Question

Cross section at the wall Problem Statement: Consider a round beam of diameter d= 25 mm, length L = 10 mm, cantilevered (clamped) to the wall (as shown in the figure), loaded by P=5.5 kN. Further pay attention to points A, B, and C (half way between A and B) in the cross section at the wall, where the bending moment is maximum. Hint: The location of the centroid of half a circle can be found in Table A-18. Hint: The first moment of area for the shaded area in the figure is Q = {(r2 - y2) and the width of the base of that shaded area is b = 2/(r2 - y2). where y = 6.2500 mm is the distance from the center of the circle to the base of the shaded area and r is the radius of the circle. Then, calculate: • o at point A • Try at point A • Tmaz at point A. • Og at point B. • Try at point B. • Tmar at point B. • O, at point C. • Try at point C. • Tmar at point C. Answer tolerance = 2%. Be sure to include units and the correct sign with your answer. The sign of the bending moment M and the shear force V are determined by the sign convention in the textbook

Answer 1

Trensverse shear :- > 7 & C =VQ I here, Vo shear force Point of interest Q- tinst proment of cared above the point of interest. I moment of inertia t- thickness of the section at the Q-A, I, + Az point of interest. Stress because of bending M = 6 6. My - 6. = M & J 2 = moment acting on the section y = distance from the centroid I - moment of inertice max) for cirerler I. I d4 is 6 & 38 M 13 Cha

For stress at - - 3M d3 - 38 x 5.5 x 104 M= pxl - 5,5 x 103 x 10 = 5.5×104 N.mm do 25 mm - (85)3 A = 35.8544 Mpu - (Tension) As the position of the point A is at the top of the section because of point t will be in tension. loud P Tension Axis CA - VQ Compression but here Q=0 because there is above the point A. no area stress element @ A -10 - 35.8544 C=0 Corax = (5x-by go + x 35.8544-01 ²4 35. 1544 Crux - 17.92 72 Mpa

& For stress at point B - Nextrelax = 3M d3 as point B is on Nextred axis 4713 Cag=vQ It = 5.5 x 16 x 1502.0833 19174.75985 (25) given in hint 8-0 B = 14.9393 Mpa Q = alm Q = 1302.0833 mm? stress element @ B Ixd4 NO = 19174.75965 mm? 12 = 14.9393 OV Tresi -by) + (Exy)² Tomax = 14.9393 Mpa

& For stresses cat poist (③ 338 M ad 3 - 35.8544 Mpa (Tension - VG . It t = 2 r²-ye 5.5 x 19 x 845.7079 191 74 75 985 x 21.65063 Koce 11.5171 Mpe here y-6.25 ron tref 20.50 -6.25 Ft -21.65063 mm Q. RU Stress element @ ③ 6 = 35.6544 Q = 845.78 79 mon Vez 11.5171 I - 19 174.75585 re 1 05.45 uu -0)" - 11.5172 (35.6541 1.5171 Crax - 21.3079 Mece

In this type of questions pay close attention to the points provided at different heights on the section of the beam, because some of them may lie on the neutral axis where stress due to bending is zero, and others may experience tension or compression according to their position. While calculating shear be careful finding Q.

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