Bnding of Prismatic Bams (Initia nots dsignd by Dr. Nazri Kamsah)
St I-bams usd in a roof construction.
5- Gnra Loading Conditions For our anaysis, w wi considr thr typs of oading, as iustratd bow. Not: Positiv oads ar shown. Ths ar: a) Distributd oad, p, ovr a ngth ; b) Concntratd oad, P m, at point m; c) Concntratd momnt, M k, at point k. Goba coordinat systm.
5- Dformation of th Nutra Axis Du to th appid oads, th bam xprincs an in-pan (x-y pan) dformation. Th dformd shap of th bam s nutra axis is shown. Not: Positiv dformation is shown. Dformation at any point is charactrizd by two paramtrs: a) Vrtica dfction, v b) Rotation or Sop, v
5-5 Emntary Bam Euations For sma dformations, mntary bam thory stats, strss, M I y strain, E My EI curvatur, d dx v M EI Not: Th bam s cross-sction is considrd symmtric with rspct to th pan of oading (x-y pan).
5- Finit Emnt Formuation Th bam is subdividd into four sctions or mnts, connctd by nods. Each nod has dgrs of frdom. At nod i, thy ar dispacmnt componnts: Q i- (vrtica dfction) and Q i (sop or rotation). Positiv dformations x Goba coordinat systm Loca coordinat systm ' ' v v v v T
5-7 Hrmit Shap Function W dfin Hrmit shap functions, so as to obtain th dformation at any point aong a sing bam mnt. It is givn in a gnra form, H i a i b c i i d i i,,, Not: a, b, c, and d ar cofficints.
Th shap functions ar xprssd in trms of, as H H H H H H H H = - = or or or or H H H H Vaus of th H i at oca nods ( = -) and ( = +) ar,
5-8 Dispacmnt Function Th dformation at any point on a bam mnt can b xprssd in trms of th Hrmit shap functions as, dv dv v Hv H Hv H d d W nd to rpac th brackt trms. Using chain ru of diffrntiation, Rca, dv d dv dx x x x x dx d dv dx x x x x x dx dξ dv Not: Th trm at oca nod is v and at nod is v. dx
Substituting th brackt trm, v() can now b xprssd as, v H v H v H v H v In a condnsd matrix form, ' ' v H whr H H H H H v ' v v ' v
5-9 Th Potntia Enrgy Approach W now dvop finit mnt formuation for a sing bam mnt using th potntia nrgy approach. Th tota potntia nrgy, p p of th ntir bam is givn by p p L EI d v dx dx L pv dx m P m v m k M k v ' k Intrna strain nrgy Potntia nrgy du to distributd oad, concntratd forc and concntratd momnt Not: p is th distributd oad; P m is point oad at point m; M k is th momnt appid at point k; v m is dfction at point m; v k is sop at point k.
5- Emnt Stiffnss Matrix Th stiffnss matrix [k] for a sing bam mnt is now drivd using th potntia nrgy approach. Th intrna strain nrgy of a sing bam mnt, U L d v EI dx dx with dx d..(i) W nd to xprss (d v/dx ) in trms of H, and. Rca, dv dv d v d v dx d dx d Substitut v = [H]{} and simpifying, yids T T d v d H d H dx d d Assignmnt: Driv th trm U =f() from n (i)..(ii)
Aso, d H d..(iii) Substitut Es.(ii), and (iii) into E.(i) and substituting dx = ( /)d, w obtain, 9 9 8 8 9 T 8EI 8 U d 9 8 Symmtric
Intgrating ach trm in th matrix and noting that, d ; d ; d ; th intrna strain nrgy can now b writtn in condnsd matrix form as, U T k whr [k] rprsnts th mnt stiffnss matrix, givn by k EI Connctivity with oca noda DOFs
Exrcis 5- An ovrhang bam carris transvrs oads as shown. Cross-sction: b = h = 9. mm; A = 8.85 mm Mod th bam using two mnts. Us E = GPa, and I = x mm. a) Writ th stiffnss matrix, [k] for ach mnt; b) Assmb th goba stiffnss matrix, [K] for th ntir bam.
5- Emnt Load Vctor Th distributd oad, p acts aong th ngth of an mnt. It has to b transformd into uivant concntratd forcs, acting at th nods. Th potntia nrgy du to th distributd oad p is L p pv dx Substitut for v H p H d and dx p d w obtain, Substituting for [H] and rarranging yids, p p H H H H d
Expanding H i, intgrating vry trms with rspct to w gt, p p p p p which can b writtn in a condnsd matrix form as, p f T whr {f} T is th uivant noda forc vctor du to th distributd oad p, givn by f p, p, p, p T
Physica Intrprtation Actua oading condition p Euivant noda forcs p p p p Vctor of th uivant noda forcs f p p p p T
Exrcis 5- Rconsidr th ovrhang bam in Exrcis 5-. Cross-sction: b = h = 9. mm; A = 8.85 mm Us E = GPa, I = x mm. For th bam undr th oading shown, a) Writ th vctor of uivant noda forcs for sction AB; b) Assmb th goba oad vctor for th ntir bam.
For a sing bam mnt with distributd oad p, th systm of inar uation can b xprssd as, 5- Systm of Linar Euations Expanding, w hav f k p p p p ' ' v v v v EI Not: Connctivity with oca noda dgrs of frdom is shown.
Exrcis 5- Rconsidr th ovrhang bam in Exrcis 5-. Cross-sction: b = h = 9. mm; A = 8.85 mm Us E = GPa, I = x mm. a) Assmb th goba systm of inar uations (SLEs); b) Impos th boundary conditions, and writ th rducd SLEs; c) Sov for th unknown noda dgrs of frdom; d) Find th vrtica dfction, v at th midd of sction AB.
Bnding Momnt 5- Shar Forc and Bnding Momnt Bnding momnt M at any point aong th bam mnt is, It was shown that, with dx v d EI M 8 8 dx v d d v d dx v d (i) (ii) v H
EI M Substituting.(ii) into.(i) and simpifying yids, Not: Rca that, ' ' v v v v
Shar Forc Shar forc V at any point aong th bam mnt, V dm dx EI It can b shown that, with d v dx 8 d dx d v d v v H (iii) d v dx 8 8 8 (iv) Substituting.(iv) into.(iii) and simpifying yids, V EI
Exrcis 5- Rconsidr th ovrhang bam in Exrcis -. Cross-sction: b = h = 9. mm; A = 8.85 mm Us E = GPa, I = x mm. a) Estimat th bnding momnt and shar forc at th midpoint of sction AB. b) Comput th raction forcs at support B.
Examp 5- For a bam with th oading shown bow, dtrmin: (a) th sops at nods and, and (b) th vrtica dfction at midpoint of sction -. Euivant noda forcs for sction -.
Soution. Stiffnss matrix for ach mnt, k 8 Q Q Q Q Q Q Q 5 Q k 8
8 8 5 K Nm N, f. Assmb goba stiffnss matrix for ntir bam,. Euivant noda forc vctor du to distributd oad p, on mnt Q Q Q Q Q 5 Q m N
F. Assmb goba oad vctor, 8 8 5 5 Q Q Q Q Q Q 5. Writ th goba systm of inar uations,
8 8 5 Q Q..79 Q Q. Impos th boundary conditions. W hav, 7. Soving for th unknown dgrs of frdom, w obtain, 5 Q Q Q Q Using th imination mthod, th goba SLEs rducd to, which ar th sops at nods and rspctivy. radians 8. Th vrtica dfction at any point on th mnt is givn by, H H H H v
At th midpoint of th mnt, =. So that, H H H H m 5 8 9 79. v.. H H H H v Aso, for mnt, =, = Q = -.79-, =, = Q =.-. Substituting ths into v(), w gt
- Pan Fram Structur Pan structurs is mad up of rigidy connctd mmbrs. Each mmbr can b tratd as a bam. Axia oads xist, thus axia dformations occur in fram structurs. Emnts of th fram structur can b in diffrnt orintation.
Considr a sing fram mnt. Not: Each oca nod has thr dgrs of frdom, i.. two dispacmnt componnts and on rotationa componnt. Noda dispacmnt vctor in goba coordinat systm is givn by: T 5 Noda dispacmnt vctor in oca coordinat systm is givn by: ' ' ' ' ' ' T ' 5
Not that th rotationa componnts, = and =, Th oca DOFs transform into th goba DOFs according to a ration whr [L] is th transformation matrix givn by, L ' m m m m L Not: and m ar th dirction cosins for th fram mnt.
Not:,, 5, and ar th bam dgrs of frdom, whi and ar simiar to th dispacmnts of a -D mnt. Combining th stiffnss matrics for a bam and -D mnts givs stiffnss matrix for a fram mnt, i.. k ' 5 EA EA EI EI EI EI EI EI EI EI EA EA EI EI EI EI EI EI EI EI
As shown in th drivation of a truss mnt, th mnt strain nrgy is givn by: T U ' k ' ' T T U L k ' L Thrfor th mnt stiffnss matrix for a sing fram mnt, in goba coordinats is: k L T k ' L
Emnt Forc Vctor Suppos a uniformy distributd oad p acts on a fram mnt. W nd to trans-form p into uiva-nt noda forcs. It can b shown that: ' T f ' T L T f ' whr f ' p p p p T (in oca coordinat) Rca, f T L f '
Examp - For a porta fram structur shown, dtrmin th dispacmnts and rotations at joints and.
Soution Stp : Emnt connctivity Emnt No st Nod nd Nod Stp : Emnt stiffnss matrics Emnt k k ' Q Q Q Q Q 5 Q. 7. 7. 78 5.. 78 5. 5. 57 5. 78. 7. 7. 78. 78 5. 5. 5. 78 5. 57
Emnt & (Loca coordinat) For mnt k ' k ' For mnt Q 7 Q 8 Q 9 Q Q Q.5.5.5 7.5 7 7 85 7.5.5.5 7.5 7 7 7 85 Q Q Q Q Q 5 Q Th transformation matrix [L] for mnt and, whr = and m =, is as foows: T L k L k ' L
Emnt & (Goba coordinat) For mnt Q 7 Q 8 Q 9 Q Q Q.5 7.5 7.5.5 7 85 7.5 7.5 7.5.5 7 7 85 For mnt Q Q Q Q Q 5 Q k k Stp : Assmb th goba stiffnss matrix K. 7.7. 5..78 5. 7 5. 5 5. 78.7. 7.78 5.. 5. 5. 78 7 5. 5
Stp : Estabish th oad vctor F 7 7 Stp 5: Writ th systm of inar uations K Q F. 7.7. 5..78 5. 7 5. 5 5. 78.7. 7.78 5.. 5. Q 5. Q 78 Q 7 7 Q 5.Q 5 5 Q 7
Stp : Soving th systm of inar uations givs us th unknown dispacmnts and rotations, i.. Q.9 in. in.9 rad.9 in.8 in 5.88 rad