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Toronto Metropolitan University *

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430

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Mechanical Engineering

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Oct 30, 2023

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Department of Mechanical and Industrial Engineering Please select your current program below: ● Mechanical Engineering Industrial Engineering Course Number 430 Course Title Solid Mechanics Semester/Year 5 th SEM 3 rd YEAR Section Number 06 Group Number N/A Assignment No. 2 Assignment Title Problem Set 2 Submission Date October 17, 2023 Due Date October 17, 2023 Student Name Student ID (xxxx1234) Signature * James Mohrhardt Xxxx97151 (Note: Remove the first 4 digits from your student ID) *By signing above you attest that you have contributed to this submission and confirm that all work you have contributed to this submission is your own work. Any suspicion of copying or plagiarism in this work will result in an investigation of Academic Misconduct and may result in a “0” on the work, an “F” in the course, or possibly more severe penalties, as well as a Disciplinary Notice on your academic record under the Student Code of Academic Conduct, which can be found online at: http://www.ryerson.ca/senate/policies/pol60.pdf .

Iz : 3 B 4 : Iz+ IG+ Iz , Izu 142 Iz , 2 (2 A(d] Iz 2[+) = 205 12 41 59 Iz , 2 (i 5 5 511 57/7 75 4 2577 2(x 5,5 5 I z T 246 81 in" . Iz , 2 [1 547 101 06] Izz 41 59 in A Iz , 205 22 in " = 2 , = 900psi M14 25) (2) in -id & 24 6 81 im" M z((5 5)(1 51(4 251) 5 5/1 5)10 25) (5 5) (1 5) (7 75) M 987241b in 2(5 5) (1 5) 2)5 5)11 5) n 4 25 ↑ -> 1 . a) of = b) = = Es = i s = = 98724(8 5 4 25) = 98724(0 75 4 25) - - -> 246 81 I A 246 81 ex 1100Psi I Er 2 , 1400 Psi intension 1 . 5 2A = -1700Psi in compression Fr F , Fz F , ex(Atop) F I (1700 2x) (Atop) 1108 (1 5) (5 5) = = (1700 1100) (1 5)(5 5) 9075 2475 F 9075 2475 Fr 11550Psi

di centroid -sin- L n a 6 (2/3) " n 13 n 5 29in Iz 3 2 z h All IzT : : - He - Iz - 6(2)(5 29 3) Izr Izi Izu Izi 52 12 98 93 Iz , 5 333 46 786 Izz 98 93 int Izi 151 05 in * Iz , 52 12 in Flexural Stress : at B : at A = e 2A = Me 23 - 10kip ft( )(0 - 5 29) 151 05 in 4 C = -(10kip ft)( ) (8-5 29) in es 4 20 Ksi in tension 15) 05 & =-2 15 Ks in compression

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E MB = IkiP 2 ki 0 1(3) 2(2) Dy (4) b ↓ d Dy 1 75 ki B c D - 64ip-ft n A - - - 4z3 - SFy = 2 By Dy 0 By 1 2 1 75 ↓ 'ki im mix By 1 25 -X -- a drex EMc = 3 I ② ↓ 1 25 M(x) 1 25(2) 1(5) & 0 25 I -i ↑ Mix = 2 5 kip-ft I f H (4) (0 251(0 125) (4) 10 251(5 875) + 15 510 2)(3) Iz : 0 2 in 6 ⑳ A (2)(4)(0 25) (6)(0 2) at 1 & 2 = I 3 n 0 125 5 875 3 Iz , 2 / + Als] d ↓ 1 - 0 25 I 1 2[5 410 25(5 875-377 2 -4 - n 3in 2[0 00521 8 266] T 16 54 a 3: Iz : max tensile stress Iz e Iz + Iz , Izz ~max - = e Iz . as IzT 16 54 2 77 Smart = 1-2 5) kip ft() (4 3) in Izz 2 77 Iz = 19 . 31 in " 19 31 in" Smax- 4 Ebks: at the top surface

Equilibrium GEMB = 25(4) Dy(8) 50(14) 0 2542s Dy 10016 - ↑ [Fy 0 25 By 100 50 R 1- By 751b By Shear : V(x) 25(x 6" 15xx 4 100xx 12 50 x 18) Moment : V(0-1 0 v(vi) 25 V(12 ) 56 M(x) 25xx -01-75xx 4 1004x- 1231- 502x-183'v10+)= 25/03v14 % = 25-75 v(12 +) - 50 100 =25 - 50 50 M(0) 0 100 ⑳ M(4) 25(4) =100 25 v(18 ) 50 V(x) ⑳ & O M(12) 25(12) 75(8) 25 L V((V") 50 50 5-0 = 300 ~max cal 100 1 . = 0 M(18) 25(18) 75(14) 100(6) -300 ⑳ e Centroid : Iz HY 0 225 - I n | ) - 0 3125 - n 125 - ↑ 2 n 0 3/25 Iz 0 00749in4 Max Flexural Stress : Sax = Max emax" - i 10 625-0 3125) in 0 . 00749144 -max= 12516 . 69 Psi Omax 12 52 KS1

Table D-1 W24 x94 z W- rokies Ma e W 0 094(20) =1 88 KP =- 8 - 42 i - 4 6 [M , Mc 20(12) 11(16) 1 88(10) 0 20 kiPS Cy Mc 29 2 2Fy 0 12 20 1 88 2y shear : cy - b 12kiP vix = -15[x a xx x] 0 094[x -> <x z0)] 20(x 83 b 11xx 207-29 2x 20 V(0) 0 V(85) = -12 752 +20 V <20) 6 12-6 12 v(8) ( 1 518) 10 094)(8) = 7 248 -0 =- 12 152 V(20 1 1 5(8) 0 094(20) 20 =b 12 V(x) 7 248 . 3 12 ⑳ . T VI f 3 ⑳ · S 20 3 12 Moment : 7 248 12 157- ⑳ M(x) - [xx x xx - 82] -[xx x xx 207] + 20xx 81 29 2x - 2078 M(x) M(0) 0 3 M(20) = -0 75(256) 1941400) 20112 can M(87 = 5783 84482 =29 2 - ⑧ ⑳ 2 M(20) 29 2-29 2 B G X - 51 008 - W 20 51 008- ⑳ Centroid : n (9 045)(0 875)(23 871 10 5151172 56)(12 156 10 8751/9 065710 2375t i (2) 19 045) (0 875t 10 515)22 50) 2 - s M 12 15 V ↓ IzT : I - 0875 T Iz , + Izz a -> I Iz a 3 2- 9 045 Iz , 22 5 e 1947 33+422 67 a 192 = 2370 in " Iz , 2 [ Ad = ⊥ 22 67in4 2[9 8751+ 9 04510 875/12 15 857 2[0 5041 10881) 1947 33 int Max flexural stress : Umax-xe Izx -i ( ) (12 15) 2370 =3 14 Ksi & max

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F W(b) 6 [MB 0 6wID -Ay14) warddad Ay W(1 5) - & Fy 0 W(1 5) -bw By ↑ - -Y - - By 4 5w 2 shear : was ex VIX) W(1 5) <X-02" w[xx -0 xx 67) w(4 5)3x 4) V10T 0 V() W11 5-WIR V(b) W31 5) -W(6) WC4 5) W(1 5) ⑧ 5 & x(m) - W(2 5) = O V(0) W(1 5) V(4 +) W11 5)-w(2) W34 5) V(x) 0 W( 5 = ( x 1 5 W(l] W(2 St- . M(x) Moment : WIl 125) - ⑳ M(x) wil 5) <x -0 [(x 0 > - xx x)] w(4 5)(x 4 ~ abolic M(0) 0 M(4) W(b) w(8) ⑳ k M(1 5) W(1 5/1 5) = (1 5) = W(2) 1 . W(I 125) M(61 w(9) -w(18) W(9) =O & Iz : Centroid pat R = (15 4)(305)(300 3) + 19 9177277 2)(154) (15 479305)(7) at 1 & 2 308mm -> 99mm (2) (15 4) (3051 19 91)(277 2) Iz , 2/ + Ad ↓ mu H 1410589 1 +423046 36166 9 2(15 4 + (3055 4/154 2.75 9394 2747 05 R 154mm at 3 : 2(92828 4 100533131 9] 105mm Izz = Iz , 201251920 7mmY 977 2 12 IzT Iz Iz , Ize max uniform distributed load =17590253 01mm * 2 72x108mm" Emax matt smax=wime ? symm ! It e 2ax w(0 308m3) 0 . 000222m + W 1500000 (0 0007208) W 1 08/kN/m

GEMB (4 8(2) (9 b)(3) Hy (6) . = b tiny not Ay b 4kN KN zmd [Fy b 4 - 9 . 6-4 8 + By ↑ - I By By 8 kN Ar ene ⊥ m V2xb shear : 6 , 4 ⑳ V(x) 6 42x 0 1 6(x 6 (x 0 84x 3 8 8. ⑧ X V(x) 0 b 4 1 4x x V(0) 0 ⑧ 3 165 I 6 V(0) 6 4 =16 x 2 19 2x-76 8 V(67 6 4 - 1 6(6) - - V(b) - 8 - 8 - =8 M(x) 8 - ⑳ 0 x 3 145 16 31 ⑳ Moment : M(x) 6 42x x 1xx 07 x x 8xx x M MC01 = 0 ⑧ I ⑧ X M(6) 6 4(6) (6) (6) D 3 165 ⊥ =0 M(3 165) 6 4(3 165) 0 813 165) 0 04444/3 16512 =16 31 kN M Stress : b dow-ars= = " favor wow. e b = swin i re b 0 . 01631m b = 254mm

F ( 5(20) = 30 ki 6 [M1 0 Mc 30(10) 20(12) dat e Mc = 60 kiP "c SFy = 2y - 30 + 20 20k Cy 10 ki Shear : 8 Vix) - 1 5[x -0 20xx 6 + 10(x 20) + 30xx- 207 - V(0) 0 ~(20) = - 1 5(8) + 20 V(x) 0 1 5(x) + 20 V(8) = - 1 5(8) 8 0 - --12 V(20) - 1 5/20) 20 +10 x 13 33 V(8t) = 1 5(8)+70 =0 =8 Moment : M(x) = 1 5 (x x + 20(x 8) 10(x - 20 + 30(x 20 I M(20-1 = -0 751204 + 20(12) M(13 32) = -0 75/13 33)" + 20113 33 -8) M(07 0 M(8) = -0 75(872 = 60 = 24 67 m(20) = 0 = 48 Stress design : 26 67 wow n In the · -G 60 Saei 36 in choose Wizx35 to be the lightest possible beam Where SIS des

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I GEMR P(2) Ay(L) Ay = e A & F = E P By I ---1 Ay L BY By = Shaar- v(x) V(x = <x - 0 P(x = Exx x ⑳ Umar is V(0- ) 0 v(2) P ⑧ z ⑧ L X v 10+) = VIL- = - - I - ⑳ v() = V(L) 0 moment : m(x) ⑧ M(x f(x 0 P2x 2 P(x x = Mmax M(0) 0 M(2) 2() P(r - E) M(z) E(E) = - ⑧ En Z - W Tallow : Fallow i & A '(y) b(E)(E) = S ↓ centroid "nor i wa . Fri are won e L n 3P Fallow - p = sch P = b) Lorip = P 3 P 1 9b p = ( . 2)(2)(b) (9 5)2 -x2(b) 1 92(b) ⊥ = 38 in P 22(b) L

① A(y) al It i = (1rY() = - + : d e "I* d e 4 ↑ As By GSMi W(L) (E) AyII greenwich By 2 By = ~max = urrra . e v(x) = (x - N2x 0 + <x 2 N VI07 0 v(2-) E W(L) VCOT Wh · s I r- i t was Wh & . b Ants e - -

centroid : h (2)((38)(76)(17) (224)(14)(r) (2) 27b)(12) (224)((4) b = 18 . 04 mm at 3 Iz , b + Ad = 16 + 226(14) (18 06 8 Iz : 77141 333 365952 22 - Lift at 1 $ 2 4 43093x105 mu * 8 Iz , 2 [en) Ad Iz : - 2 2[ 7612)(38 18 0634 Iz Iz , Izw mine Iz + 2 044 =10 "mm" 226 = 2(438976 362414 48] 1 6032x10"mm" shear Stress ⑨ a point b 12 e - A -> I de in 2048 a [A(y) Ie I - = (76)(12)( 8 04) A A = 26420 64 & = [A xy) & 38(((b))8 - 15 . 06) + (73))12)(38 - 18 . 04/ ~ - & = 12068 , 8 mm Y = * I = obro e TB 10 . 74 MPa -iii YA 3 . 64MPa

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Iz : If Ide T : Mid : - 2 I I 10 02") 14 It 160 73 in " Shear stress Max [ @y 0 ↑ - i ↓ - or Qa : SAYY) I [C 4( ) -( * )(5) - = [1 - I - a - I - (5 " - ) 10 75 = 19 4904 in ~mar-am e Year an Bus] YMax= 6 7 1ksi