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

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Beverley Haynes
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mmetric leder Bems i Bedig Lodig Coditios o

resultts t sectio re: the bedig momet () d z

sleder bems stresses d deformtio due to the

()log the br (bedig momet digrm) b cuttig the

For the emple show, equilibrium t left of B

() F B F C Σ ( z ) 0 () + D + L B F B L C F C!

2 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 due to the sher force () re egligible] Fid ()log the br (sher force digrm) b cuttig the br t d imposig force_ equilibrium. ()log the br (bedig momet digrm) b cuttig the br t d imposig momet_z equilibrium. For the emple show, equilibrium t left of B gives: Σ F 0 () + F B F C! () F B F C Σ ( z ) 0 () + D + L B F B L C F C! () D + L B F B L C F C Note pivot for z-momet is t to elimite ukow () from equtio For distributed lodig q (), with q () i N/m + log obti () d () b itegrtio Differetil reltioships: d ( ) q ( ) d ( ) ( )

3 Emples () d () digrms strt t L with ()0, ()0 wlk the bem from right to left: (i directio) the verticl forces (pplied d rectios t supports) poit to where the digrm is goig: cocetrted lod up (positive) mes the digrm hs step up distributed lod up (positive) mes the digrm is slopig up cocetrted lod dow (egtive) mes the digrm hs step dow distributed lod dow (egtive) mes the digrm is slopig dow the digrm d cocetrted momets (pplied d rectios t supports) poit to where the digrm is goig: CCW cocetrted momet (positive) mes the digrm hs step up digrm up (positive) mes the digrm is slopig up CW cocetrted momet (egtive) mes the digrm hs step dow digrm dow (egtive) mes the digrm is slopig dow

Kiemtics costrit Cross sectios : st flt, rotte roud z b θ () ectio displcemet ectio rottio bem slope:

θ ()+dθ Locl mesure of deformtio t sectio dθ ρ ( ) dθ ( ) ρ Rdius t eutrl is tructurl respose

( ')' 0 deflectio v( ) v( 0 )+ dv ' ' v 0 + θ( ') ' 0 0 with θ ( 0 )θ 0 d v( 0 )v 0 determied b BC (e.

4 Kiemtics costrit Cross sectios : st flt, rotte roud z b θ () ectio displcemet ectio rottio bem slope: i -directio v(): deflectio θ ( ) dv ectio deformtio ectio t hs rottio θ () ectio t + hs rottio θ (+) θ ()+dθ Locl mesure of deformtio t sectio dθ ρ ( ) dθ ( ) ρ Rdius t eutrl is tructurl respose curvture sig covetio dθ NEUTRL X: o chge i -legth with deformtio slope θ( ) θ( 0 )+ d θ ' ' θ ρ ( ')' 0 deflectio v( ) v( 0 )+ dv ' ' v 0 + θ( ') ' 0 0 with θ ( 0 )θ 0 d v( 0 )v 0 determied b BC (e.g., θ 0 0 d v 0 0 t wll) Note tht for simpl supported bems, must use isted two BCs o v() (zero t supports)

5 sectio deformtio!" tri sectios rigidl rotteà stri vries lierl with for positive curvture, top of sectio shortes, bottom elogtes for cross sectios with top-bottom smmetr the mid-ple is eutrl o chge i -legth o eutrl mid-pleà set 0 t midple ε (, ) ds ( ρ( ) ) dθ ρ( ) dθ ρ( ) dθ ρ( ) Note: ε is fuctio of : lrger stris w from the eutrl is ε h ε X ρ

ectio equilibrium The Bedig omet () t sectio is obtied b itegrtig the cotributios of ech elemetl

f the mteril is lier elstic d the Youg s modulus of elemetl re d is E the stress c be obtied s

(E) E() (); E(,, d σ (,, E(,, ε (, ) ( E ) E(,, ε (, ) d ( )( E ) ρ ( ) ( ) Costt over cross

6 ectio equilibrium The Bedig omet () t sectio is obtied b itegrtig the cotributios of ech elemetl re, d t distce from the is, which crries orml stress σ (,, ( ) σ (,, z ) d Costitutive Properties f the mteril is lier elstic d the Youg s modulus of elemetl re d is E the stress c be obtied s ectio Respose () () ( ) ρ Effective ectio tiffess: σ (,, d f ol mteril, E()! (E) E() (); E(,, d σ (,, E(,, ε (, ) ( E ) E(,, ε (, ) d ( )( E ) ρ ( ) ( ) Costt over cross sectio ivert E(,, d E(,, ( ) ρ ( ) d ρ () ( E ) ( ) f mterils (E, E )! (E) E + E NOTE:,, ll clculted with respect to 0 i i d i

ectio momet of ierti d omet of ierti of rectgle d circle with

superpositio: + + bh πr 4 4 Cutio: for simple superpositio, ll

theorem d + d + d " + + bt "( + h + t % ) # ' & # i ( i c +( i c

' # ' + bt " & + ( h + t % ) # ' & & # BH bh % ' ( bt )' ' & BH

7 ectio momet of ierti d omet of ierti of rectgle d circle with respect to their cetroid Obti for more comple sectios b superpositio: + + bh πr 4 4 Cutio: for simple superpositio, ll res for, must hve cetroid o 0 if ot à eed to use prllel is theorem d + d + d " + + bt "( + h + t % ) # ' & # i ( i c +( i c ) i ) 4 πr O πr " Prllel is theorem % " ' " ( bt )' + th % ' # ' + bt " & + ( h + t % ) # ' & & # BH bh % ' ( bt )' ' & BH bh + i c i c omet of ierti of i with respect to its ow cetroid -coordite of cetroid of re i

8 pecil cse: homogeeous bem (modulus E); costt cross sectio with momet of ierti Bedig momet () obtied from Σ z 0 à () (m be NOT costt log shft) ρ ( ) ( ) E ε (, ) curvture log the bem (t eutrl is) ρ( ) ( ) E σ (, ) E ε (, ) ( ) il stri orml stress ρ() π R 4 d bh 4

Uiil Lodig Torsio Bedig ΣF 0 à N Σ 0 à T Σ z 0 à ectios

ε ( ) N() σ (,, d du( ) N() ( E) ( ) ectios : st flt,

τ (, G(, γ (, dϕ( ) T() ( G ) ( ) p ectios : st flt,

) ρ( ) σ (,, E(,, ε (, ) dθ ( ) () ( ) ρ ( E ) ( ) ( E)

) u( 0 )+ ( E ) ( ') ' φ( ) φ( 0 )+ 0 0 T (' ) (G p ) (

9 Uiil Lodig Torsio Bedig ΣF 0 à N Σ 0 à T Σ z 0 à ectios : st flt, trslte i b u() du ( ) du( ) ε ( ) σ (,, E(,, ε ( ) N() σ (,, d du( ) N() ( E) ( ) ectios : st flt, rotte roud b ϕ() T () r τ ( r, ) d d ϕ( ) dϕ( ) γ (, r τ (, G(, γ (, dϕ( ) T() ( G ) ( ) p ectios : st flt, rotte roud z b θ() () σ (,, d dθ ( ) ( ) ρ dθ ( ) ε (, ) ρ( ) σ (,, E(,, ε (, ) dθ ( ) () ( ) ρ ( E ) ( ) ( E) ( ) E(,, d ( Gp) ( ) r G(, d ( E ) ( ) E(, d N (' ) u( ) u( 0 )+ ( E ) ( ') ' φ( ) φ( 0 )+ 0 0 T (' ) (G p ) ( ') ' θ( ) θ( 0 )+ 0 (' ) ( E ) ( ') ' v( ) v( 0 )+ θ( ')' 0

10 olutio Pth for tticll Determite (D) Bems i Bedig ) For simpl supported bems, first obti rectios t the supports ) Obti b FBD, Σ( 0 terl Bedig omet resultt (lso use q () -d / d () -d /) ) Obti (E) ech sectio : (E) () E () + E () + Effective ectio tiffess 4) ech sectio : Curvture t sectio 5) Obti il stri Norml stress lope field Deflectio Field

11 Force ethod for tticll determite () Bems i Bedig elect compio tticll Determite (D) problem: Remove redudt costrits t the supports: Replce redudt displcemet costrit with redudt rectio force Replce redudt rottio costrit with redudt rectio momet olve the compio tticll Determite (D) problem: Obti the displcemet d/or rottio t the redudt costrit(s) i terms of the redudt rectio force d/or redudt rectio momet mpose zero displcemet d/or rottio t the redudt costrit: Obti the redudt rectio force d/or rectio momet Bck-substitute the redudt rectio force d/or momet i the solutio to the compio D probem: Fid the solutio to the problem. TP: use superpositio whe ou solve the D compio problem

S. Socrate 2013 K. Qian

S. Socrate 2013 K. Qian S. Socrte 213 K. Qi 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