yed fnay: {S} g = {S} = [K] {u} + {F } [K] g {F } g {}}{{}}{ [T ] T [K] [T ]{u} g + [T ] T {F } (9) Equatons (4) and (9) dever the end forces and moments of a prsmatc beam n any poston n terms of ts end transatons and rotatons, n oca and goba cartesan systems, respectvey. It s worth mentonng that the reverse s not true, snce stffness matrces [K] and [K] g are snguar. Indeed, rgd body moton cannot be recovered from the end forces. It s worth mentonng that equaton (9) s vad for any coordnate transformaton even n genera D (space) frames, grds etc. 5

6 . Equbrum equatons Equbrum equatons are set up qute anaogousy to the sope defecton method. Contrbutons of the connectng members are added for each ont. In most computer codes ths process s mpemented n a oop whch runs on a members and for ths purpose members are assgned unque ndces whch appear as superscrpts n ths secton. Ths technque has become more or ess standard n a fnte eement codes. There are three standard equbrum condtons at each ont and contrbutons of the members can thus be assembed n matrx form. The beam stffness matrx s perceved as composed of four submatrces for the purpose. [K], = [ [K],, [K], T, [K],, [K],, where, s the member ndex. From these submatrces, the goba stffness matrx and oad (coumn) matrx are assembed. The process s demonstrated on a smpe structure n fgure 5 4 Fg. 5 Smpe frame wthout supports, member ndces n crces. ont 4 [K],, [K],, [K], T, [K],, ] () Goba stffness matrx has 4 = rows and coumns and s graduay fed wth submatrces of the members. Correspondng dspacements are not shown n the fgure. Ths s how the matrx ooks after beam, has been nserted (eft) and when the assemby s compete (rght): ont 4 [K],, [K],, [K],, [K],, [K], T, [K], T, [K],, + [K],, + [K],4, [K],4,4 4 A submatrces must be transformed to the goba coordnate system pror to the aocaton n the goba matrx. The oad coumn matrx s assembed anaogousy from the contrbutons of the member matrces. Supports present homogeneous knematc boundary condtons n the anguage of the fnte eement method and mathematcs. Such condtons are mpemented n the matrx form of the equbrum condtons smpy by eavng out the correspondng coumns and rows of the goba stffness matrx and oad coumn matrx. 4 Fg. 6 Smpe frame wth supports 4 [K],4 T,4 [K],4 4,4 For the supports n fgure 6, a three dspacement components are restraned at ont, both transatons are restraned at ont and horzonta transaton s restraned at ont 4. Accordngy, rows and coumns,,,4,5 and are eft out from the goba stffness matrx and oad coumn matrx. The stffness matrx shrnks to 6 6 sze and becomes reguar n ths step. The process descrbed above appears n countess modfcatons n actua structura anayss codes but ts subtetes are beyond the scope of these notes. Unfortunatey, t s aso mpractca to demonstrate t on numerca exampes, snce manua, per matrx eement, operatons are tedous. 6

7 . Exampes.. Frame x g.5 α x, Constant EJ everywhere, E = 6.5 Cross-secton.98.98, A =.96, EJ =.5, AE = 6.5 EA =.5, EJ = k =.4 Stffness matrx of eement, n the oca coordnate frame (matrx of eement, s the same): [K],, = symm Rotaton ϕ can be emnated from the condton M, =. It shows n matrx [K],, as the emnaton of the ast coumn wth the ad of the ast row (the process s caed condensaton n the fnte eement method context): [K],, (condensed) =.6.4 symm Note (and possby check) that ths s equvaent to usng the modfed beam stffness matrx specfed n sec.... The two ast coumns and rows of the condensed matrx can be dsmssed snce the assocated dspacements at ont are constraned: [K],, (condensed and strpped) = symm..6 The two steps (condensaton and strppng) can be carred out n reversed order, too. Ths oca stffness matrx s vad for beam, as we snce the same supports (boundary condtons) appy (that s, [K],, (condensed and strpped) = [K],, (condensed and strpped)). The transformaton matrx for beam, : [K],,g (cond. and strp.) = [T ] T, [T ], = = symm..6

8 Ths stffness matrx s vad for the transaton components of ont n the goba coordnate system and ts rotaton. The transformaton matrx for beam, s [T ], = Transformaton of [K],, (condensed and strpped) woud dever the other stffness matrx. The assemby smpfes to a smpe sum of the two matrces n ths partcuar exampe because the remanng DOFs at ont are the same for both members. The matrx equbrum equaton [K]{u} + {F } = {} has rank, the oad term apparenty s {F } T = {,, } and standard souton devers the dspacements {u}. The reader s recommended to carry ths out for exercse. Instead of t, symmetry of the structure s utzed here to factate manua souton. Note that the actua oad case s antsymmetrc. The spt technque repaces the actua structure wth ts symmetrc haf. The stffness matrx of eement, then s suffcent for the souton. Owng to.5 the vertca constrant at ont, the respectve coumn and row of the matrx can be dsmssed. The fna matrx and the rght hand sde of the equbrum equatons read: wth souton u,x =., ϕ =.75. It turns out that bendng moments and shear forces vansh. It can be deduced mmedatey from the dagram of the antsymmetrc oad case snce the symmetrc haf of the structure s statcay determnate. The bendng moments and shear forces obvousy vansh n ths structure. Grdwork frame structures Grdwork frames are a drect generazaton of statcay determnate panar frames oaded excusvey ateray (bacony grders) to statcay ndetermnate frames. In these notes, ust frames consstng of straght beams are consdered athough n prncpe curved beams aso can appear n grdworks. Ony atera oads are consdered, see fgure 7. Vectors of the externa oad coupes, f present, must e n the grdwork pane. z y x,y,y Fg. 7 A sampe rectanguar grdwork wth goba coordnate system Wth ths mtaton, grdwork beams can aso be perceved as pane beams oaded n ther panes whch addtonay resst and bear torque. Both the force and the dspacement methods can be apped to grd frames. The number of redundants usuay s hgh (tweve n fgure 7) and the dspacement method generay s preferred. The treatment s thus confned to the dspacement method here. Torsona deformaton occurs when dfferent rotatons are mposed around the beam axs at two neghbourng onts. An nstance s ndcated n fgure 7 for beam,. 8

9 It s apparent that ont rotaton n a grd s a vector wth two components x,,y n the grd pane. The dspacement of ont s determned by the two rotaton components and vertca defecton w. These three knematc varabes are the standard degrees of freedom (DOFs) and prmary unknowns n the dspacement method. Rotaton vectors at onts of a beam are decomposed nto the components t n the beam axs and n norma to t, see fgure 8. The postve t axs goes from ont to ont. Postve sense of axs n s seected so that t, n and z bud a rght-handed coordnate system, the oca coordnate system of the beam. Symbos t and n are nterm n the next two subsectons n order to smpfy notaton. More consstent symbos x and y for these axes are used from sec.. on. Standard transformaton can then be apped from x, y to t, n (x, y ) components and back. Two mportant restrctons are adopted: Beams are prsmatc. Prncpa axes of ther cross-sectons e n the t n and t z panes. These restrctons mpy that bendng occurs n the member s t z pane ony and axa rotaton components t nduce pure torson. Equbrum condtons can be assembed wth the beam stffness characterstcs known from the panar frames anayss, wth one excepton the beam torsona stffness must be derved.. Beam torsona stffness The reatve torsona rotaton of the two end onts of beam, n fgure 8 s = t t. y z n (<ο) n n t t t x Fg. 8 Torque of a snge beam. The torque expectedy depends on ths reatve rotaton and on the torsona stffness of the beam. The theory s mted to prsmatc beams here and then ( ) GJt T, = () Here GJ t s the cross-secton torsona stffness. It s derved n the theory of eastcty and s the product of the shear eastcty moduus G and the cross-secton moment of torque stffness J t. The atter equas the cross-secton poar moment of nerta J p for crcuar sectons, otherwse J t J p. Another symbo, the beam torsona stffness g, = (GJ t /),, s ntroduced for brevty. Torque fnay s converted to the end axa moments M t, and M t,. M t, = T, = g, ( t t ) = g, ( t t ) () It s easy to show that ndces, can be swapped n the formua: M t, = T, = g, ( t t ), 9

10 Postve senses of a rotatons and moments vectors n equaton () are from to, see fgure 9. M, t M, z M, n n t t M, t Fg. 9 End moment notaton n sometrc vew. A smpe rectanguar grdwork Manua assemby of the equbrum condtons s ustrated n a smpe exampe. The grd frame n fgure features one free ont, number. The prmary unknowns are defecton w and two rotaton components x and y of the ont. Jont subscrpt s omtted for brevty. Coordnate transformaton can be spared n rectanguar grds when the oca beam axes concde wth the goba ones. Care shoud be gven to proper sgns when addng the bendng and axa end moments. Fexura and torsona stffnesses of the cross-secton are EJ = and GJ t =.5 for smpcty. The beam fexura stffnesses then are k, = f.5, k, =. The torsona stffness s g, =.5. The next step y vew beam, n the sope defecton or dspacement methods paradgm s to set up the expressons for the end forces of beams. The formuas revewed n 4 x secton are utzed together wth equatons (). In order to conform w to the sgn conventon ntroduced n secton, beams shoud be vewed Fg. A smpe grd frame. aganst the goba x or y axes n order to obtan compatbe end moments from the bendng. A three equbrum condtons refer to ont, the subscrpt s thus omtted. Moment around the goba x-axs conssts of the axa moment from beam, and the bendng end moment from beam,. vew beam, The frst moment foows from equaton () x : M t, + M, = M t, = g, ϕ x The second moment can be evauated wth the ad of equaton () when the beam s vewed aganst the goba x axs as ndcated n fgure. M, = /k, x /k, ψ, + /8f,

11 ψ= w/ w x Keepng the vew drecton and notng that postve w s upwards, t turns out that ψ, = w/, Fg. Vew at beam, aganst goba x axs Smary n the goba y drecton y : M t, + M, = M, = k, y k, ψ, ψ, = w/, M t, = The end moment at ont competes the formuas for end moments Vertca equbrum reads M, = k, y k, ψ, : V, + V, = when postve sense of the shear force and vew drectons n fgure are taken nto account. It s deveoped further wth the knowedge of the sope defecton method V, = M, + M, V, = M, + f,,, After substtutons the equbrum condtons become : =.5k, (ϕ x ψ, ) + f, + f, 8, x : g, x + /k, x + /8f, + /k, /, w = y : k, y + k, /, w = ( k, y + 6k ) (, w +,, k, x + k ), w + 5,, 8 f, = The fna form of the system after the numbers have been substtuted: w x y f f.75 It s nstructve to dentfy the shares of the fexura and torsona stffnesses n the equatons. A snge {}}{ contrbuton of torque appears n eement (, ) of the matrx,.65 = It has a mnute effect on the souton and coud be negected n ths specfc case. Ths s qute typca n practce. Smpfed modes of grdworks are frequent n whch the torsona stffnesses are negected atogether. The ont dspacements and nterna forces are shown n fgure. fex. {}}{ tors.

12 V: M: Fg. Jont dspacement and rotatons components and the nterna forces dagrams for the smpe grdwork Torque 5.9 n beam, s not shown.. Genera grdworks, matrx formuaton The above manua approach s acceptabe for very smpe grds. For genera grds, appcaton can be made of the dspacement method paradgm, n partcuar, the aocaton of ndvdua beam contrbutons n the goba matrx. For that purpose, the standard stffness matrx of the dspacement method (4) s recaed here, smpfed by eavng out the rows and coumns pertanng to axa dspacements and forces: M M w 6k/ symm. k/ k w 6k/ k/ k/ k 6k/ k/ k () x y x z Fg. A sampe genera grdwork wth a goba coordnate system and a oca system of z y x y x y In the context of a genera grdwork structure, ths stffness matrx refers to the rotaton components y n the oca coordnate system of each beam as ndcated n fgure for beam. Note, that components y, M y n the oca coord. system have opposte postve sense than ont rotatons and end moments M n matrx () when x axs runs from to! Reca aso that x, y are denoted t, n n prevous two subsectons. Matrx () s extended to ncude the rotaton x and the assocated torsona stffness g (n the wake of

13 the above note, eements k/ get opposte sgns than n matrx ()): [K b ] : M x M y M x M y w x y 6k/ k/ g k/ k symm. w x y 6k/ k/ g k/ k 6k/ k/ g k/ k (4) Ths matrx can be mutped by the tranformaton matrx [T ] qute anaogousy as the stffness matrx of the genera panar frame beam was mutped n equatons 4-8. The submatrces [T ] of the transformaton matrx apparenty are organzed dfferenty now snce rotatons, not dspacements, are subect to transformaton: [T ] = cos α sn α sn α cos α (5) Most grdworks are rectanguar n the ground pane. The tranformaton can then be repaced by proper aocaton of the eements of the matrx () n the goba stffness matrx. A grdwork n fgure 4 beongs to ths category. The exampe s used here to demonstrate the subtetes of ths haf way from manua to matrx souton and dscourage from ts use. The goba coordnate system seected s shown n yg the fgure as we as the oca coordnate systems 4 of ndvdua beams. The oca x x y rotatons can be g x dentfed wth rotatons n matrx () ncudng y x Fg. 4 A sampe grdwork wth rectanguar members y y the same postve sense whereas the oca x concde wth rotatons x n (4). Bendng stffness factors from () of beams and are then aocated n the goba matrx (.) but obvousy n the coumns and rows x /M x whereas the torque stffness factors from (4) go n the y rows/coumns! Note aso that z g s upwards but z are downwards n a beams. The beam numbers are dstngushed by crces n fgure 4 and matrx (.). Beam s more ntrcate to assembe. When postve rotatons n matrx () are to agree wth postve sense of the goba x rotaton, the oca x axs s mped as shown n fgure 4. Rght-handed coordnate system entas then the ndcated sense of the oca y axs, and, fnay, the assgnement = and = of the ont numbers n (). For nstance, note that [K, ] submatrx of matrx () s ocated to [K, ] submatrx of (.). The dfference n respect to [K, ] s soey n the sgn of the off-dagona eement whch s underned n (.) for easy reference. Ths manua assemby s ntrcate and prone to errors. In computer codes, the matrces of ndvdua beams smpy are a transformed wth the ad of [T ] and the agorthm s much smper.

14 M x w x y 6k + 6k k k M y k symmetrc k k +g g + k M x M y w x y 6k k g k k 6k + 6k k.4 Grdwork matrx formuaton, exampes.4. grdexampe Ground pane of the grdwork s n fgure 5. k k k k + g k + g Matera and cross-secton (.4.4) data: y g xg F= α 5 4 E = 9., J =.56, EJ = G = E/ (ν = ), GJ t = Beam stffnesses: k =.4, g =. Fg. 5 Grdwork exampe, ground pane Note: the order of ont ndces n beam matrces subscrpts defnes the orentaton of the oca x axes of the beams - from to n beam, and from to n beam,. Accordng to that the transformaton matrx of beam, s set up. [K, ] = [K, ] =

15 Interna forces dagrams:.5.5 [T, = 5 [K, ],, [T, ] = 5 [K, ],,g = [T, ] T [K, ],, [T, ] = [K] g = [K, ],,g + [K, ],, = {F } = [K] g {u} + {F } = {} 8. {u} = S, = [K, ],, {u} =.5 S, = [K, ],, [T ]{u} =.5.5 V: T: M: Fg. 6 Grdwork exampe, resuts. In shear force dagram the vew drectons at ndvdua beams are ndcated by arrows 5