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If Normal stresses of same nature px and py and shear stress q is acting on two perpendicular planes and q = (pxpy)1/2, then the major and minor principal stresses respectively are
- a)px + py and px − py
- b)px and px − py
- c)0.5 (px + py)and 0.5 (px − py )
- d)px + py and zero
Correct answer is option 'D'. Can you explain this answer?
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If Normal stresses of same nature px and py and shear stress q is act...
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If Normal stresses of same nature px and py and shear stress q is act...
Major and Minor Principal Stresses
The given question is asking for the major and minor principal stresses when normal stresses of the same nature (px and py) and a shear stress (q) are acting on two perpendicular planes. The relationship between these stresses is given as q = sqrt(px * py).
To determine the major and minor principal stresses, we need to understand the concept of principal stresses and how they relate to the given normal and shear stresses.
Principal stresses are the maximum and minimum values of normal stresses that act on a particular point in a material. They are represented by sigma1 (major principal stress) and sigma2 (minor principal stress).
To find the major and minor principal stresses, we can use the following equations:
sigma1 = (px + py)/2 + sqrt(((px-py)/2)^2 + q^2)
sigma2 = (px + py)/2 - sqrt(((px-py)/2)^2 + q^2)
Now, let's solve the given problem step by step:
Step 1: Given normal stresses:
px and py are normal stresses acting on two perpendicular planes.
Step 2: Shear stress:
q = sqrt(px * py)
Step 3: Major Principal Stress (sigma1):
sigma1 = (px + py)/2 + sqrt(((px-py)/2)^2 + q^2)
= (px + py)/2 + sqrt((px^2 - 2pxpy + py^2)/4 + pxpy)
= (px + py)/2 + sqrt((px^2 + 2pxpy + py^2)/4)
= (px + py)/2 + sqrt((px + py)^2)/2
= (px + py)/2 + (px + py)/2
= px + py
Therefore, the major principal stress (sigma1) is equal to px + py.
Step 4: Minor Principal Stress (sigma2):
sigma2 = (px + py)/2 - sqrt(((px-py)/2)^2 + q^2)
= (px + py)/2 - sqrt((px^2 - 2pxpy + py^2)/4 + pxpy)
= (px + py)/2 - sqrt((px^2 + 2pxpy + py^2)/4)
= (px + py)/2 - sqrt((px + py)^2)/2
= (px + py)/2 - (px + py)/2
= 0
Therefore, the minor principal stress (sigma2) is equal to zero.
Hence, the major and minor principal stresses are px + py and zero respectively, which corresponds to option D.
The given question is asking for the major and minor principal stresses when normal stresses of the same nature (px and py) and a shear stress (q) are acting on two perpendicular planes. The relationship between these stresses is given as q = sqrt(px * py).
To determine the major and minor principal stresses, we need to understand the concept of principal stresses and how they relate to the given normal and shear stresses.
Principal stresses are the maximum and minimum values of normal stresses that act on a particular point in a material. They are represented by sigma1 (major principal stress) and sigma2 (minor principal stress).
To find the major and minor principal stresses, we can use the following equations:
sigma1 = (px + py)/2 + sqrt(((px-py)/2)^2 + q^2)
sigma2 = (px + py)/2 - sqrt(((px-py)/2)^2 + q^2)
Now, let's solve the given problem step by step:
Step 1: Given normal stresses:
px and py are normal stresses acting on two perpendicular planes.
Step 2: Shear stress:
q = sqrt(px * py)
Step 3: Major Principal Stress (sigma1):
sigma1 = (px + py)/2 + sqrt(((px-py)/2)^2 + q^2)
= (px + py)/2 + sqrt((px^2 - 2pxpy + py^2)/4 + pxpy)
= (px + py)/2 + sqrt((px^2 + 2pxpy + py^2)/4)
= (px + py)/2 + sqrt((px + py)^2)/2
= (px + py)/2 + (px + py)/2
= px + py
Therefore, the major principal stress (sigma1) is equal to px + py.
Step 4: Minor Principal Stress (sigma2):
sigma2 = (px + py)/2 - sqrt(((px-py)/2)^2 + q^2)
= (px + py)/2 - sqrt((px^2 - 2pxpy + py^2)/4 + pxpy)
= (px + py)/2 - sqrt((px^2 + 2pxpy + py^2)/4)
= (px + py)/2 - sqrt((px + py)^2)/2
= (px + py)/2 - (px + py)/2
= 0
Therefore, the minor principal stress (sigma2) is equal to zero.
Hence, the major and minor principal stresses are px + py and zero respectively, which corresponds to option D.
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If Normal stresses of same nature px and py and shear stress q is acting on two perpendicular planes and q = (pxpy)1/2, then the major and minor principal stresses respectively area) px + py and px − pyb) px and px − pyc) 0.5 (px + py)and 0.5 (px − py )d) px + py and zeroCorrect answer is option 'D'. Can you explain this answer?
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If Normal stresses of same nature px and py and shear stress q is acting on two perpendicular planes and q = (pxpy)1/2, then the major and minor principal stresses respectively area) px + py and px − pyb) px and px − pyc) 0.5 (px + py)and 0.5 (px − py )d) px + py and zeroCorrect answer is option 'D'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about If Normal stresses of same nature px and py and shear stress q is acting on two perpendicular planes and q = (pxpy)1/2, then the major and minor principal stresses respectively area) px + py and px − pyb) px and px − pyc) 0.5 (px + py)and 0.5 (px − py )d) px + py and zeroCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If Normal stresses of same nature px and py and shear stress q is acting on two perpendicular planes and q = (pxpy)1/2, then the major and minor principal stresses respectively area) px + py and px − pyb) px and px − pyc) 0.5 (px + py)and 0.5 (px − py )d) px + py and zeroCorrect answer is option 'D'. Can you explain this answer?.
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