Development of Truss Equations

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MANE & CIV Introuction to Finite Eeents Prof.

-.9 + ecture notes Suar: Stiffness atri of a bar/truss eeent Coorinate

anasis (incuing utipoint constraints) D Truss eeent Trusses: Engineering

, briges, roof supports Actua trusses: Air structures copose of sener ebers

1 MANE & CIV Introuction to Finite Eeents Prof. Suvranu De Deveopent of Truss Equations Reaing assignent: Chapter : Sections ecture notes Suar: Stiffness atri of a bar/truss eeent Coorinate transforation Stiffness atri of a truss eeent in D space Probes in D truss anasis (incuing utipoint constraints) D Truss eeent Trusses: Engineering structures that are copose on of two-force ebers. e.g., briges, roof supports Actua trusses: Air structures copose of sener ebers (I-beas, channes, anges, bars etc) joine together at their ens b weing, rivete connections or arge bots an pins Gusset pate A tpica truss structure

2 Iea trusses: Assuptions Iea truss ebers are connecte on at their ens. Iea truss ebers are connecte b frictioness pins (no oents) The truss structure is oae on at the pins Weights of the ebers are negecte Frictioness pin A tpica truss structure These assuptions aow us to ieaize each truss eber as a two-force eber (ebers oae on at their etreities b equa opposite an coinear forces) eber in copression eber in tension Connecting pin FEM anasis schee Step : Divie the truss into bar/truss eeents connecte to each other through specia points ( noes ) Step : Describe the behavior of each bar eeent (i.e. erive its stiffness atri an oa vector in oca AND goba coorinate sste) Step : Describe the behavior of the entire truss b putting together the behavior of each of the bar eeents (b assebing their stiffness atrices an oa vectors) Step : App appropriate bounar conitions an sove

3 Stiffness atri of bar eeent E, A Broos/Coe Pubishing / Thoson earning : ength of fbar A: Cross sectiona area of bar E: Eastic (Young s) ouus of bar û(ˆ) :ispaceent of bar as a function of oca coorinate ˆ of bar The strain in the bar at ˆ û ε( ˆ) ˆ The stress in the bar (Hooe s aw) ( ˆ) E ε(ˆ) Tension in the bar T( ˆ) ε ˆ ˆ ˆ ˆ ˆ û(ˆ) ˆ ˆ Assue that the ispaceent û(ˆ) is varing inear aong the bar ˆ ˆ û(ˆ) ˆ ˆ û ˆ ˆ Then, strain is constant aong the bar: ε ˆ E Stress is aso constant aong the bar: Eε ˆ ˆ Tension is constant aong the bar: T ε ˆ ˆ The bar is acting ie a spring with stiffness ˆ Reca the ecture on springs E, A Broos/Coe Pubishing / Thoson earning Two noes:, Noa ispaceents: ˆ ˆ Noa forces: fˆ fˆ Spring constant: Eeent stiffness atri in oca coorinates fˆ - ˆ fˆ ˆ ˆ fˆ - ˆ Eeent force fˆ ˆ ˆ vector Eeent stiffness atri Eeent noa ispaceent vector

4 E, A E, A What if we have bars? This is equivaent to the foowing sste of springs Eeent Eeent PROBEM Probe : Fin the stresses in the two-bar asseb oae as shown beow E, A E, A P Soution: This is equivaent to the foowing sste of springs Eeent Eeent We wi first copute the ispaceent at noe an then the stresses within each eeent The goba set of equations can be generate using the technique eveope in the ecture on springs F F F here an F P Hence, the above set of equations a be epicit written as F () ( ) P () F () P P Fro equation ()

5 To cacuate the stresses: For eeent # first copute the eeent strain () P an then the stress as () () P E AA (eeent in tension) Siiar, in eeent # () P () () P E (eeent in copression) A Broos/Coe Pubishing / Thoson earning Inter-eeent continuit of a two-bar structure Bars in a truss have various orientations eber in copression eber in tension Connecting pin 5

6 ŷ ˆ,fˆ ˆ, fˆ, f, f ˆ,fˆ At noe : At noe : ˆ, f ˆ ˆ, fˆ, f ˆ ˆ ˆ ˆf ˆf ˆf f ˆf f f f In the goba coorinate sste, the vector of noa ispaceents an oas f f ; f f f Our objective is to obtain a reation of the for f Where is the eeent stiffness atri in goba coorinate sste The e is to oo at the oca coorinates ˆ ŷ ˆ,f ˆ ˆ, fˆ fˆ ˆ ˆ fˆ -,f ˆ, fˆ Rewrite as fˆ fˆ fˆ fˆ - - ˆ ˆ ˆ ˆ - ˆ ˆ fˆ ˆ ˆ 6

7 NOTES. Assue that there is no stiffness in the oca ^ irection.. If ou consier the ispaceent at a point aong the oca irection as a vector, then the coponents of that vector aong the goba an irections are the goba an ispaceents.. The epane stiffness atri in the oca coorinates is setric an singuar. NOTES 5. In oca coorinates we have fˆ ˆ But or goa is to obtain the foowing reationship f Hence, nee a reationship between ˆ an an between fˆ an f ˆ Nee to unerstan ˆ how the coponents ˆ ˆ of a vector change ˆ ˆ ˆ with coorinate transforation ˆ ˆ ˆ Transforation of a vector in two iensions ŷ ˆv v v v ˆv v sin v cos v cos v sin ˆ Ange is easure positive in the counter cocwise irection fro the + ais) The vector v has coponents (v, v ) in the goba coorinate sste an (v^, v^ ) in the oca coorinate sste. Fro geoetr ˆv v cos v sin ˆv v sin v cos 7

8 In atri for ˆv cos sin v ˆv sin cos v Or ˆv v ˆv v where Direction cosines cos sin Transforation atri for a singe vector in D * * T reates ˆv T v ˆv v where ˆv an v are coponents of the sae ˆv v vector in oca an goba coorinates, respective. Reationship between ˆ an for the truss eeent At noe At noe ˆ ˆ ˆ ˆ * T Putting these together * T ˆ T ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ T * T T T * Reationship between fˆ an f for the truss eeent At noe At noe Putting these together fˆ * f T fˆ f fˆ * f T fˆ f fˆ Tf f ˆf f ˆf f ˆf ˆf f fˆ f fˆ f fˆ f fˆ f fˆ T f * T T T * 8

9 Iportant propert of the transforation atri T The transforation atri is orthogona, i.e. its inverse is its transpose T T T Use the propert that + = Putting a the pieces together ŷ ˆ, fˆ ˆ, fˆ ˆ fˆ T f ˆ T ˆ, fˆ ˆ, ˆ fˆ The esire reationship is f fˆ ˆ ˆ Tf ˆ T f T ˆ T Where T T ˆ T is the eeent stiffness atri in the goba coorinate sste T ˆ - - T T ˆT 9

10 Coputation of the irection cosines cos sin (, ) (, ) What happens if I reverse the noe nubers? ' cos ' sin (, ) Question: Does the stiffness atri change? (, ) Eape Bar eeent for stiffness atri evauation Broos/Coe Pubishing / Thoson earning E psi A in 6 in b in cos sin Coputation of eeent strains Broos/Coe Pubishing / Thoson earning Reca that the eeent strain is ˆ ˆ ˆ ˆ ε ˆ ˆ ˆ T

11 ε Coputation of eeent stresses stress an tension Reca that the eeent stress is E Eε E ˆ ˆ Reca that the eeent tension is T ε Steps in soving a probe Step : Write own the noe-eeent connectivit tabe ining oca an goba noes; aso for the tabe of irection cosines (, ) Step : Write own the stiffness atri of each eeent in goba coorinate sste with goba nubering Step : Assebe the eeent stiffness atrices to for the goba stiffness atri for the entire structure using the noe eeent connectivit tabe Step : Incorporate appropriate bounar conitions Step 5: Sove resuting set of reuce equations for the unnown ispaceents Step 6: Copute the unnown noa forces

12 Noe eeent connectivit tabe EEMENT Noe Noe E 6 E 6 6 E (, ) (, ) Stiffness atri of eeent () Stiffness atri of eeent () Stiffness atri of eeent () There are egrees of freeo (of) per eeent ( per noe) Goba stiffness atri () K 66 () () How o ou incorporate bounar conitions?

13 Eape E# P P E# 5 o Soution Step : Noe eeent connectivit tabe EEMENT Noe Noe The ength of bars an are equa () E: Young s ouus A: Cross sectiona area of each bar Sove for () an () Stresses in each bar Tabe of noa coorinates Noe cos5 sin5 sin5 Tabe of irection cosines EEMENT ength ength ength cos5 sin5 -cos5 sin5 Step : Stiffness atri of each eeent in goba coorinates with goba nubering Stiffness atri of eeent ()

14 Stiffness atri of eeent () Step : Assebe the goba stiffness atri K The fina set of equations is K F Step : Incorporate bounar conitions Hence reuce set of equations to sove for unnown ispaceents at noe P P

15 Step 5: Sove for unnown ispaceents P E A P E A Step 6: Obtain stresses in the eeents For eeent #: () E E P ( P ) A For eeent #: () E E P ( P ) A Muti-point constraints Broos/Coe Pubishing / Thoson earning Figure -9 Pane truss with incine bounar conitions at noe (see probe wore out in cass) 5

16 Probe : For the pane truss P= N, =ength of eeents an = P E# E= GPa A = 6 - for eeents an = 6 - for eeent E# E# Deterine the unnown ispaceents 5 o an reaction forces. Soution Step : Noe eeent connectivit tabe EEMENT Noe Noe Tabe of noa coorinates Noe Tabe of irection cosines EEMENT ength ength ength / / Step : Stiffness atri of each eeent in goba coorinates with goba nubering Stiffness atri of eeent () 9 - ( )(6 ) 6

17 Stiffness atri of eeent () 9 - ( )(6 ) Stiffness atri of eeent () ( )(6 ) Step : Assebe the goba stiffness atri K N/ The fina set of equations is K F Eq() Step : Incorporate bounar conitions P E# E# E# o 5 Aso, in the oca coorinate sste of eeent How o I convert this to a bounar conition in the goba (,) coorinates? 7

18 F F F P F F F P E# E# E# 5 o Aso, F in the oca coorinate sste of eeent How o I convert this to a bounar conition in the goba (,) coorinates? Using coorinate transforations (Muti-point constraint) Eq () Siiar for the forces at noe F F F n F F F F F F F F F F F F F F F Eq () 8

19 Therefore we nee to sove the foowing equations siutaneous K F Eq() Eq() F F Eq() Incorporate bounar conitions an reuce Eq() to P 5 F F Write these equations out epicit 5 6 ( ) P 5 F 5 F 6 (.5.5 ) 6 (.5.5 ) Eq() Eq(5) Eq(6) A Eq (5) an (6) 5 6 ( ) F F using Eq() using Eq() 5 6 ( ) Eq(7) 5 6 ( ) P Pug this into Eq() Copute the reaction forces F.5.5 F F 6 F.5.5 F N 5 5 9

20 Phsica significance of the stiffness atri In genera, we wi have a stiffness atri of the for K An the finite eeent force-ispaceent reation F F F Phsica significance of the stiffness atri The first equation is F Force equiibriu equation at noe Couns of the goba stiffness atri What if =, =, =? Whie.o.f an are he fie F Force aong.o.f ue to unit ispaceent at.o.f F Force aong.o.f ue to unit ispaceent at.o.f F Force aong.o.f ue to unit ispaceent at.o.f Siiar we obtain the phsica significance of the other entries of the goba stiffness atri In genera ij = Force at.o.f i ue to unit ispaceent at.o.f j eeping a the other.o.fs fie

21 Eape E# P P The ength of bars an are equa () E: Young s ouus A: Cross sectiona area of each bar Sove for an using the phsica interpretation approach E# 5 o Soution Notice that the fina set of equations wi be of the for P P Where,, an wi be eterine using the phsica interpretation approach To obtain the first coun.cos(5) F = E# F = E#.cos(5) = app T F = T F = Force equiibriu F Tcos(5) T cos(5) F Tsin(5) T sin(5) Force-eforation reations T T Cobining force equiibriu an force-eforation reations T T T T Now use the geoetric (copatibiit) conitions (see figure).cos(5).cos(5) Fina ( )

22 To obtain the secon coun E# E#.cos(5) =.cos(5) app T F = T F = Force equiibriu F Tcos(5) T cos(5) F Tsin(5) T sin(5) Force-eforation reations T T Cobining force equiibriu an force-eforation reations T T T T Now use the geoetric (copatibiit) conitions (see figure).cos(5).cos(5) This negative is ue to copression Fina ( ) Broos/Coe Pubishing / Thoson earning D Truss (space truss)

23 In oca coorinate sste fˆ fˆ fˆ fˆ fˆ fˆ z z fˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ z z The transforation atri for a singe vector in D * ˆ T * T n n n Broos/Coe Pubishing / Thoson earning, an n are the irection cosines of ^ cos cos n cos z Transforation atri T reating the oca an goba ispaceent an oa vectors of the truss eeent ˆ T fˆ Tf * T T 6 6 T * Eeent stiffness atri in goba coorinates T 66 T ˆ T

24 ˆT T n n n n n n n n n n n n n n T n n n n n n Notice that the irection cosines of on the oca ais enter the atri ^

14 - OSCILLATIONS Page 1

14 - OSCILLATIONS Page 1 14.1 Perioic an Osciator otion Motion of a sste at reguar interva of tie on a efinite path about a efinite point is known as a perioic otion, e.g., unifor circuar otion of a partice.