This set of Finite Element Method Multiple Choice Questions & Answers (MCQs) focuses on “Three Dimensional Problems – Stress Calculations”.

1. What are the stress invariants used for?

a) Calculate principal stresses

b) Calculate axial stresses

c) Calculate bending stresses

d) Calculate torsional stresses

View Answer

Explanation: Stress invariants are used for the calculation of unknown principal stresses. These stresses are three in number; one along each axis – x, y, z. There are a total of three stress variants for a (3 * 3) stress tensor. These help to calculate the unknown values of the principal stresses.

2. Which of the following is one of the three invariants of a (3 * 3) stress tensor?

a) I = u_{x}u_{y} + u_{y}u_{z} + u_{z}u_{x} + z^{2}_{xy} + z^{2}_{yz} + z^{2}_{zx}

b) I = u_{x}u_{y} + u_{y}u_{z} – u_{z}u_{x} + z^{2}_{xy} + z^{2}_{yz} + z^{2}_{zx}

c) I = u_{x}u_{y} + u_{y}u_{z} + u_{z}u_{x} – z^{2}_{xy} – z^{2}_{yz} – z^{2}_{zx}

d) I = u_{x}u_{y} – u_{y}u_{z} + u_{z}u_{x} + z^{2}_{xy} + z^{2}_{yz} + z^{2}_{zx}

View Answer

Explanation: I = u

_{x}u

_{y}+ u

_{y}u

_{z}+ u

_{z}u

_{x}– z

^{2}

_{xy}– z

^{2}

_{yz}– z

^{2}

_{zx}is the second invariant of a (3 * 3) stress tensor. This is used to calculate the principal stress of the matrix. Here, u

_{x}, u

_{y}and u

_{z}refer to the stresses normal to the surface; whereas the rest of them correspond to the stresses parallel to the surface.

3. If σ_{1}, σ_{2}, and σ_{3} are the roots of the three invariant equations, which of the following corresponds to the relation between them?

a) σ_{1} / σ_{2} = σ_{3}

b) σ_{1} > σ_{2} > σ_{3}

c) σ_{1} < σ_{2} < σ_{3}

d) σ_{1} * σ_{2} > σ_{3}

View Answer

Explanation: By general convention, it is assumed that σ

_{1}> σ

_{2}> σ

_{3}. Where, σ

_{1}corresponds to the highest tensile stress, σ

_{3}corresponds to the highest compressive stress, σ

_{2}corresponds to the intermediate value of stress obtained. This relation helps to differentiate between the three values of principal strain obtained on solving the stress invariant equation.

4. The stress tensor is usually decomposed into three parts.

a) True

b) False

View Answer

Explanation: The given statement is false. The stress tensor is usually decomposed into two parts. They are hydrostatic component and the deviatoric stress component. This helps us simplify the given matrix and allows us to arrive at the values of the principal stresses.

5. Which of the following is the definition of a hydrostatic stress?

a) Stress whose components are equal

b) Stress whose components are only shear

c) Stress whose components are unequal

d) Stress whose components are only normal

View Answer

Explanation: Hydrostatic stress is defined as a component of stress that contains only uniaxial / normal stresses. This does not take into consideration the shear stresses; in other words no shear stress components are present.

6. Which of the following is the definition of a deviatoric stress?

a) Stress component that contains unequal principal stresses

b) Stress component that contains equal principal stresses

c) Stress component that contains unequal axial stresses

d) Stress component that contains unequal bending stresses

View Answer

Explanation: Deviatoric stress is defined as a component of stress that contains unequal stresses. These stresses can be obtained by subtracting the hydrostatic stresses from the principal stresses. In other words, the summation of hydrostatic and deviatoric stresses is equal to principal stresses.

7. The value of principal stress σ_{3} can be found out by solving the determinant of the stress tensor matrix.

a) True

b) False

View Answer

Explanation: The given statement is true. The expression for the third and final stress variant is given by σ

_{x}σ

_{y}σ

_{z}+ 2z

_{xy}z

_{yz}z

_{zx}– σ

_{x}z

^{2}

_{xy}– σ

_{y}z

^{2}

_{yz}– σ

_{z}z

^{2}

_{xz}. This also happens to be the determinant of the stress tensor matrix.

8. Find the value of I1, given stress tensor matrix = [1, 5, 4, 3, 6, 7, 9, 8, 2]

a) 8

b) 11

c) 9

d) 13

View Answer

Explanation: We know that I

_{1}= σ

_{x}+ σ

_{y}+ σ

_{z}

= 1 + 6 + 2

= 9 MPa

9. Given stress tensor matrix = [5, 3, 1, 3, 2, 0, 1, 0, 4], Calculate the value of the second stress invariant(I_{2}).

a) 11

b) 13

c) 15

d) 12

View Answer

Explanation: We know that I

_{2}= u

_{x}u

_{y}+ u

_{y}u

_{z}+ u

_{z}u

_{x}– z

^{2}

_{xy}– z

^{2}

_{yz}– z

^{2}

_{zx}

= 5 * 2 + 2 * 4 + 4 * 5 – 3

^{2}– 1

^{2}– 4

^{2}

= 10 + 8 + 20 – 9 – 1 – 16

= 38 – 26

= 12 MPa

10. What is the definition of an octahedral stress?

a) Stresses that act on the octahedron plane

b) Stresses that act on the polyhedron plane

c) Stresses that act on the hexahedron plane

d) Stresses that act on the decahedron plane

View Answer

Explanation: Planes whose normal vector makes equal angles to the axes of principal stresses are referred to as octahedron planes. The normal and shear stresses that act on these planes are referred to as octahedral stresses. Usually there are eight such planes present in a conventional system.

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