Explanation Given information: The rod has a circular cross section with a polar moment of inertia J and moment of inertia I . A vertical force P is applied at the point A . Consider only the strain energy due to bending. The material constants are modulus of elasticity is E and modulus of rigidity G . Explanation: Consider a section from free end of the circular rod. Show the free-body diagram of the section as in Figure 1. Geometry of the section: Show the free-body diagram of the geometry of the section as in Figure 2. Torsion about the section: Determine the torsion at the section by taking torsion about the section. ∑ T x = 0 T − P r ( 1 − cos θ ) = 0 (1) Moment about the section: Determine the moment at the section by taking moment about thesection. ∑ M x = 0 M − P r sin θ = 0 (2) Show the calculation of values as follows: Conclusion: Solve Equation (1). T = P r ( 1 − cos θ ) Solve Equation (2). M = P r sin θ Strain energy due to Bending: Determine the strain energy in the circular rod due to bending using the equation. ( U i ) b = ∫ 0 s M 2 d s 2 E I (3) Here, d s = r d θ , M is the moment in the beam, and I is the moment of inertia of the beam. Show the calculation of values as follows: Conclusion: Substitute P r sin θ for M and r d θ for d s in Equation (3), and integrate. ( U i ) b = ∫ 0 θ M 2 r d θ 2 E I = ∫ 0 π ( P r sin θ ) 2 r d θ 2 E I = P 2 r 3 2 E I ∫ 0 π ( sin 2 θ ) d θ = P 2 r 3 2 E I ∫ 0 π ( 1 − cos 2 θ 2 ) d θ ( U i ) b = P 2 r 3 4 E I ∫ 0 π ( 1 − cos 2 θ ) d θ = P 2 r 3 4 E I [ θ − sin 2 θ 2 ] 0 π = P 2 r 3 4 E I [ π − sin 2 ( π ) 2 − 0 + 0 ] = π P 2 r 3 4 E I Strain energy due to Torsion: Determine the strain energy in the circular rod due to torsion using the equation. ( U i ) T = ∫ 0 s T 2 d s 2 G J (4) Here, d s = r d θ , T is the torsion in the circular rod, and J is the polar moment of inertia. Show the calculation of values as follows: Conclusion: Substitute P r ( 1 − cos θ ) for T and r d θ for d s in Equation (4), and integrate

11th Edition

ISBN: 9780137605521

Author: HIBBELER

Publisher: RENT PEARS

See similar textbooks

Concept explainers

Strain energy

The term strain indicates the variation in an object's length, area, and volume when the object is subjected to an external force. The external force may be of normal, shear, moment, torque, etc. The strain in an object depends on the types of external f…

Videos

Question

Chapter 14.3, Problem 40P

To determine

The vertical displacement at point C.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 14.3, Problem 39P

Chapter 14.3, Problem 41P

Students have asked these similar questions

-6- 8 من 8 Mechanical vibration HW-prob-1 lecture 8 By: Lecturer Mohammed O. attea The 8-lb body is released from rest a distance xo to the right of the equilibrium position. Determine the displacement x as a function of time t, where t = 0 is the time of release. c=2.5 lb-sec/ft wwwww k-3 lb/in. 8 lb Prob. -2 Find the value of (c) if the system is critically damping. Prob-3 Find Meq and Ceq at point B, Drive eq. of motion for the system below. Ш H -7~ + 目 T T & T тт +

Q For the following plan of building foundation, Determine immediate settlement at points (A) and (B) knowing that: E,-25MPa, u=0.3, Depth of foundation (D) =1m, Depth of layer below base level of foundation (H)=10m. 3m 2m 100kPa A 2m 150kPa 5m 200kPa B

W PE 2 43 R² 80 + 10 + kr³ Ø8=0 +0 R²+J+ kr200 R² + J-) + k r² = 0 kr20 kr20 8+ W₁ = = 0 R²+1) R²+J+) 4 lec 8.pdf Mechanical vibration lecture 6 By: Lecturer Mohammed C. Attea HW1 (Energy method) Find equation of motion and natural frequency for the system shown in fig. by energy method. m. Jo 000 HW2// For the system Fig below find 1-F.B.D 2Eq.of motion 8 wn 4-0 (1) -5- m

Chapter 14 Solutions

MECHANICS OF MATERIALS

Ch. 14.2 - A material is subjected to a general state of...Ch. 14.2 - The strain-energy density for plane stress must be...Ch. 14.2 - The A-36 steel bar consists of two segments, one...Ch. 14.2 - If P = 10 kip, determine the total strain energy...Ch. 14.2 - Determine the maximum force P and the...Ch. 14.2 - Consider the thin-walled tube of Fig.5-26 . Use...Ch. 14.2 - Determine the bending strain energy in the 2-in...Ch. 14.2 - Determine the bending strain energy in the...Ch. 14.2 - Determine the bending strain energy in the simply...Ch. 14.3 - Determine the horizontal displacement of joint A....

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.

The vertical displacement at point C .

Solution Summary: The author explains how to determine the vertical displacement at point C. The free-body diagram shows the geometry of the section.

Concept explainers

Videos

The vertical displacement at point C.

Want to see the full answer?

Chapter 14 Solutions