The principal stresses of the state of stress.
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
Related questions
Question
Where did-53.13 and -26.6 and 63.4 come from please show the work
![# Explanation
## Given Information:
- The stress component along the x direction, σₓ = 150 MPa.
- The stress component along the y direction, σᵧ = 30 MPa.
- The shear stress component, τₓᵧ = -80 MPa.
## Calculation:
Calculate the principal plane (θₚ) as shown below.
\[
\tan(2θₚ) = \frac{2τₓᵧ}{σₓ - σᵧ}
\]
Substitute 150 MPa for σₓ, 30 MPa for σᵧ, and -80 MPa for τₓᵧ.
\[
\tan(2θₚ) = \frac{2 \times (-80)}{150 - 30}
\]
\[
\tan(2θₚ) = -1.333
\]
\[
2θₚ = -53.13°
\]
\[
θₚ = -26.6° \quad \text{and} \quad 63.4°
\]
Hence, the principal planes of the state of stress are -26.6° and 63.4°.
## To Determine:
The principal stresses of the state of stress.
## Answer:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F92e48f65-fa69-4461-ad79-2a8d07f543e5%2F6f9ab8f7-a865-4948-ae67-570b09564c62%2Fq4bdzgn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:# Explanation
## Given Information:
- The stress component along the x direction, σₓ = 150 MPa.
- The stress component along the y direction, σᵧ = 30 MPa.
- The shear stress component, τₓᵧ = -80 MPa.
## Calculation:
Calculate the principal plane (θₚ) as shown below.
\[
\tan(2θₚ) = \frac{2τₓᵧ}{σₓ - σᵧ}
\]
Substitute 150 MPa for σₓ, 30 MPa for σᵧ, and -80 MPa for τₓᵧ.
\[
\tan(2θₚ) = \frac{2 \times (-80)}{150 - 30}
\]
\[
\tan(2θₚ) = -1.333
\]
\[
2θₚ = -53.13°
\]
\[
θₚ = -26.6° \quad \text{and} \quad 63.4°
\]
Hence, the principal planes of the state of stress are -26.6° and 63.4°.
## To Determine:
The principal stresses of the state of stress.
## Answer:
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 1 images
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.Recommended textbooks for you
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
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