Explanation Given information: A material is subjected to stresses σ x and σ y . The material constants are E and ν . Calculation: Apply the normal stress formula for x shown below. σ x = σ x + σ y 2 + σ x − σ y 2 cos 2 θ + τ x y sin 2 θ Here, σ x and σ y are the normal stresses in the x and y direction, τ x y is the shear stress, and θ is the orientation. The shear stress τ x y = 0 . Substitute 0 for τ x y . σ x = σ x + σ y 2 + σ x − σ y 2 cos 2 θ (1) Consider cos 2 θ = 2 cos 2 θ − 1 . For the normal direction. Substitute 2 cos 2 θ − 1 for cos 2 θ in Equation (1). σ n = σ x + σ y 2 + σ x − σ y 2 ( 2 cos 2 θ − 1 ) = σ x + σ y 2 + ( σ x − σ y ) cos 2 θ − σ x − σ y 2 = σ x 2 + σ y 2 + σ x cos 2 θ − σ y cos 2 θ − σ x 2 + σ y 2 = σ y + σ x cos 2 θ − σ y cos 2 θ = σ x cos 2 θ + σ y ( 1 − cos 2 θ ) = σ x cos 2 θ + σ y sin 2 θ Apply the normal stress formula for y shown below. σ y = σ x + σ y 2 − σ x − σ y 2 cos 2 θ − τ x y sin 2 θ The shear stress τ x y = 0 . Substitute 0 for τ x y . σ y = σ x + σ y 2 − σ x − σ y 2 cos 2 θ (2) For y direction. Substitute 2 cos 2 θ − 1 for cos 2 θ in Equation (1)

9th Edition

ISBN: 9780133254426

Author: Russell C. Hibbeler

Publisher: Prentice Hall

See similar textbooks

Videos

Question

Chapter 10.6, Problem 10.45P

To determine

The orientation θ of the strain gage.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 10.6, Problem 10.44P

Chapter 10.6, Problem 10.46P

Students have asked these similar questions

For the state of a plane strain with εx, εy and γxy components: (a) construct Mohr’s circle and (b) determine the equivalent in-plane strains for an element oriented at an angle of 30° clockwise. εx = 255 × 10-6 εy = -320 × 10-6 γxy = -165 × 10-6

The piece of plastic is originally rectangular. Determine the average normal strain that occurs along the diagonals AC and DB. Determine the shear strain at corners A and B if the plastic distorts as shown by the dashed lines.

(a) Determine the shear strain at corner A if the plate distorts as shown by the dashed line. (b) Determine the average normal strain that occurs along the diagonal AC and DB. 5 mm 2 mm 4 mm 2 mm IB 300 mm $2 mm D A 400 mm 3 mm

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.

The orientation θ of the strain gage.

Solution Summary: The author explains that the strain gage's orientation is underset_mathrmtan-1left.

Videos

The orientation θ of the strain gage.

Want to see the full answer?

Chapter 10 Solutions