Lecture Fundamental Cncepts in Structural Plasticit Prblem -: Stress ield cnditin Cnsider the plane stress ield cnditin in the principal crdinate sstem, a) Calculate the maximum difference between the Vn-Mises and Tresca ield cnditin b) Shw the lcatins n the plane stress graph where the maximum difference ccurs Prblem - Slutin: Thrugh bservatin, the maximum difference ccurs at either A r B, as indicated in the belw figure Mises ield cnditin can be expressed as
The difference at A At Tresca ield surface Tresca A At Mises Yield surface Substitute int Mises ield cnditin, we have Mises A.5 The difference at A is Mises Tresca A A A A.5 0.5 0.5 The difference at B Tresca ield surface in the secnd quadrant is The distance frm Tresca ield surface B t the rigin is d Tresca B Mises ield surface in the secnd quadrant is
Als, at Mises ield surface 0 Cmbining the abve tw equatins, we have the crdinate f Mises ield surface B, The distance frm Tresca ield surface B t the rigin is d Mises B The difference at B is Mises Tresca d d B B B 0. B 0. Cmpare A and, the maximum difference ccurs at A, where B 0.5 max
Prblem -: In the earl twenties, passenger cars did nt have electric starters. crank t start the engine. The driver had t use a The crank is a slid rd f radius r and the gemetr f the crank is shwn belw. Define the equivalent stress b where sigma is the stress prduced b bending and is the shear stress due t trsin. a) Find the relatinship between the maximum equivalent stress in the crank and the magnitude f the crank lad P. (Use the principle f superpsitin) b) Derive a frmula fr the elastic deflectin under the lad P in the directin f the lad P. Prblem - Slutin: a) Determine the magnitude f the lad P causing first ield in the mstl stressed pint n the crank We will use the Mises ield cnditin which simplified t We calculate frm the mment, and frm the trsin Mr r where I I 4 Tr r 4 where J J Because we are asked t find P causing first ield at the maximum stressed pint, we 4 need t find max and max Frm phsics we can see that the M max and T max will ccur at the supprt 4
Mmaxr Plr Mmax Pl max I r 4 4 Tmaxr PRr Tmax PR J r 4 4 Pl max r PR max r ma x max 4 Pl PR r r 4 P 4l 6 r R r P l R 4 r r b) Calculate the elastic deflectin in the directin f the lad P Beam CD Beam AB T 0 M M Px 0 MA M PL R PR0 M PL M Px at 0 x R M x P LPx 0 M x PL Px at 0 x L R 5
Beam CB M PR x at 0 x R T PRat 0 x R In summar, here are the mment distributins Trsin is cnstant alng Beams AB and BC: T PR Beam CD: T 0 6
M U bending dx EI l R x dx PL x Px P R L R dx 0 EI 0 dx EI 0 EI R R LR x x Lx x Rx Lx 0 0 0 P x R EI P 6EI after lengthl algebra R L U bending P R L 6EI T U trsin dx GJ R LR PR PR dx GJ dx 0 0 GJ PR GJ U trsin RlR PR L GJ Use Castiglian s Therem t calculate the deflectin where the lad is applied, in the directin f the lad U P Utrsin Ubending P P PR L R L P 6E I GJ R L P EI RL GJ 7
r 4 r 4 Recall that fr slid circular crss-sectin I and J 4 E Als, G R L RL P EI GJ R L RL P 4 4 Er 4 Er 4 4 P R L RL Er r r 4 P R L RL Er r r Nte, if R 0, we have a cantilever beam whse deflectin is PL EI 8
Prblem -: Cnsider a thin-walled tube f radius r, thickness t and length L. The tube is full clamped n ne end and free n the ther. It is twisted at the free end b an axial trque T. (a) Derive an expressin between trsinal mment and the relative end rtatin. (b) Assuming L/R=0 and R/t=0, give the expressin fr the critical trque that will cause the tube stress t reach ield in shear. Prblem - Slutin: (a) Express T as a functin f T x r da d Gr dx da d G r da dx d GJ dx x0 0 xl max T x GJ L (b) The distributin f trsinal shear stress can be expressed as Tr x J Where 9
J r r 4 4 i r r t 4 4 Given r r 0 t t 0 4 J r 4 r r 0 J 0.07 r 4 Tr 5.8T x 4 0.07r r Plane stress ield cnditin states: xx x x x In pure shear xx 0 x x 5.8T r T. r 0
Prblem -4: Cnsider the fllwing ke ring prblem a) Derive the ut f plane displacement where the frce is applied. b) Determine the magnitude and distributin f the bending stress and the shear stress alng the ring. c) Find the lcatin f the maximum equivalent plastic strain. d) Determine the critical pening frce fr which first ield wuld ccur. Cnsider the plane stress Prblem -4 Slutin: a) Derive the ut f plane displacement where the frce is applied. We can use Castiglian s Therem t calculate the displacement where the frce is applied. In additin, t calculating the strain energ cntributin frm the mment, we must als accunt fr the cntributin frm trsin U M T dx EI GJ dx. Calculate M, T Frm the gemetr abve, we can see that M Pa T Pb where a Rsin b RRcs
S M PRsin T P R cs. Calculate strain energ U M T dx dx EI GJ PRsin PR cs Rd EI GJ 0 0 sin Rd PR R PR R sin sin EI 4 GJ 0 4 R PR R EI GJ PR PR U EI GJ 0. Appl Castiglian Therem, the ut f plane displacement where the frce is applied is U PR P EI GJ b) Determine the magnitude + distributin f bending stress + shearing Mz PR sin r 4 PRsin 4 I r 4 r s r PR 4 cs J r r Tr PR c c) Find the lcatin f maximum plastic strain Recall that relatin f plastic strain and plastic stress is A n The maximum plastic strain ccurs at lcatin f the maximum Mises equivalent stress,. Calculate the Mises equivalent stress The stress n the uter surface is planar, s we will use the plane stress cnditin xx x x x Where is Mises equivalent stress
In ur case, 0 S 4 PRsin PRcs r r. Calculate the Maximum Mises equivalent stress max max will ccur when we have max. We can find the maximum b taking the derivative, setting it equal t 0, find where max ccurs and then g back t get max d 4 PR sin cs d r r PR 8 s incs r PR cs sin Set d d 0, we have sincs 0 0, Minimum Mises equivalent stress ccurs at 0, where 0 4 Maximum Mises equivalent stress ccurs at, where PR r d) The critical pening frce fr which first ield wuld ccur when, that is S we have 4 PR cr r r P cr 4 R
Prblem -5: Plasticit Cnsider the fur pint bending f a beam f length L. The beam is laded b tw rllers parted b a distance f L/. The material f the beam is rigid, perfectl plastic. Determine the lad capacit f the beam under tw different end cnditins. a) Write an expressin fr full plastic bending mment f a beam f rectangular crss-sectin b h. b) Ends f the beam are simpl supprted c) Ends f the beam are clamped Prblem -5 Slutin: a) M hb 4 b) Mment distributin Px l M 0 x Pl l l M x 6 P l M lx xl l l We can assume that within x M M hb 4 4
The rate f change f internal energ is l l d x l MKdx M dxm l l dx xl U M w w Where gemetricall 6 l l Rate f wrk balance M P c w w M 6 Pcw l 6M P c l c) If the ends are clamped, we have tw plastic hinges at the supprts w w l l Rate f wrk balance i M i Pc w M M M P w M w Pcw l P c M l c 5
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