MACHINE DESIGN - An Integrated Approach, 4th Ed. 4-1a-1 PROBLEM 4-1a Statement: A differential stress element has a set of applied stresses on it as indicated in each row of Table P4-1. For row a, draw the stress element showing the applied stresses, find the principal stresses and maximum shear stress using Mohr's circle diagram, and draw the rotated stress element showing the principal stresses. Given: σx := 1000 σy := 0 σz := 0 τxy := 500 τyz := 0 τzx := 0 Solution: See Figure 4-1a and Mathcad file P0401a. 500 y 1. Draw the stress element, indicating the x and y axes. 1000 x 2. Draw the Mohr's circle axes, indicating the τ and σ axes with CW up and CCW down. 3. Plot the positive x-face point, which is (+1000, -500), and label it with an "x." FIGURE 4-1aA 4. Plot the positive y-face point, which is (0, +500), and label it with a "y." Stress Element for Problem 4-1a 5. Draw a straight line from point x to point y. Using the point where this line intersects the σ-axis as the center of the Mohr circle, draw a circle that goes through points x and y. 6. The center of the circle will be at σc := σx + σy σc = 500 2 7. The circle will intersect the σ-axis at two of the principal stresses. In this case, we see that one is positive and the other is negative so they will be σ1 and σ3. The third principal stress is σ2 = 0. 2 8. Calculate the radius of the circle σx − σy 2 + τxy 2 R := R = 707.1 τ CW τ CW τ1-3 τ 1-2 500 500 y -500 500 σ3 1000 0 1500 σ1 τ2-3 -500 σ σ3 500 σ2 0 1000 1500 σ1 σ 2φ 500 x τ CCW 500 τ CCW FIGURE 4-1aB 2D and 3D Mohr's Circle Diagrams for Problem 4-1a © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, P0401a.xmcd mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. MACHINE DESIGN - An Integrated Approach, 4th Ed. 9. Calculate the principal stresses 4-1a-2 σ1 := σc + R σ1 = 1207 σ3 := σc − R σ3 = −207 σ2 := 0 10. Draw the three Mohr's circles to represent the complete 3D stress state. y 11. Calculate the principal shear stresses τ12 := 0.5⋅ ( σ1 − σ2) τ12 = 603.6 τ23 := 0.5⋅ ( σ2 − σ3) τ23 = 103.6 τ13 := 0.5⋅ ( σ1 − σ3) τ13 = 707.1 207 1207 22.5° x The maximum principal stress is always τ13. 12. Determine the orientation of the principal normal stress (σ1) with respect to the x-axis. From the 2D Mohr's circle diagram, we see that the angle 2φ from x to σ1 is CCW and is given by σx − σc ϕ := ⋅ acos 2 R 1 ϕ = 22.5 deg FIGURE 4-1aC Rotated Stress Element for Problem 4-1a 13. Draw the rotated 2D stress element showing the two nonzero principal stresses. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, P0401a.xmcd mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. MACHINE DESIGN - An Integrated Approach, 4th Ed. 4-3-1 PROBLEM 4-3 Statement: For the bicycle pedal-arm assembly in Figure 4-1 with rider-applied force of 1500 N at the pedal, determine the maximum principal stress in the pedal arm if its cross-section is 15 mm in dia. The pedal attaches to the pedal arm with a 12-mm screw thread. What is the stress in the pedal screw? Given: Distances (see figure) Rider-applied force a 170 mm Frider 1.5 kN Pedal arm diameter d pa 15 mm Screw thread diameter d sc 12 mm b 60 mm z Solution: See Figure 4-3 and Mathcad file P0403. a 1. From the FBD in Figure 4-3A (and on the solution for Problem 3-3), we see that the force from the rider is reacted in the pedal arm internally by a moment, a torque, and a vertical shear force. There are two points at section C (Figure 4-3B) that we should investigate, one at z = 0.5 d pa (point A), and one at y = 0.5 d pa (point B). Frider a Mc = 0 M x: Frider b Tc = 0 Mc b Arm y Fc Pedal x 2. Refering to the FBD resulting from taking a section through the arm at C, the maximum bending moment Mc is found by summing moments about the y-axis, and the maximum torque Tc is found by summing moments about the x-axis. M y: Tc C Frider FIGURE 4-3A Free Body Diagram for Problem 4-3 z Section C A Maximum bending moment: Mc Frider a Mc 255 N m Maximum torque: Tc Frider b Fc Frider 3. x Tc 90 N m Vertical shear: B Arm y FIGURE 4-3B Fc 1.500 kN Points A and B at Section C Determine the stress components at point A where we have the effects of bending and torsion, but where the transverse shear due to bending is zero because A is at the outer fiber. Looking down the z-axis at a stress element on the surface at A, Distance to neutral axis cpa 0.5 d pa cpa 7.5 mm © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. MACHINE DESIGN - An Integrated Approach, 4th Ed. 4 π d pa Moment of inertia of pedal-arm Ipa Bending stress (x-direction) σx Stress in y-direction σy 0 MPa Torsional stress due to Tc τxy Principal stresses at A, equation (4.6a) Mc cpa Tc cpa τxy 135.8 MPa 2 Ipa σx σy 2 σx σy 2 CW 2 σx σy 2 τxy 2 2 σx σy 2 τxy 2 σ2A 0 MPa σ3A 23 MPa Determine the stress components at point B where we have the effects of transverse shear and torsion, but where the bending stress is zero because B is on the neutral plane. Looking down the y-axis at a stress element at B, 2 π d pa Cross-section area of pedal-arm Apa Torsional stress due to Tc and shear stress due to Fc τzx Normal stresses σx 0 MPa Principal stresses at B σ1B 124 MPa Apa 176.7 mm 4 4 Fc τxy 3 Apa σ1B σ3B 5. 4 σx 769.6 MPa Ipa σ1A σ1A 793 MPa 3 Ipa 2.485 10 mm 64 σ3A 4. 4-3-2 τzx 124.5 MPa CW σz 0 MPa σx σz 2 σx σz 2 2 σx σz 2 τzx 2 2 σx σz 2 τzx 2 σ2B 0 MPa The maximum principal stress is at point A and is 2 σ3B 124 MPa σ1A 793 MPa © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. MACHINE DESIGN - An Integrated Approach, 4th Ed. 6. 4-3-3 Determine the stress in the pedal screw. Bending moment Msc Frider b Msc 90 N m Distance to neutral axis csc 0.5 d sc csc 6 mm Moment of inertia of pedal screw Isc Bending stress (y-direction) σy Stress in z-direction σz 0 MPa Torsional stress τxy 0 MPa 4 π d sc 3 Isc 1.018 10 mm 64 Msc csc Isc 4 σy 530.5 MPa Since there is no shear stress present at the top of the screw where the bending stress is a maximum, the maximum principal stress in the pedal screw is σ1 σy σ1 530.5 MPa © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. MACHINE DESIGN - An Integrated Approach, 4th Ed. 4-33a-1 PROBLEM 4-33a Statement: For the bracket shown in Figure P4-14 and the data in row a of Table P4-3, determine the bending stress at point A and the shear stress due to transverse loading at point B. Also the torsional shear stress at both points. Then determine the principal stresses at points A and B. Given: Tube length L 100 mm F y Arm length a 400 mm Arm thickness t 10 mm Arm depth h 20 mm Applied force F 50 N Tube OD OD 20 mm A B T T x M L R Tube ID ID 14 mm Modulus of elasticity E 207 GPa FIGURE 4-33 Free Body Diagram of Tube for Problem 4-33 Solution: See Figure 4-33 and Mathcad file P0433a. 1. Determine the bending stress at point A. From the FBD of the tube in Figure 4-33 we see that Reaction force R F R 50.0 N Reaction moment M F L M 5.00 N m Distance from NA to outside of tube ct 0.5 OD ct 10.0 mm Moment of inertia It Bending stress at point A σxA 64 π 4 OD ID 4 M ct It 5968 mm 4 σxA 8.38 MPa It 2. Determine the shear stress due to transverse loading at B. Cross-section area A π 4 2 OD ID Maximum shear V R Maximum shear stress (Equation 4.15d) τVmax 2 2 V A 160.2 mm 2 τVmax 0.624 MPa A 3. Determine the torsional shear stress at both points. Using equation 4.23b and the FBD above Torque on tube T F a Polar moment of inertia J Maximum torsional stress at surface τTmax T 20.0 N m 32 π 4 OD ID T ct J 4 J 11936 mm 4 τTmax 16.76 MPa 4. Determine the principal stress at point A. Stress components σxA 8.378 MPa σzA 0 MPa τxz τTmax τxz 16.76 MPa © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. MACHINE DESIGN - An Integrated Approach, 4th Ed. 4-33a-2 Principal stresses σ1 σxA σzA 2 2 σxA σzA 2 τxz 2 σ1 21.46 MPa 2 σxA σzA 2 τxz 2 σ3 13.08 MPa σ2 0 MPa σ3 τ13 σxA σzA 2 σ1 σ3 τ13 17.27 MPa 2 5. Determine the principal stress at point B. Stress components σxB 0 MPa σyB 0 MPa τxy τTmax τVmax τxy 16.13 MPa Principal stresses σ1 σxB σyB 2 2 σxB σyB 2 τxy 2 σ1 16.13 MPa 2 σxB σyB 2 τxy 2 σ3 16.13 MPa σ2 0 MPa σ3 τ13 σxB σyB 2 σ1 σ3 2 τ13 16.13 MPa © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. MACHINE DESIGN - An Integrated Approach, 4th Ed. 4-46-1 PROBLEM 4-46 Statement: The V-links of Figure P4-17 are rotated by the crank arm through a shaft that is 60 mm dia by 3.23 m long. Determine the maximum torque applied to this shaft during motion of the V-linkage and find the maximum stress and deflection for the shaft. See Problem 4-43 for more information. Given: Paper roll dimensions OD 900 mm ID 220 mm Shaft dims d 60 mm Lshaft 3230 mm Lroll 3230 mm 3 ρ 984 kg m Roll density G 79 GPa Modulus of rigidity Assumptions: The greatest torque will occur when the link is horizontal and the paper roll is located as shown in Figure P4-17 or Figure 4-46. Solution: See Figure 4-46 and Mathcad file P0446. y 1. Determine the weight of the roll on the V-arms. 4 π W 2 2 OD ID Lroll ρ g W 18.64 kN 2. Summing moments about the shaft center, T OD 2 W T 8.390 kN m 3. Calculate the polar moment of inertia. J π d 4 W 6 J 1.272 10 mm 32 4 T Ry 4. The maximum torsional stress will be at the outside diameter of the shaft. The radius of the OD is, r d r 30 mm 2 60-mm-dia shaft 450.0 FIGURE 4-46 Free Body Diagram used in Problem 4-46 5. Determine the maximum torsional stress using equation (4.23b). τmax Tr J τmax 197.8 MPa 6. Use equation (4.24) to determine the angular shaft deflection. θ T Lshaft J G θ 15.447 deg © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.