S. Socrate 2013 K. Qian

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Godwin Chambers
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1 S. Socrte 213 K. Qi

odig Coditios o ech Sectio () pplied lodig oly log the is () of the br.

d the etire il force digrm is: à N() F C For distributed lodig f () (with f () i N/m + log ), obti the

2 odig Coditios o ech Sectio () pplied lodig oly log the is () of the br. The oly iterl resultt t y sectios is the il force N() Fid N()log the br (il force digrm) by cuttig the br t ech d imposig -equilibrium. For the emple show, equilibrium t gives: for < B :Σ F N() +F C +F B à N() F C + F B for > B :Σ F N()+F C d the etire il force digrm is: à N() F C For distributed lodig f () (with f () i N/m + log ), obti the force by itegrtig f () log the br. For the br show: N( ) f ( ) d The differetil reltioship betwee the distributed lod f () d the il force N() is dn( ) d f () This c be directly obtied from ΣF o d slice of the br

à Stri ß à sectio deformtio cross sectios sty flt à Sme ε t ll poits of sectio Structurl respose Elogtio: Displcemet

3 Kiemtics costrit (geometry of deformtio) Cross sectios : sty flt, trslte by u () Sectio deformtio Sectio t hs displcemet u () Sectio t +d hs displcemet u (+d) u + ocl mesure of deformtio t sectio : (chge i legth)/(origil legth) à Stri ß à sectio deformtio cross sectios sty flt à Sme ε t ll poits of sectio Structurl respose Elogtio: Displcemet field : with u ( )u ε ( ) ( ) d δ d d d ε ( ) u d d u + ε ( ) u( ) + d ( ) d determied by Boudry coditios (e.g., u t support)

Sectio equilibrium The il Force N() t sectio is obtied by

stress σ N() σ (, d Costitutive Properties If the mteril is

obtied s: σ (,, ε ( ) Sectio Respose N() N() σ (, d d, d

If oly 1 mteril, )à (E) eff )(); Costt over cross sectio, ivert

4 Sectio equilibrium The il Force N() t sectio is obtied by itegrtig the cotributios of ech elemetl re d, which crries orml stress σ N() σ (, d Costitutive Properties If the mteril is lier elstic, d the molus of elemetl re d is E, the stress c be obtied s: σ (,, ε ( ) Sectio Respose N() N() σ (, d d, d Effective Sectio Stiffess:, ε ( ) d d ( E) ( E) eff eff ( ) ( ) If oly 1 mteril, )à (E) eff )(); Costt over cross sectio, ivert d, d d d ( ) N() ( E) ( ) eff If 2 mterils (E 1, E 2 ) à (E) eff E E 2 2

Specil cse: homogeeous br (molus E) ; costt cross

N σ : orml stress δ ε : il stri σ E : Youg's Molus

5 Specil cse: homogeeous br (molus E) ; costt cross sectio ; costt il force Equilibrium () ΣF à N P (costt log br) N σ 1 1 K E structure δ mteril δ - : elogtio of the br N P K : δ δ il stiffess of the br N σ : orml stress δ ε : il stri σ E : Youg's Molus of the mteril ε K mteril, geometry δ P E 1/ K E ; P E δ K ε

6 Solutio Proceres: 1) Force Method for Stticlly Idetermite (SI) Brs i il odig Remove redt support à Replce with redt rectio Solve the compio Stticlly Determite (SD) problem Obti the displcemet t the redt support i terms of the redt rectio Impose zero displcemet t the redt support d obti the redt rectio Bck-substitute the redt rectio i the solutio to the compio SD probem to fid the solutio to the SI problem. 2) Method of Joits for Stticlly Determite trusses Determie the free d costried DOFs of the joits i the truss Drw FBDs of the joits for the free DOFs. Drw the ukow il forces, N, positive, i.e., comig out of the joits. Impose equilibrium of the joits log the free DOFs (ΣF d/or ΣF y ). Obti ll the il forces i the brs. Drw FBDs of the joits for the costried DOFs. Now the il forces re kow: drw them the wy they ct (pushig the joit if N is compressive, pullig if N is tesile) idictig their mgitude i the FBDs. Drw the crtesi compoets of the ukow rectios positive ( log d y) Impose equiibrium of the joits log the costried DOFs (ΣF d/or ΣF y ). Obti the crtesi compoets of the rectios t the supports. Check globl equilibrium!

7 3. Geometry of deformtio i trusses 1. For displcig joit, J, with displcemet vector u J : {u J, uj y }, subjected to ukow lod P J : {P J,PJ y }: 2. Drw ech br, e.g., J, coected to joit J for ech br obti the crtesi compoets of the uit vector log the br poitig towrd the movig joit, J (e.g., for J : r J : {r J, r J y }) obti the elogtio of ech br by sclr proct betwee the vectors e.g., for J : δ J r J u J r J uj + r J y uj y obti the il forces i ech br e to the displcemet of joit J, e.g., N J K J δ J 3. Drw the FBD of the movig joit. Drw the crtesi compoets of the ukow eterl lod P J d impose equilibrium log d y à obti {P J,PJ y } 4. Drw FBDs of ech support pi (e.g., ) log the costried DOFs. Drw the crtesi compoets of the ukow rectioà obti {R,R y } 5. If more th oe joit moves: use SUPERPOSITIONà solve problem for ech movig joit d superpose the solutios.

S(x)along the bar (shear force diagram) by cutting the bar at x and imposing force_y equilibrium.

S(x)along the bar (shear force diagram) by cutting the bar at x and imposing force_y equilibrium. mmetric leder Bems i Bedig Lodig Coditios o ech ectio () pplied -Forces & z-omets The resultts t sectio re: the bedig momet () d z re sectio smmetr es the sher force () [ for sleder bems stresses d deformtio