Question

3. Consider a thin elastic plane of thickness t with a given in-plane displacement of ſu(x,y) = $dz 5)2 + +d22 (v(x,y) = {2,5)2 +d2 (1) a) Find an expression for all the strain components on this plane. (10 points) b) Find an expression for all the stress components on this plane. (10 points) c) Write down the integral equation to calculate the force on edge A of this plane. (you do not need to solve the integral). (5 points) d) Write down the integral equation to calculate the moment on edge A of this plane around z axis. (you do not need to solve the integral). (5 points) h E, G, v Constant

Answer 1

Scuben:- Criven dat a ucury) – 4zdı (9lb)?+llydz (94) v exky) = 42d9 (4b)*+1/3 del2 (916)? (216) 44 (90) di (Mb)°107 (916) h. A u= .b alow we have strain constants in acy & xy direchons EIGIN costant Ex Qy = 212 2y Ey Pag - Ou + Qe ay Here stress components are given by the Shren. Strain relahon Shep wing Elash a constants IE, & Strain components Ex: au ak 53 cy 2u 24 - ch you che non te Vxy = au Que axtay = { 2 } [de e di a bev tala u har

be Here stress components on this plane Now we have Ex = 07 - voy Ey = ty E - E Vxy = tay again we have g G=E 2(1+1) a I Do - E -gy Exy II Ex Ey Day 001-V for a plane Stress Condition : . q= T-VUCEy) x= Tor f 5) (de vermin 2 l-on (Ex)+ El E (dian 7-07 E b3 to dzy h3 diav 63 dix 1-1V [ Tror C to day h3 E y = C Ey) + 1-7 (Ex) Oy E (de 1/3) 21 l-ov igre Can 33 For Cargo todox 63 g = For Cdign glede na 2 h3 to dix

Zay= (xy) 211-9) Try= 2 ( a sort de 92) 5- Here force on edge A on edge 'A' (x=6) consider a cenist thicknen' dy' on face 'Al force : Shress x area : area= (t dy) at= thickness of the plate We have Normal force on 'A! pn = Et l-ov 1Pn = R x C Edy) [ a lot o da yv by the day A Loties - 1 day h Totul normal force, Pn=. Et S l-ov Shear force 'Al pt It Et 211-10) (% exy (tdy) 3- +ds y Jay Total shear force Per Et 201-0) (do the 4 dy de moment on edge Air ↑ Phy Pro Apes moment on face A pad A • Prea B Mo = Preth / * Pop xh - Proxb Shear force on illz Exy (Ads) PROV PTO

Hore yah Pro lectus statement les de 20(1-1) Shoar force on' al Txy (t.de ), Here Y=o TA s Care Jax 20170) Shear force 'B' 1 Try Ct.dy) Here 220 -4 Pro Et 2 (tv) (42 44 0

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