Question

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Question 1. For each of the plane-stress conditions given below, using the matrix transformation law, determine the state of stress at the same point for an element rotated in the x-y plane 30° clockwise from its original position: (a) Ox = 200 MPa Oy = 400 MPa Txy = - 60 MPa (b) Ox = 300 MPa Oy = -180 MPa Txy = 320 MPa Question 2 my x2 The state of stress at a point of an elastic solid is given in the x-y-z coordinates by: 0 90 180 0 [0]=0 0 = 180 240 0 MPa 0 0 120 Lºxx VX vy a. b. Using the matrix transformation law, determine the state of stress at the same point for an element rotated about the x-axis (in the y-z plane) 60° clockwise from its original position. Calculate the stress invariants and write the characteristic equation for the original state of stress, Calculate the deviatoric invariants for the original state of stress, Calculate the principal stresses and the absolute maximum shear stress at the point. What are the stress invariants and the characteristic equation for the transformed state of stress, C. d. e

Answer 1

52 = 200 mpa 400 ms oy Zry =-60 mpa ♡30° stressen in new plane x41 on cost +2 sino Cos Oy Sayo exy since oy' = Ox sin cotoy cos co - 22xy sino co o Zalgia (Oy-ry) sinocou + 2xy (cobo-sisco) Qald kloostcoitat c0360 since 2coscosmos Ox' Since cosce -2 Sino como oy! -Since como Smoococo Boc-no Lay cos30 Sig²30 2 60 530 51230 sin 30 los ago - 2 singolo J.80 cosso-sm30 2xy - Sin3660530 Sin30 C0330 2 0.866 032 2x 866x0s 0.52 0.4662 -2*0.866X0-5 - 056x0.5 0-86620's 0.8662-0.5? Qrey 0.75 0-25 0.866 doo 400 0.25 075 -0.866 JE -60 -0433 0.433 0.5

ox' 198.04 oyl ho 1.96 hely 56.6 ox'= 198.04 mda ogla 401.96m Da 2aly! 2 56.6 19 Da our 300mya og 3-180 maa hay 32ompc 2 Q 2 30° 075 0.25 0-866 . 300 oy! ods 0.75 -0-866 -180 0433 320 L anlyn 0 433 O'S ol 457.12 2 -337.12 oy! -47.84 2wyl onla 487.12 MPa ogle -337-12 MPa 2x412-47.84 mpa

2 О) југ) Чо 166 о (c) - во оцо MPC о 120 5) Stress in variants, Т: Oxt у 162 40tdo+ I2 о +30 с. 1, 2 зву+ су 2+2 бу - 25y z qox a4ot dox(20+ (2o xo - («о? 2 360 бро 3 2 teo ачо Тад то(«чбх120) — Lwo (воx120) - о grqa oo o - 388 8 00 0 3 - - (24Ыrto Characterishic equation is Бо 3_ Io-+T,0– Т. 3_ 450 + 360 c + (246xto'ѕо кутск

solving the characteristic equatios we gel, the pnncapal stress o = 442.57 MPa 22 257.957 MPa 3 z :-50.527 MPa Absolute maximum shear stres 10-02) which is namin 442-57-57.957 57.957450.522 11-50-527-442.59 2 2 2 192.31 51/54.au/ 246.55 Absolute man im um shea sorens = 246.55 MPa I, a 450 p = I 3 150 P 0 o on op 2y 2уу any x-p exy ry 0 Р on-P 62 Hydrostatu state Pure shear deviatoric streng [uso 90 180 -60 80 240 180 90 isa 0 150 120 - 30 Deviatonic stress