101-305 Theor of Strutures 1-1 Instrutor: ST, P, J nd TPS Struturl Engineering 101-305 Theor of Strutures 1 - Instrutor: ST, P, J nd TPS Struturl Sstems Struture the tion of building: ONSTUTION Struturl Sstem Struturl embers/elements - Primr lod bering omponents, e.g. truss nd frme members - teril behvior limited to linerelsti onl : something (s building) tht is onstruted b: something rrnged in definite pttern of orgniztion mnner of onstrution : the rrngement of prtiles or prts in substne or bod b: orgniztion of prts s dominted b the generl hrter s whole onnetions/joints - Internl onstrution onditions where djent members re tied to eh other - Tpes of onnetions depend on the epeted performne of the joining members, e.g. rigid, internl hinge, nd tpil onnetion kθ the ggregte of elements of n entit in their reltionships to eh other igid Internl hinge Tpil θ Supports/oundr onditions - onstrution onditions t the lotions where the pplied lods re finll trnsmitted to - esisting fores ginst trnsltion/rottion t the support re lled support retions - Pths t whih the pplied lods re trnsferred to the sstem to its supports re lled lod pths
101-305 Theor of Strutures 1-3 Instrutor: ST, P, J nd TPS Supports/oundr onditions 101-305 Theor of Strutures 1-4 Instrutor: ST, P, J nd TPS Tpes of Elements in Struturl Sstems lssifition of struturl sstems is bsed on mn onsidertions Generll, the most fundmentl omponent or elements of struturl sstem re lssified bsed on their mode(s) of deformtion nd lod-resisting behvior + = inge or Pin struturl sstem n be omposed of vrious tpes of elements in order to stisf the funtionl requirements There is no unique w to lssif the struturl sstems oller Struturl Elements Trusses z z Frmes bles rhes Fied Guide Shells Solids
101-305 Theor of Strutures 1-5 Instrutor: ST, P, J nd TPS Trusses 101-305 Theor of Strutures 1-6 Instrutor: ST, P, J nd TPS Frmes Truss elements, the simplest form of struturl elements, rr pure tension nd ompression Truss elements re onneted to other elements with pin onnetions The elements, when loded, re subjeted to onl il fores nd il deformtion Frme elements generll rr sher fores nd bending moments in ddition to il fores Frme elements re onneted to other elements with rigid, semi-rigid, or pin onnetion Pin onnetions do not llow n moment trnsfer between the onneting elements (the onneting elements n rotte freel reltive to eh other) igid onnetions impl omplete moment trnsfer Frme elements n hve multiple modes of deformtion, i.e. il, sher, nd bending
101-305 Theor of Strutures 1-7 Instrutor: ST, P, J nd TPS bles 101-305 Theor of Strutures 1-8 Instrutor: ST, P, J nd TPS rhes bles re slender members in whih the rosssetionl dimensions re reltivel smll ompred with the length Primr lod-rring mehnism is il tension The defleted shpe of ble depends on the pplied loding For disrete loding, the deformed shpe of the ble is series of pieewise stright lines between the points of the pplied lods rhes re urved members tht re supported t eh end suh tht the primr lod-rring mehnism is il ompression rh elements m be subjeted to seondr fores suh s bending nd sher depending on the pplied loding nd support onditions In n ses, il ompression still remins the dominnt mode of tion For distributed loding, the deformed shpe is urve
101-305 Theor of Strutures 1-9 Instrutor: ST, P, J nd TPS Pltes 101-305 Theor of Strutures 1-10 Instrutor: ST, P, J nd TPS Shells Pltes re flt struturl elements whose thikness is reltivel smll ompred with the lterl dimensions The internl tions of plte elements inlude il (membrne), trnsverse sher, nd bending Shells, similr to pltes, re struturl elements whose thikness is smll ompred to the lterl dimensions The differene of shells from pltes is tht the surfe of shell element is not flt but urved ending nd trnsverse sher re rried either norml to or within the plne of the element depending on the pplied loding nd the element orienttion
101-305 Theor of Strutures 1-11 Instrutor: ST, P, J nd TPS Solids 101-305 Theor of Strutures 1-1 Instrutor: ST, P, J nd TPS Struturl nlsis Solid elements re three-dimensionl struturl elements used to model three-dimensionl ontinuum strutures suh s dms, thik bridge dek, et. The internl tion of solid element is usull desribed in terms of stress nd strin in three dimensions relted b onstitutive reltionship sstemti stud of the response of struture under the speified tions (eternl lodings, therml epnsion/ontrtion of struturl members, settlement of supports) Usull rried out b estblishing the reltionships of fores nd deformtions throughout the struture equires knowledge of element behvior, struturl mterils, support onditions, nd pplied lods Epressed s mthemtil lgorithms whih n be lssified s lssil methods nd numeril methods (or mtri methods) lssil ethods Numeril ethods Erl sophistited methods invented to nlze speifi lsses of problems eveloped bsed on priniples of struturl mehnis nd suitble pproimtions Essentil for understnding the fundmentl priniples of struturl mehnis ore generlized mtri-bsed numeril methods developed to nlze omple sstems Generll requires the use of omputers
101-305 Theor of Strutures 1-13 Instrutor: ST, P, J nd TPS Struturl nlsis Proess Strutures 101-305 Theor of Strutures 1-14 Instrutor: ST, P, J nd TPS Struturl esign Proess oneptul esign Struturl odel (pproprite pproimtion) Preliminr esigns Option 1 Option Option N Two-imensionl odel (Plne) Three-imensionl odel (Spe) Struturl nlsis Toolbo Struturl nlsis Toolbo Intermedite esigns Seletion of est Preliminr esign ejetion of ll Preliminr esigns One of the most importnt steps in struturl nlsis is the seletion of n pproprite struturl model bsed on the vilble informtion of the struture In generl, the seletion of struturl model is linked to the determintion of the method of nlsis Finl esign Struturl nlsis Toolbo esign hnges onstrution
101-305 Theor of Strutures 1-15 Instrutor: ST, P, J nd TPS egree of Stti etermin 101-305 Theor of Strutures 1-16 Instrutor: ST, P, J nd TPS Eternl Stti etermin lssifition efinition: r = No. of independent (support) retion omponents No. of equtions of ondition n = Struture Independent etion omponents r Number of Eqs. of ondition n r = + 3 n dditionl stti equtions from speil internl onditions of onstrution For plnr strutures: r r r < 3 + n = 3 + n > 3 + n Sttill unstble eternll Sttill determinte eternll Sttill indeterminte eternll imum no. of stti equilibrium equtions tht n be written
101-305 Theor of Strutures 1-17 Instrutor: ST, P, J nd TPS Eternl & Internl Stti etermin 101-305 Theor of Strutures 1-18 Instrutor: ST, P, J nd TPS Overll Stti etermin lssifition efinition: m = No. of members in struture j = No. of joints Struture j Struture hrteristis n m r For plne frmes nd bems: 3m+ r < 3j+ n Sttill unstble overll 3m+ r = 3j+ n 3m+ r > 3j+ n Sttill determinte overll Sttill indeterminte overll Totl no. of stti equtions tht n be written Totl no. of unknown fore omponents F = 0 F = 0 = 0 z
101-305 Theor of Strutures 1-19 Instrutor: ST, P, J nd TPS Free-od igrm 101-305 Theor of Strutures 1-0 Instrutor: ST, P, J nd TPS Guidelines to rwing F of Plnr Struture Free-od igrm (F) Struturl odeling Sketh of struturl omponent with ll of the pproprite fores ting on it Entire Struture Portion of Struture Use line digrms to represent portion of struture under onsidertion dethed from its supports & disonneted from ll other prts of the struture Selet orienttion of referene - oordinte sstem, nd show ll fore omponents ssuming the positive diretion w.r.t. the referene oordinte sstem Usull gives reltionships between eternl fores Usull gives reltionships between internl nd eternl fores Y X z
101-305 Theor of Strutures 1-1 Instrutor: ST, P, J nd TPS 101-305 Theor of Strutures 1 - Instrutor: ST, P, J nd TPS omputtion of etions Emple 1 5 T rw F of the struture 60 o 5. T hek for eternl stti determin lulte the unknown retions hek the results b using lterntive equilibrium equtions Wh? isonnet struture into rigid portions Write equtions of equilibrium & ondition To void solving simultneous equtions, write the bove equtions s.t. eh eqution involves onl 1 unknown t time 300. 00. 300. 5 T 5. T r = 3 nd struture is stble F = 0: = 5 T 5 = = =. 887 T o tn60 3 = ( ) + ( ) = 5. 774 T F = 0 : + = 5. =. 5. 887 = 0. 387 T = 0 : + 5 3+ = 0 = 5 5 5 3= 10 T-m hek: = 0 5 5+ 3= 10 5+ 5 3= 0 O.K.
101-305 Theor of Strutures 1-3 Instrutor: ST, P, J nd TPS 101-305 Theor of Strutures 1-4 Instrutor: ST, P, J nd TPS Emple Emple 3 10 T 10 T 6 T/m E F 300. r = 5 nd n = r = 3 + n sttill determinte eternll (stble) 00. 00. 00. 00. 00. 00. 00. 3 T 060. 10. F = 0 results in 4 retion unknowns Smmetr of struture & loding redues no. of unknowns to nd 0 10 T 6 T/m E E E E 6 T/m F F 3.6 T-m 7. T-m 3 T 3 T = 0 : 10 + 4 6 5 6 6 = 0 1 = ( 0 + 60 + 36 ) = 9 T 4 F = 0: 10 + 9 6 6 = 0 = 10 9 + 1 + 6 = 1 T hek: = 0 E 6+ 10 4 + 6 1= 6+ 40 58+ 1= 0 O.K. 3 T 3.6 T-m 7. T-m 3 T 3.6 T-m 1.8 T-m 1.8 T-m 3 T 3 T 3 T 1.8 T-m 3 T 1.8 T-m 3.6 T-m
101-305 Theor of Strutures 1-5 Instrutor: ST, P, J nd TPS ems nd Plne Frmes 101-305 Theor of Strutures 1-6 Instrutor: ST, P, J nd TPS eltionships between Lods: & w( ) il Fore F F Sher Fore F Internl fore in the diretion of the entroidl is of the element Internl fore in the diretion perpendiulr to the entroidl is F = 0: s 0: w α w w+ w + w ( + ) = 0 = w d w d = (1) Slope of SF t n point = lod intensit t tht point O + + ending oment Internl ouple ting on the ross setion of the element The vlues of these internl fores n vr long the length of eh struturl member depending on the loding nd boundr (or support) onditions = 0: O ( )( α ) ( ) 0 + + w + = 0 s 0 = + w α d d = () Slope of t n point = sher t tht point
101-305 Theor of Strutures 1-7 Instrutor: ST, P, J nd TPS 101-305 Theor of Strutures 1-8 Instrutor: ST, P, J nd TPS Equtions (1) & () re.e. of equilibrium. Substitution of (1) into () gives d d = w (3) This eqution reltes to w t setion. Eqution (1) n be written Similrl, from Eq. () d = wd = wd (4) = wd (5) hnge in sher between setions nd = re under lod intensit digrm between nd = d (6) = d (7) hnge in moment between setions nd = re under sher fore digrm (SF) between nd emrks For distributed lod, slope of SF t n point = lod intensit t tht point For point lod, slope of SF For distributed lod, hnge in sher over segment = re under lod intensit digrm over the segment For point lod, hnge in sher = lod mgnitude Slope of t n point = sher t tht point nd is onstnt for onstnt sher Slope of = zero t point of zero sher nd is m or min Slope of hnges bruptl t point lod hnge in moment over segment = re under SF over the segment + onve member - onve member zero hnge in urvture t point of infletion
101-305 Theor of Strutures 1-9 Instrutor: ST, P, J nd TPS 101-305 Theor of Strutures 1-30 Instrutor: ST, P, J nd TPS Emple 4 Emple 5 10 T 10 T 6 T/m T/m G E F.00 00. 00. 00. 00. 00. 00. 00. 1 T 9 T 9 T 1 T.00 ( T) ( T-m) 1 18 11 6 3 6 11 18 1 3.00 5 T 4 T-m 4 T 4 T 4 T-m 4 4 3 T-m 4 T-m 4 T-m 1 T G G = 1 = T-m = 1 4 10 = 4 T-m 9 T 4 9 T 4 T 4 4 = 1 7 10 5+ 9 3 6 3 1 5= 3 T-m L. 10 T 1 T 9 T L 9 9 3 ( T-m) ( T) 4 4
101-305 Theor of Strutures 1-31 Instrutor: ST, P, J nd TPS 101-305 Theor of Strutures 1-3 Instrutor: ST, P, J nd TPS Emple 6 3.60 8 T 853.8 T-m 4.00 1.6 T/m 1940.6 T-m 0. m ( T) 9.05.65 8.00.37 5.35 11.75 egree of indetermin: ( 3 + 5) ( 3 3+ 0) = 853.8 T-m = 0 : 853. 8 = = =. 37 T 360 ( T-m) 853.8 853.8 1486. 1940.6 853.8 T-m 8 T 1.6 T/m 1940.6 T-m.37 T.37 T = 0: 1 853. 8 8 8 1940. 6 = + 8 + 16. 8 = 905. T 8 100 100 F = 0: = 8+ 1. 6 8 9. 05= 1175. T
101-305 Theor of Strutures 1-33 Instrutor: ST, P, J nd TPS 101-305 Theor of Strutures 1-34 Instrutor: ST, P, J nd TPS Emple 7 rw the nd sketh the elsti urve for the frme shown. 3 T/m Trnsformtion of oordintes Y' Y 4 T E.00.00 ' tnφ ' u φ ' X ' X.00.00 5.00 ' ' tnφ = 0 : E 5 = = 3 = 75. T 3 T/m 4 T 13.5 4 6 7.5 T = 0 : = 75. + 3 1 3 1+ 4 = 3 T-m = 0 : = 3 4+ 75. = 195. T = 0: = 4 T 7.5 3 T/m E E ( T ) ( T-m) 6 1 15 3 9.375 onsider the trnsformtion of the oordintes of vetor u rotted through n ngle φ in plne ( ) ( ) = ' ' tnφ osφ = ' osφ ' sinφ = ' tnφ + ' osφ = ' sinφ + ' osφ osφ sinφ ' u= = = sinφ osφ ' ene the rottion mtri in - problem is defined s osφ sinφ = sinφ osφ n vetor u in globl oordintes n therefore be obtined from its omponents in lol oordintes b the reltionship u= T u ' onversel, for the trnsformtion to globl oordintes u' = u Note: ' T u T 1 = orthogonl mtri T = I
101-305 Theor of Strutures 1-35 Instrutor: ST, P, J nd TPS 101-305 Theor of Strutures 1-36 Instrutor: ST, P, J nd TPS Emple 8 nlze the frme shown nd sketh the nd the elsti urve. 3 T/m = 0 : (F ) = 3 5. 5. 83 5 = 3. 33 T-m 3 T/m.83 T 1.17 T 1.5 T/m 4.00 1.5 T/m 3.33 T-m 3.33 T-m '.83 T ' 4.00 = 0 : 9 + 15. 4 3 5 5. = 0 = 83. T = 0 : 9 15. 4 3 5 65. = 0 = 1. 17 T = = 0: (F ) = 0 = 5.00 = 0: (F Struture) = 6 T hek: = 0 (F ) 83. 9+ 6 4 15. 4 3 5 5. = 0 O.K..83 T 6 T 1 1 ' osθ sinθ 6 T = ' sinθ osθ = 1 1. 83 T 1 1 ' osθ sinθ 0 T = ' sinθ osθ = 1 1. 83 T 3.33 θ ' '.4 T = 6.65 T T = T (T-m)
101-305 Theor of Strutures 1-37 Instrutor: ST, P, J nd TPS 101-305 Theor of Strutures 1-38 Instrutor: ST, P, J nd TPS Three-hinged rhes Y X w f Y w X N N φ d tnφ = ' = d sinφ = tnφ 1+tn φ osφ = 1 1+tn φ Prboli rh: L L The eqution of the rh tkes the form N osφ sinφ N = sinφ osφ = + + 0 1 with = 0 t = 0 nd = L L nd = f t = 4 f 4 f 0 = 0, 1 =, = L L N = = w N = N osφ sinφ ( ) = osφ w sinφ = N sinφ + osφ ( ) = sinφ + w osφ 4 f = 1 L L The four retion omponents n be found from four equtions of equilibrium L w L f = 0 : = 0 : = = wl L wl L = f + 4 wl = = 8 f
101-305 Theor of Strutures 1-39 Instrutor: ST, P, J nd TPS 101-305 Theor of Strutures 1-40 Instrutor: ST, P, J nd TPS d 4 f tn φ = L d = L ( ) 4 f ( L ) wl 1 L L N = w 8f 4f 4f 1+ ( L ) 1+ ( L ) L L w 4 = L + 16 f ( L ) 8 f 4 f ( L wl ) L L 1 = + w 8f f f 1+ 1+ L L = 0 gi!! 4 4 ( L ) ( L ) w = wl wl 4 f 1 = w 8f L L = 0 gi!! Therefore the lod is rried entirel b il ompression whih vries prbolill long the spn.