Question

Tranquier Plane - Element thickness = .200" {= 15410° psi ,u=30 3 (1.0, 1.0) 27 10,0) (2.0, a) Compute strain displacement Matrix, B. b) Compute element stiffress matrx, kle) c) If .2 va U3 Lv3 .001 Looo5 / compute d) Compute & and ł. sodal forces: Fle)- kle) des

Answer 1

Consider al 3(10 ro) Plane Stress element is thickness = t 0.2.0" E = 15x iopsi 0 OBO (0,0) The area of the (2.0,0) Islangale is denoted by A and is given by : 2A = det 2 x x = (x2y₃ - xgY₂) + (xay, x, y). + (x,ya-kay,) It is positive, if the wines are numbered in yelle deurtel Llockwise aldo. using the provided triangular plans o elementi * (x1 syo) = (0,0) (x.sya) = (2-0, mo) flessy) = (-0, 1-0) 8. 2nd = (asy - sz. Ya) + (x2), - x, ya) + love, Yz - szy.) 24 = (2x1 - x0) + (1x 0-0x1) + (0x0 - 2x0) numbered in cyllic wounter- 2A = 42 A =+1 1oo the winers are clockwise order Que Stain - Displaumut Matix , B: Using Strain le) B d = - Displacement Equation: Jos o sio yao 1 20 o 32 0 23 0 % LR 32 Yaz 213 Yai Y 42,

of where Matrix • Stree, B B: Stain - Displau mert and J j k = Y; - Yk & xgk = x3 - 2K 906 B = Strain - Displacement Matrix B ² B (4 - Y3) 0 lys y,) o ly, y) o 2A (22-23) (21-23) o (-2, 1 (23-24) (y2 - Y3) (x - 2y) (y2 - 81) (23-24) (y - 3) Sines 2A = 2 & (2,, gy) = (0,0); (22) ₂) = (20, 1.0) (3ya) = (0,0) No: 2 to - 1 (1-2 (1-1) (0-1) (1-6) (2-0) (0-1) 10 0 0 -1 01 t o -1 0 -1 0 2 Ang - 0 -1 1 2 - Element Stiffness Matrix tel & Kle) - At BTEB where ait are E are constant K AFE (B'B) • Finding At EBB Since A 21 t= 0.200" ; Ex 15 x 108 psi o AtE = x 0.2 x 15 x 100 = 3 x 100

Stree; Elem 2 0-10] 0- 102 0 1 2 oo - TO Too-OTO -1 as grans pose # ?! 200 -OT Ooo To Todo- - T Theutole, To o 0 -107 1 lo - on - i od 200 !! 10- 2 - o o - 0 - 0 1 0 - 1 -1 -2 17 0 0 -2 2 -1 -3 - 2 2 -3 - 3 2 5 -2 -3 -2 5 - o - - - < d : BB 16 - 1-2 L s 0 -2 -

• Element ا ذ Stress matrix, K = AE (g) - ا ه م ا ا 2 ب 100 ع 0 ا ات | - 2 3 ا - 22 = 2x texl - ر - ط کر بر موم ا - 2 | 0 2 2 | 2 3 5 -2 - 067SR 0 || 0 2 - 3 - 82 | 2 - - ا 2 - 2 کی ۔ ای که در تم -2 0 -3 2 | 5 -2 | -3 اس بر س س با - 002 | | . 00e5 | ہج * 5 - 0 ا ا 15 (عام - - 0 0 0 - 0 2 - - ۴۵۵ 3x6 Ne05 و )60، سه 002 • 681 23 ، 0005 - 8062 + 1000 oo) - 00005

w? Io.001 0.001 L-0.0005 A = E 0.001 8 = 15x108 1 0.001 -0.0005 = 158 103 Li Ang L -0.5 Nodal folkes : 0 Fle) 1 N - 1 1 0 -1 -2 0 L1 -2 2 -1 -3 -1 2 2 -3 -3 2 5 -2 i cê - 001 0005 5 OK OX6 0.0005. - 0.001 0.0015 -Ang -0.0015 -0.002 0.0025