Mechanics of Materials, 7th Edition
7th Edition
ISBN: 9780073398235
Author: Ferdinand P. Beer, E. Russell Johnston Jr., John T. DeWolf, David F. Mazurek
Publisher: McGraw-Hill Education
expand_more
expand_more
format_list_bulleted
Videos
Textbook Question
Chapter 11.3, Problem 35P
Show by integration that the strain energy in the tapered rod AB is
where Jmin is the polar moment of inertia of the rod at end B.
Fig. P11.35
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
The normal strain in a suspended bar of material of varying cross section due to its own weight is given by the expression γy/3E where γ = 2.9 lb/in.3 is the specific weight of the material, y = 3.4 in. is the distance from the free (i.e., bottom) end of the bar, L = 17 in. is the length of the bar, and E = 24000 ksi is a material constant. Determine,
(a) the change in length of the bar due to its own weight.
(b) the average normal strain over the length L of the bar.
(c) the maximum normal strain in the bar.
In many situations physical constraints prevent strain from occurring in a given direction. For example, εz= 0 in the case shown, where longitudinal movement of the long prism is prevented at every point. Plane sections perpendicular to the longitudinal axis remain plane and the same distance apart. Show that for this situation, which is known as plane strain, we can express σz, εx, and εy as follows
12. A single strain gage is cemented to
solid 96-mm-diameter aluminum
shaft at an angle B = 20° with a line
parallel to the axis of the shaft.
Knowing that G = 27 GPa, determine
the torque T corresponding to a gage
reading of 400u.
48 mm
Figure P12
Chapter 11 Solutions
Mechanics of Materials, 7th Edition
Ch. 11.3 - Determine the modulus of resilience for each of...Ch. 11.3 - Determine the modulus of resilience for each of...Ch. 11.3 - Determine the modulus of resilience for each of...Ch. 11.3 - Determine the modulus of resilience for each of...Ch. 11.3 - The stress-strain diagram shown has been drawn...Ch. 11.3 - The stress-strain diagram shown has been drawn...Ch. 11.3 - Prob. 7PCh. 11.3 - Prob. 8PCh. 11.3 - Using E = 29 106 psi, determine (a) the strain...Ch. 11.3 - Using E = 200 GPa, determine (a) the strain energy...
Ch. 11.3 - A 30-in. length of aluminum pipe of...Ch. 11.3 - A single 6-mm-diameter steel pin B is used to...Ch. 11.3 - Prob. 13PCh. 11.3 - Prob. 14PCh. 11.3 - The assembly ABC is made of a steel for which E =...Ch. 11.3 - Show by integration that the strain energy of the...Ch. 11.3 - Prob. 17PCh. 11.3 - Prob. 18PCh. 11.3 - Prob. 19PCh. 11.3 - 11.18 through 11.21 In the truss shown, all...Ch. 11.3 - Prob. 21PCh. 11.3 - Each member of the truss shown is made of aluminum...Ch. 11.3 - Each member of the truss shown is made of aluminum...Ch. 11.3 - 11.24 through 11.27 Taking into account only the...Ch. 11.3 - Prob. 25PCh. 11.3 - 11.24 through 11.27 Taking into account only the...Ch. 11.3 - 11.24 through 11.27 Taking into account only the...Ch. 11.3 - Prob. 28PCh. 11.3 - Prob. 29PCh. 11.3 - Prob. 30PCh. 11.3 - 11.30 and 11.31 Using E = 200 GPa, determine the...Ch. 11.3 - Assuming that the prismatic beam AB has a...Ch. 11.3 - Prob. 33PCh. 11.3 - The design specifications for the steel shaft AB...Ch. 11.3 - Show by integration that the strain energy in the...Ch. 11.3 - The state of stress shown occurs in a machine...Ch. 11.3 - Prob. 37PCh. 11.3 - The state of stress shown occurs in a machine...Ch. 11.3 - Prob. 39PCh. 11.3 - Prob. 40PCh. 11.3 - Prob. 41PCh. 11.5 - A 5-kg collar D moves along the uniform rod AB and...Ch. 11.5 - The 18-lb cylindrical block E has a horizontal...Ch. 11.5 - The cylindrical block E has a speed v0 =16 ft/s...Ch. 11.5 - Prob. 45PCh. 11.5 - Prob. 46PCh. 11.5 - The 48-kg collar G is released from rest in the...Ch. 11.5 - Prob. 48PCh. 11.5 - Prob. 49PCh. 11.5 - Prob. 50PCh. 11.5 - Prob. 51PCh. 11.5 - The 2-kg block D is dropped from the position...Ch. 11.5 - The 10-kg block D is dropped from a height h = 450...Ch. 11.5 - Prob. 54PCh. 11.5 - A 160-lb diver jumps from a height of 20 in. onto...Ch. 11.5 - Prob. 56PCh. 11.5 - A block of weight W is dropped from a height h...Ch. 11.5 - 11.58 and 11.59 Using the method of work and...Ch. 11.5 - 11.58 and 11.59 Using the method of work and...Ch. 11.5 - 11.60 and 11.61 Using the method of work and...Ch. 11.5 - 11.60 and 11.61 Using the method of work and...Ch. 11.5 - 11.62 and 11.63 Using the method of work and...Ch. 11.5 - 11.62 and 11.63 Using the method of work and...Ch. 11.5 - Using the method of work and energy, determine the...Ch. 11.5 - Using the method of work and energy, determine the...Ch. 11.5 - The 20-mm diameter steel rod BC is attached to the...Ch. 11.5 - Torques of the same magnitude T are applied to the...Ch. 11.5 - Prob. 68PCh. 11.5 - The 20-mm-diameter steel rod CD is welded to the...Ch. 11.5 - The thin-walled hollow cylindrical member AB has a...Ch. 11.5 - 11.71 and 11.72 Each member of the truss shown has...Ch. 11.5 - 11.71 and 11.72 Each member of the truss shown has...Ch. 11.5 - Each member of the truss shown is made of steel...Ch. 11.5 - Each member of the truss shown is made of steel....Ch. 11.5 - Each member of the truss shown is made of steel...Ch. 11.5 - The steel rod BC has a 24-mm diameter and the...Ch. 11.9 - 11.77 and 11.78 Using the information in Appendix...Ch. 11.9 - 11.77 and 11.78 Using the information in Appendix...Ch. 11.9 - 11.79 through 11.82 For the beam and loading...Ch. 11.9 - 11.79 through 11.82 For the beam and loading...Ch. 11.9 - 11.79 through 11.82 For the beam and loading...Ch. 11.9 - 11.79 through 11.82 For the beam and loading...Ch. 11.9 - 11.83 through 11.85 For the prismatic beam shown,...Ch. 11.9 - 11.83 through 11.85 For the prismatic beam shown,...Ch. 11.9 - 11.83 through 11.85 For the prismatic beam shown,...Ch. 11.9 - 11.86 through 11.88 For the prismatic beam shown,...Ch. 11.9 - 11.86 through 11.88 For the prismatic beam shown,...Ch. 11.9 - 11.86 through 11.88 For the prismatic beam shown,...Ch. 11.9 - For the prismatic beam shown, determine the slope...Ch. 11.9 - For the prismatic beam shown, determine the slope...Ch. 11.9 - For the beam and loading shown, determine the...Ch. 11.9 - For the beam and loading shown, determine the...Ch. 11.9 - 11.93 and 11.94 For the beam and loading shown,...Ch. 11.9 - 11.93 and 11.94 For the beam and loading shown,...Ch. 11.9 - For the beam and loading shown, determine the...Ch. 11.9 - For the beam and loading shown, determine the...Ch. 11.9 - Prob. 97PCh. 11.9 - For the beam and loading shown, determine the...Ch. 11.9 - 11.99 and 11.100 For the truss and loading shown,...Ch. 11.9 - 11.99 and 11.100 For the truss and loading shown,...Ch. 11.9 - 11.101 and 11.102 Each member of the truss shown...Ch. 11.9 - 11.101 and 11.102 Each member of the truss shown...Ch. 11.9 - 11.103 and 11.104 Each member of the truss shown...Ch. 11.9 - 11.103 and 11 104 Each member of the truss shown...Ch. 11.9 - A uniform rod of flexural rigidity EI is bent and...Ch. 11.9 - For the uniform rod and loading shown and using...Ch. 11.9 - For the beam and loading shown and using...Ch. 11.9 - Two rods AB and BC of the same flexural rigidity...Ch. 11.9 - Three rods, each of the same flexural rigidity EI,...Ch. 11.9 - Three rods, each of the same flexural rigidity EI,...Ch. 11.9 - 11.111 through 11.115 Determine the reaction at...Ch. 11.9 - 11.111 through 11.115 Determine the reaction at...Ch. 11.9 - 11.111 through 11.115 Determine the reaction at...Ch. 11.9 - 11.111 through 11.115 Determine the reaction at...Ch. 11.9 - 11.111 through 11.115 Determine the reaction at...Ch. 11.9 - For the uniform beam and loading shown, determine...Ch. 11.9 - 11.117 through 11.120 Three members of the same...Ch. 11.9 - 11.117 through 11.120 Three members of the same...Ch. 11.9 - 11.117 through 11.120 Three members of the same...Ch. 11.9 - 11.117 through 11.120 Three members of the same...Ch. 11.9 - 11.121 and 11.122 Knowing that the eight members...Ch. 11.9 - 11.121 and 11.122 Knowing that the eight members...Ch. 11 - Rod AB is made of a steel for which the yield...Ch. 11 - Each member of the truss shown is made of steel...Ch. 11 - The ship at A has just started to drill for oil on...Ch. 11 - Collar D is released from rest in the position...Ch. 11 - Each member of the truss shown is made of steel...Ch. 11 - A block of weight W is placed in contact with a...Ch. 11 - Two solid steel shafts are connected by the gears...Ch. 11 - A 160-lb diver jumps from a height of 20 in. onto...Ch. 11 - For the prismatic beam shown, determine the slope...Ch. 11 - A disk of radius a has been welded to end B of the...Ch. 11 - A uniform rod of flexural rigidity EI is bent and...Ch. 11 - The steel bar ABC has a square cross section of...
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.Similar questions
- A thin wire, lying along the x axis, is strained such that each point on the wire is displaced Δx = kx2 along the x axis. If k is constant, what is the normal strain at any point P along the wire?arrow_forwardProve that the sum of the normal strains in perpendicular directions is constant, i.e., Px + Py = Px′ + Py′arrow_forwardQ7. Determine the torsional strain energy in the steel shaft (G= 75 GPa). The shaft has a radius of 30 mm. [Ans. 26.2 J] 3 kN.m 4 kN.m 0.5 m 0.5 m Fixed end 0.5 marrow_forward
- Determine the shortening of point B with respect to the ground at D. Segment BC weighs 1.8 kN/m and segment CD weighs 2.5 kN/m. Both segments have values of E = 200 GPa. The cross-sectional areas of segment BC and CD are 100 mm2 and 200 mm2 respectively. | 5 kN . B 3 m 3 kN|| 3 kN 1 m 1marrow_forward3. Invert the plane stress form of Hooke's law and express the stress in terms of the strain components. E O. = 1- u E - (ε, + με,) 1-u O, = E -Yxy 1+ uarrow_forward7.38 Determine the x, y, z components of internal loading in the rod at point D. There are journal bearings at A,B, and C. Take F={7i−12j−5k}kN Determine the x, y, z components of internal force at point D. Determine the x, y, z components of internal moment at point D.arrow_forward
- A fabric used in air-inflated structures is subjected to a biaxial loading that results in normal stresses ox = 18 ksi and oz = 24 ksi.Knowing that the properties of the fabric can be approximated as E = 12.6 x 10 psi and v = 0.34, determine the change in length of (a) side AB, (b) side BC, (c) diagonal AC.arrow_forwardA steel shaft is subjected to the two loads shown. Knowing that G=80 GpA, determine the total strain energy in the shaft. 1 m A 1.5 m 35 kN-m B 160 mm C 2m 100 mm 20 kN-marrow_forward15.3 (B). Show that the strain energy per unit volume of a material under a single direct stress is given by (stress x strain). Hence show that for a material under the action of the principal stresseso,,02 and o, the strain energy per unit volume becomes 1 Toi+o +o3 - 2v(0,02+0,03+0203)] 2E A thin cylinder 1m diameter and 3 m long is filled with a liquid to a pressure of 2 MN/m?. Assuming a yield stress for the material of 240 MN/m2 in simple tension and a safety factor of 4, determine the necessary wall thickness of the cylinder, taking the maximum shear strain energy as the criterion of failure. For the cylinder material, E = 207 GN/m2 and v 0.286. [14.4 mm.]arrow_forward
- A spherical gas container having an inner diameter of 5 m and a wall thickness of 24 mm is made of steel for which E = 200 GPa and v = 0.29. Knowing that the gage pressure in the container is increased from zero to 1.8 MPa, determine (a) the maximum normal stress in the container, (b) the corresponding increase in the diameter of the container. Hint: refer back to earlier notes for relationship between strain, Poisson's ratio, and stress in two directions Sm The unpressurized cylindrical storage tank shown has a 5-mm wall thickness and is made of steel having a 400-MPa ultimate strength in tension. Determine the maximum height h to which it can be filled with water if a factor of safety of 4.0 is desired. (Density of water = 1000 kg/m³.) 14.5 m Hint: recall pressure (p) due to a column of water is p = yharrow_forwardA steel cable is used to support an elevator cage at the bottom of a 2100-ft-deep mineshaft. A uniform normal strain of 240 μin./in. is produced in the cable by the weight of the cage. At each point, the weight of the cable produces an additional normal strain that is proportional to the length of the cable below the point. Assume D = 2100 ft and d = 500 ft. If the total normal strain in the cable at the cable drum (upper end of the cable) is 790 μin./in., determine (a) the strain in the cable at a depth of 500 ft. (b) the total elongation of the cable. Drum Cable Answers: (a) ε = (b) 8 = i i d D Elevator cage X uin./in. in.arrow_forwardIn many situations it is known that the normal stress in a given direc-tion is zero. For example, σz= 0 in the case of the thin plate shown. For this case, which is known as plane stress, show that if the strains εx and εy have been determined experimentally, we can express σx, σyand εz as follows:arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
Elements Of ElectromagneticsMechanical EngineeringISBN:9780190698614Author:Sadiku, Matthew N. O.Publisher:Oxford University Press
Mechanics of Materials (10th Edition)Mechanical EngineeringISBN:9780134319650Author:Russell C. HibbelerPublisher:PEARSON
Thermodynamics: An Engineering ApproachMechanical EngineeringISBN:9781259822674Author:Yunus A. Cengel Dr., Michael A. BolesPublisher:McGraw-Hill Education
Control Systems EngineeringMechanical EngineeringISBN:9781118170519Author:Norman S. NisePublisher:WILEY
Mechanics of Materials (MindTap Course List)Mechanical EngineeringISBN:9781337093347Author:Barry J. Goodno, James M. GerePublisher:Cengage Learning
Engineering Mechanics: StaticsMechanical EngineeringISBN:9781118807330Author:James L. Meriam, L. G. Kraige, J. N. BoltonPublisher:WILEY
Elements Of Electromagnetics
Mechanical Engineering
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Oxford University Press
Mechanics of Materials (10th Edition)
Mechanical Engineering
ISBN:9780134319650
Author:Russell C. Hibbeler
Publisher:PEARSON
Thermodynamics: An Engineering Approach
Mechanical Engineering
ISBN:9781259822674
Author:Yunus A. Cengel Dr., Michael A. Boles
Publisher:McGraw-Hill Education
Control Systems Engineering
Mechanical Engineering
ISBN:9781118170519
Author:Norman S. Nise
Publisher:WILEY
Mechanics of Materials (MindTap Course List)
Mechanical Engineering
ISBN:9781337093347
Author:Barry J. Goodno, James M. Gere
Publisher:Cengage Learning
Engineering Mechanics: Statics
Mechanical Engineering
ISBN:9781118807330
Author:James L. Meriam, L. G. Kraige, J. N. Bolton
Publisher:WILEY
An Introduction to Stress and Strain; Author: The Efficient Engineer;https://www.youtube.com/watch?v=aQf6Q8t1FQE;License: Standard YouTube License, CC-BY