Test 9 - Principal Stresses and Stress Transformation

Question 1 ప్రశ్న 1

In a two-dimensional state of stress, what is the defining characteristic of a principal plane?

A The normal stress is zero on the plane. B The shear stress is zero on the plane, and the normal stress is a principal stress. C Both normal and shear stresses are maximum on the plane. D The plane must be inclined at 45° to the original x-axis.

Question 2 ప్రశ్న 2

For a plane-stress state with tensile normal stress taken as positive, which expression gives the two in-plane principal stresses?

A σ1,2 = (σx - σy)/2 ± √[((σx + σy)/2)^2 + τxy^2] B σ1,2 = σx + σy ± τxy C σ1,2 = (σx + σy)/2 ± √[((σx - σy)/2)^2 + τxy^2] D σ1,2 = (σx + σy)/2 ± (σx - σy)/2

Question 3 ప్రశ్న 3

At a point, σx = 70 MPa tension, σy = 30 MPa tension and τxy = 0. What are the in-plane principal stresses and principal directions?

A 70 MPa and 30 MPa, with the x- and y-directions already principal B 100 MPa and 0 MPa, with principal planes at 45° C 50 MPa and 50 MPa, with every direction principal D 70 MPa and -30 MPa, with principal planes at 90° to x only

Question 4 ప్రశ్న 4

A point is subjected to a uniaxial tensile stress of 120 MPa, with all other in-plane stress components zero. What are the in-plane principal stresses and the maximum in-plane shear stress?

A σ1 = 120 MPa, σ2 = 120 MPa, τmax = 0 B σ1 = 60 MPa, σ2 = -60 MPa, τmax = 60 MPa C σ1 = 120 MPa, σ2 = -120 MPa, τmax = 120 MPa D σ1 = 120 MPa, σ2 = 0, τmax = 60 MPa

Question 5 ప్రశ్న 5

A plane-stress state has σx = 60 MPa, σy = 60 MPa and τxy = 25 MPa. What are the principal stresses and maximum in-plane shear stress?

A σ1 = 60 MPa, σ2 = 60 MPa, τmax = 25 MPa B σ1 = 85 MPa, σ2 = 35 MPa, τmax = 25 MPa C σ1 = 85 MPa, σ2 = 25 MPa, τmax = 30 MPa D σ1 = 60 MPa, σ2 = 10 MPa, τmax = 50 MPa

Question 6 ప్రశ్న 6

At a point, σx = 80 MPa, σy = 20 MPa and τxy = +30 MPa. What are the in-plane principal stresses approximately?

A 80 MPa and 20 MPa B 100 MPa and 0 MPa C 92.43 MPa and 7.57 MPa D 110 MPa and -10 MPa

Question 7 ప్రశ్న 7

A plane-stress state has σx = 100 MPa, σy = -20 MPa and τxy = 40 MPa. What are the principal stresses and maximum in-plane shear stress approximately?

A σ1 = 112.11 MPa, σ2 = -32.11 MPa, τmax = 72.11 MPa B σ1 = 120 MPa, σ2 = -40 MPa, τmax = 80 MPa C σ1 = 100 MPa, σ2 = -20 MPa, τmax = 40 MPa D σ1 = 72.11 MPa, σ2 = -72.11 MPa, τmax = 40 MPa

Question 8 ప్రశ్న 8

Use the convention that positive τxy acts in the +y direction on the +x face, and let θ be the counterclockwise rotation of the x'-axis from the x-axis. For σx = 80 MPa, σy = 20 MPa and τxy = +30 MPa, what are the smallest positive orientations of the x'-axis for the major principal stress and for one maximum-magnitude in-plane shear stress, respectively?

A 15° and 45° B 22.5° and 45° C 45° and 67.5° D 22.5° and 67.5°

Question 9 ప్రశ్న 9

Use the stress-transformation equations with positive τxy acting in the +y direction on the +x face. For σx = 80 MPa, σy = 20 MPa and τxy = +30 MPa, the x'-axis is rotated 30° counterclockwise from the x-axis. What are σx' and τx'y' approximately?

A σx' = 50.00 MPa and τx'y' = +42.43 MPa B σx' = 90.98 MPa and τx'y' = -10.98 MPa C σx' = 80.00 MPa and τx'y' = +30.00 MPa D σx' = 39.02 MPa and τx'y' = +10.98 MPa

Question 10 ప్రశ్న 10

Which quantity remains invariant when the coordinate axes are rotated in a two-dimensional stress state?

A The individual value of σx B The individual value of τxy C The sum σx + σy D The angle of the principal plane measured from the x-axis

Question 11 ప్రశ్న 11

For plane stress, which expression equals the product of the two in-plane principal stresses?

A σxσy - τxy^2 B σxσy + τxy^2 C (σx + σy)^2 D (σx - σy)^2/4

Question 12 ప్రశ్న 12

A material point is in a state of pure in-plane shear with τxy = 50 MPa and σx = σy = 0. What are the principal stresses and their orientation relative to the original axes?

A Both principal stresses are zero. B Principal stresses are 100 MPa and 0, at 45°. C Principal stresses are +25 MPa and -25 MPa, along the original axes. D Principal stresses are +50 MPa and -50 MPa, on directions rotated 45° from the original axes.

Question 13 ప్రశ్న 13

At a point, σx = σy = 60 MPa and τxy = 0. Which statement is correct regarding in-plane stress transformation?

A Only the x- and y-directions are principal. B Every in-plane orientation has normal stress 60 MPa and zero shear stress. C Maximum in-plane shear stress is 60 MPa. D The principal stresses are 120 MPa and 0.

Question 14 ప్రశ్న 14

For a general plane-stress state, what normal stress acts on the planes of maximum in-plane shear stress?

A Always zero B Always equal to the larger principal stress C The average normal stress (σx + σy)/2 D The difference σx - σy

Question 15 ప్రశ్న 15

The mean in-plane normal stress at a point is 35 MPa and the maximum in-plane shear stress is 25 MPa. What are the two in-plane principal stresses?

A 60 MPa and 10 MPa B 35 MPa and -25 MPa C 50 MPa and 20 MPa D 70 MPa and 0 MPa

Question 16 ప్రశ్న 16

The in-plane principal stresses at a point are 120 MPa and 40 MPa. What are the mean normal stress and maximum in-plane shear stress?

A Mean = 40 MPa, τmax = 80 MPa B Mean = 60 MPa, τmax = 60 MPa C Mean = 120 MPa, τmax = 40 MPa D Mean = 80 MPa, τmax = 40 MPa

Question 17 ప్రశ్న 17

Use the convention that positive τxy acts in the +y direction on the +x face. A stress state has σx = 60 MPa, σy = -20 MPa and τxy = 0. If the x'-axis is rotated 45° counterclockwise from x, what are σx' and τx'y'?

A σx' = 40 MPa and τx'y' = +20 MPa B σx' = 20 MPa and τx'y' = -40 MPa C σx' = 20 MPa and τx'y' = +40 MPa D σx' = 60 MPa and τx'y' = -20 MPa

Question 18 ప్రశ్న 18

Consider the following statements about principal stresses in plane stress:
1. Shear stress is zero on a principal plane.
2. Principal stresses are extreme values of normal stress obtainable by rotating the plane orientation.
3. Planes of maximum in-plane shear coincide with the principal planes.
4. The two principal directions in plane stress are perpendicular when the principal stresses are distinct.

Which statements are correct?

A 1 and 3 only B 2 and 3 only C 1, 2 and 4 only D 1, 2, 3 and 4

Question 19 ప్రశ్న 19

Match the plane-stress quantity with its analytical expression:
1. Sum of principal stresses
2. Product of principal stresses
3. Mean normal stress
4. Maximum in-plane shear stress

P. σx + σy
Q. σxσy - τxy^2
R. (σx + σy)/2
S. √[((σx - σy)/2)^2 + τxy^2]

Which matching is correct?

A 1-P, 2-Q, 3-R, 4-S B 1-Q, 2-P, 3-S, 4-R C 1-R, 2-S, 3-P, 4-Q D 1-S, 2-R, 3-Q, 4-P

Question 20 ప్రశ్న 20

At a point, σx = 80 MPa, σy = 20 MPa and τxy = 40 MPa. Which statement about the principal stresses is correct?

A The principal stresses are 80 MPa and 20 MPa. B The principal stresses are 90 MPa and 10 MPa. C Both principal stresses are tensile and equal. D The principal stresses are 100 MPa and 0 MPa.

Question 21 ప్రశ్న 21

At a point, σx = 50 MPa, σy = 10 MPa and the larger principal stress is 70 MPa. What is the magnitude of τxy?

A 20.00 MPa B 28.28 MPa C 34.64 MPa D 40.00 MPa

Question 22 ప్రశ్న 22

Use the convention that positive τxy acts in the +y direction on the +x face. If σx = σy = 40 MPa and τxy = +20 MPa, which orientation statement is correct?

A The original x- and y-directions are principal directions. B The major principal direction is at 45° counterclockwise from x, while the original x- and y-faces are planes of maximum-magnitude in-plane shear. C The major principal direction is at 22.5° counterclockwise from x. D No principal directions exist because σx = σy.

Question 23 ప్రశ్న 23

For a plane-stress state, rotate the coordinate axes by exactly 90° counterclockwise. Under the standard tensor transformation convention, how do the stress components change?

A σx' = σy, σy' = σx and τx'y' = -τxy B σx' = σx, σy' = σy and τx'y' = τxy C σx' = -σy, σy' = -σx and τx'y' = τxy D σx' = σy, σy' = σx and τx'y' = τxy

Question 24 ప్రశ్న 24

A plane-stress state has in-plane principal stresses of 100 MPa and 40 MPa. Since plane stress also has the third principal stress σ3 = 0, which statement correctly distinguishes maximum in-plane shear stress from the absolute maximum shear stress at the point?

A Both are 30 MPa. B Maximum in-plane shear is 50 MPa and absolute maximum shear is 30 MPa. C Both are 50 MPa. D Maximum in-plane shear is 30 MPa, while the absolute three-dimensional maximum shear stress is 50 MPa.

Question 25 ప్రశ్న 25

Consider the following statements about analytical plane-stress transformation:
1. The mean in-plane normal stress is invariant under rotation of axes.
2. The maximum in-plane shear stress equals one-half of the difference between the two in-plane principal stresses.
3. A plane of maximum in-plane shear must always be inclined exactly 45° to the original x-axis.
4. Principal-plane orientation equations involve 2θ because the transformed stress components contain sin2θ and cos2θ terms.

Which statements are correct?

A 1 and 3 only B 2 and 4 only C 1, 2 and 4 only D 1, 2, 3 and 4

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