Question

Please draw each point

PROBLEM 4 (25%) Given the following state of stress, 20 ksi 16 ksi 8 ksi Given the following state of stress, using the Mohr's circle: Determine the principal normal stresses and show their sense on a properly oriented element, “PRINCIPAL STRESS”. Show stress invariance with the original stress condition. Find the maximum shear stress with their associate normal stresses and show the results on a properly oriented element, "MAX SHEAR STRESS”. Show stress invariance with the original stress condition. Determine stresses if the element is rotated 35° counterclockwise (CCW) and show the results on a properly oriented element, “STRESS @ 35 deg CCW”. Show stress invariance with the original stress condition. On the circle, label the following points, lines, and angles [including coordinates (0,7) when applicable]: Point A Point B Point C (center) Point 01 Point 02 Point Tmax (positive and negative) Angle 20p Angle 20 and the points, lines and angles corresponding to the rotated element (35° CCW). Point R1 Point R2 O O On the stress element, show the element angle of rotation, and the stress magnitudes with their corresponding directions.

Answer 1

20 ksi stress element *8 ksi" 8 ksi 16 ks, 10. 20 ksi Coordinate of ** Face of stress element, (5x, exy) (8,16) A B = ( Oy, Try) cause * shear stress element Clock wise rotation of stress element taken as positive (-20,-16) coordinate of center © = (0,0) Ontby 8 +20 2 - 6 (-6,0)

Normal to x 35 x 2 = 70° 3x2-x of pulona Emay = 21.26 616.45,18.63) Ri Al 8,16) 41.189 (20p) 48.820 max. principal stress 0 : 15.26ksi 62 = 27.26 ksi 26.0) 10,0) Principal stress, +3+.98 (180+08.82) (180 +298) 41.190 B(-20,-16) (4.247,-18.65) A Emay Normal to y-axis Normal to У!

Resulty from mohrle circle 5,= max, principal stress o+R Radius - COA) (3%-(-6)) + (06-0}} 0,- -6+21.26 R 2 1.26 ksi 15,= 15,26 ksi 62= T-R -6 - 21.26 162 = -27,26 ksi min. principal stress,

stress element 2 stress in varience 15.26 —27.2. = , 1 1-2ч.ч)° 0 : 15.24 6. : 27. 24 6, + бу - 8+(-20) = - 2 15.24 +(-27, 24) C+02 - - - 2 Иe Исе битбу = 6,16,

Maximum shear stress = Tmax=+R = + 21.26 KSI Tomax e + 21,26 ksi maximum shear stress normal stress associated with plane -6ksi (6 (max) is (-60) (0,oo 5= -6ksi (6, emax)

Grigel 2 р 20.26 1с Д...24 -21,2 6 6, +6y = 8-20 = Cl2 644 ! – – – 6 — -12 Srix'+ Smy' б, toy = иелло Onx tony'

20p 4 20s ) (31) tant () 200= 이 • 4 3.82 | 20 , 4 2-2u.u CI) AG 20 2. 2092 180 + 207) 아6) - 228.32 Or, < 114.1°C 20 A(8,16) UI. 18. 20, 20 8s.= 20.59°CY C(-6) 14

coordinate of Rotated plane (R2gR 2) (0,',7,') (RJI O 70 R = 21,26 61.18 48.82 (1.80 66,0) (-60) 에 sin (61.18) o' = cos( 81.18°) R R a = + 18,63ksi o' -10.25 - - 6 T = 18.63 ksi 8,'= -16,25 R, = -16.25, 18.63

coordinate of Rotated plane (R2gR 2) (0,',7,') (RJI O 70 R = 21,26 61.18 48.82 (1.80 66,0) (-60) 에 sin (61.18) o' = cos( 81.18°) R R a = + 18,63ksi o' -10.25 - - 6 T = 18.63 ksi 8,'= -16,25 R, = -16.25, 18.63

(0 م ) 5- 5.26 ks (1.176 R = 21,28 « ( 6, 7) Tmax وا (61.125)، ما Rx + 6 - 16, 9. 24' ks : 'ج R sin( 600) 89) - 21.2 x sin (61./89) |-- او k 4.3 R2 : (ره ر ره ) (8,03/- ر2 ,1)

stress element * stress invarionce Б. си Ч,ч - 4,25 Byla 350 1861 - 12.01 Ч,241 : 2-20 бу б) +бу 6. - 16,25 -2 -16,25 +4,247 Ox' tfyl= - 12. Иew би «ву : Б. +6, 1