EMA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 4 Pure Bending Homework Answers

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1 EA 3702 echanics & aterials Science (echanics of aterials) Chapter 4 Pure Bending Homework Answers

2 100 mm Homework 4.1 For pure bending moment of 5 kn m on hollow beam with uniform wall thickness of 10 mm and cross-section dimension as shown, please calculate the stress at point E and point F, respectively. For the hollow beam, the cross-section moment of inertia I: I = 1 12 (b 1h 1 3 b 2 h 2 3 ) = m 4 For E, y =?? mm σ x = 69.3Pa z y E C F 50 mm For F, y =?? mm σ x = 86.6Pa

3 40 mm 20 mm Homework 4.2 The beam shown is made of polymer which has allowable tensile stress of 25 Pa and compression stress of 30 Pa. Please calculate the largest value for the bending couple that can be applied to the plane of symmetry to the beam. To determine the cross-section moment of inertia, break it into two parts Part #1 on the top, Area: A 1 =?? m 2 Location of centroid 1 from the bottom y 1 =? m Part #2 on the bottom, Area: A 2 =?? m 2 Location of centroid 2 from the bottom y 2 =? m EA 3702 echanics & aterials Science Zhe Cheng (2018) 4 Pure Bending y ave 50 mm 25 mm #1 #2 y 2 y 1

4 40 mm 20 mm Homework 4.2 For the entire cross-section, location of the centroid from the bottom y ave = (A 1 y 1 + A 2 y 2 )/(A 1 +A 2 ) 50 mm 25 mm y ave #1 #2 y 2 y 1 = m For the entire cross-section, the total moment of inertia I = (I i + A i d i 2 ) I = i 1 12 b 1h A 1 d b 2h A 2 d 2 2 = m 4

5 40 mm 20 mm Homework 4.2 aximum tensile stress occurs at top surface σ max _tension = c σ I tension_allowable 275N m aximum compressive stress occurs at 23.3mm bottom surface σ max _compression = c σ I compression_allowable EA 3702 echanics & aterials Science Zhe Cheng (2018) 4 Pure Bending 50 mm 25 mm 236N m Therefore, maximum bending moment that can be applied is the smaller of the two, which is 236 N m #1 #2 y 2 y 1

6 Homework 4.3 A flat 10 mm wide long strip of steel is bent into part of a circle with radius of curvature of 100 mm by two bending couples as shown. Calculate (a) the maximum thickness of the steel strip if allowable stress is 400 Pa, (b) with the dimension designed, the corresponding moment applied to reach maximum stress of 400 Pa knowing E = 200 GPa. (a) Knowing E and bending radius, max stress for bending: σ max = Ec ρ σ allowable h = 2c 0.4mm A (b) aximum stress for bending σ max = c σ I allowable 0.107N m EA 3702 echanics & aterials Science Zhe Cheng (2018) 4 Pure Bending A A A 10 mm

7 Homework 4.4 A composite bar having aluminum plate (E Al = 70 GPa, allowable stress = 100 Pa) sandwiched between two copper plates (E Brass = 105 GPa, allowable stress = 150 Pa). Please calculate the largest permissible bending moment when the composite bar is subject to bending moment about a horizontal axis, as illustrated. Transform the Al plate in the center into Cu plate: n = E Al /E Cu =????? cm 4 cm EA 3702 echanics & aterials Science Zhe Cheng (2018) 4 Pure Bending 4 cm

8 h 2 = h 1 = 4 cm Homework 4.4 b 2 =??? cm 4 cm b 1 = 4 cm For transformed cross-section with all Cu, centroid is the center oment of inertia for the transformed cross-section I transf I transf = 1 12 (b 1h 3 1 2b 2 h 3 2 ) I transf = m 4 EA 3702 echanics & aterials Science Zhe Cheng (2018) 4 Pure Bending

9 Homework 4.4 ax stress in Cu occur when c Cu =? cm σ max_cu = c Cu I transf σ allowable_cu 1530N m ax stress in Al occur when c Al =? cm σ max_al = n c Al I transf 3070N m σ allowable_al ax bending moment can be applied is 1530 N m cm 4 cm

10 Homework 4.5 A plastic cylinder support with radius of 4 in is subjected to 5000 lb eccentric axial force as shown. Determine the axial normal stress at point B when (a) a = 0, (b) a = 2 in. Knowing moment of inertia for the cross-section is I z 1 r 4 (a) When a = 0, centric loading σ x_b = 99.5psi 4 B z y 5000 lbs a x

11 Homework 4.5 (b) When a = 2 in, eccentric axial loading Bending moment: =?? lb in Axial normal stress at B have two contributions σ x_b = F A c I B z y 5000 lbs 2 in x σ x_b = 298.5psi

12 Homework 4.6 A member is subject to loading force in the vertical plane of symmetry as illustrated. The allowable stress in horizontal crosssection FGHI is 100 Pa. Please calculate the largest force P that can be applied. For cross-section FGHI, P is an eccentric axial loading P P 5 cm F G G H 4 cm Bending moment =? P cm I H 5 cm P F G

13 Homework 4.6 P P 5 cm Compressive stress due to centric loading σ x_centric = P? cm 2 ax compressive stress due to F G G H pure bending moment 4 cm σ x_bend =?? P I H?? cm 2 Total max compressive stress 5 cm P 4 cm σ x = σ x_centric + σ x_bend σ allowable F G P 4.21kN

14 Homework 4.7 The bending moment couple is applied to a beam cross-section in a plane forming an angle =30 o from the horizontal xz plane. Please calculate stress at points of E, F, and G. Resolve the bending moment Bending moment around z axis z =??lb in ax stress caused by z alone σ m_z =?? psi = 500 lb in z E G y 0.5 in F 0.5 in 0.5 in Bending moment around y axis y =?? lb in ax stress caused by y alone σ m_y =?? psi

15 Homework 4.7 E y F For point E, z causes compression, y causes tension σ x_e =- σ m_z + σ my = 801psi For point F, z causes compression, y causes compression σ x_f = 11193psi = 500 lb in z z y G 0.5 in 0.5 in 0.5 in For point G, z causes tension, y causes tension σ x_g = 11193psi

EMA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 4 Pure Bending

EMA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 4 Pure Bending EA 3702 echanics & aterials Science (echanics of aterials) Chapter 4 Pure Bending Pure Bending Ch 2 Aial Loading & Parallel Loading: uniform normal stress and shearing stress distribution Ch 3 Torsion: