To determine To derive: An expression for an equivalent bending moment M e . Explanation Given information: A solid bar with a circular cross section, would cause the shear stress as the combination of an applied moment M and torque T . Calculation: Find the moment of inertia of the solid bar as shown below. I = π c 4 4 Here, c is the radius of the bar. Find the polar moment of inertia of the solid bar as shown below. J = π c 4 2 Find the bending stress as shown below. σ = M c I Here, I is the moment of inertia and M is the bending moment. Substitute π c 4 4 for I . σ = M c π c 4 4 = M c × 4 π c 4 = 4 M π c 3 Find the shear stress as shown below. τ = T c J Here, T is the torque and J is the polar moment of inertia. Substitute π c 4 2 for J . τ = T c π c 4 2 = T c × 2 π c 4 = 2 T π c 3 Find the principal stresses in complex system as shown below. σ 1 , 2 = σ x + σ y 2 ± ( σ x − σ y 2 ) 2 + τ x y 2 (1) Here, σ x is the normal stress in x direction, σ y is the normal stress in y direction, and τ x y is the shear stress in xy plane. Substitute 4 M π c 3 for σ x , 0 for σ y , and 2 T π c 3 for τ x y in Equation (1). σ 1 , 2 = 4 M π c 3 + 0 2 ± ( 4 M π c 3 − 0 2 ) 2 + ( 2 T π c 3 ) 2 = 2 M π c 3 ± 4 M 2 π 2 c 6 + 4 T 2 π 2 c 6 = 2 M π c 3 ± 4 π 2 c 6 ( M 2 + T 2 ) = 2 M π c 3 ± 2 π c 3 M 2 + T 2 (2) Find the maximum and minimum principal stresses using Equation (2) as shown below

9th Edition

ISBN: 9780133254426

Author: Russell C. Hibbeler

Publisher: Prentice Hall

See similar textbooks

Concept explainers

Combined Loading

Any functioning component, such as machine parts, structural members, pressure vessels, and so on, are all under the influence of more than one force. When anybody is under the influence of more than one force such as shear forces, bending moments, axial…

Videos

Question

Chapter 10.7, Problem 10.70P

To determine

To derive: An expression for an equivalent bending moment Me .

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 10.7, Problem 10.69P

Chapter 10.7, Problem 10.71P

Students have asked these similar questions

Refrigerant-134a is expanded isentropically from 600 kPa and 70°C at the inlet of a steady-flow turbine to 100 kPa at the outlet. The outlet area is 1 m2, and the inlet area is 0.5 m2. Calculate the inlet and outlet velocities when the mass flow rate is 0.65 kg/s. Use the tables for R-134a. The inlet velocity is m/s. The outlet velocity is m/s.

A container filled with 70 kg of liquid water at 95°C is placed in a 90-m3 room that is initially at 12°C. Thermal equilibrium is established after a while as a result of heat transfer between the water and the air in the room. Assume the room is at the sea level, well sealed, and heavily insulated. NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Determine the final equilibrium temperature. Use the table containing the ideal gas specific heats of various common gases. The final equilibrium temperature is °C.

Steam at 100 psia and 650°F is expanded adiabatically in a closed system to 10 psia. Determine the work produced, in Btu/lbm, and the final temperature of steam for an isentropic expansion efficiency of 80 percent. Use steam tables. The work produced is Btu/lbm. The final temperature of steam is °F.

Chapter 10 Solutions

Mechanics of Materials

Ch. 10.3 - Prove that the sum of the normal strains in...Ch. 10.3 - The state of strain at the point on the arm has...Ch. 10.3 - Prob. 10.3P Ch. 10.3 - Prob. 10.4P Ch. 10.3 - 10-5. The state of strain at the point on the gear...Ch. 10.3 - Use the strain transformation equations and...Ch. 10.3 - Use the strain transformation equations and...Ch. 10.3 - Prob. 10.8P Ch. 10.3 - Use the strain transformation equations to...Ch. 10.3 - Use the strain- transformation equations to...

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.