General

Question 1

What is the equation for maximum shear stress in terms of the principle stresses?

Answer 1

(σ₁ - σ₃)/2

Question 2

What is the equation for maximum shear strain in terms of the principle strains?

Answer 2

(ɛ₁ - ɛ₃)/2

Question 3

What is the determinant of a rotation matrix that maps a right-handed coordinate system?

Answer 3

1

Question 4

What is the determinant of a rotation matrix that maps a left-handed coordinate system?

Answer 4

-1

Question 5

How is it possible to change the handedness of a rotation matrix?

Answer 5

By multiplying one of its rows by -1

Question 6

0 0 10^-3

0 0 0

10^-3 0 0

Is a small strain matrix. For an elementary cube of material, what is the angle between the face pointing in the 1-direction and the 3-axis? Likewise, what is the angle between the face pointing in the 3-direction and the 1-axis?

Answer 6

10^-3 radians

10^-3 radians

Question 7

Where do invariants get their name?

Answer 7

From the fact that they are independent of coordinate-system orientation.

Question 8

Describe how you would construct a Mohr’s circle from a single stress state.

Answer 8

1. Draw 2D illustrations of the elementary cube of material’s stress state.
2. Shear axis is positive downward, normal is abscissa.
3. For each drawing, plot the stress state of each face.
4. The normal stress in between the normal stress states of the each diagram is the centre of each circle.
5. The radius of the circles can then easily be calulcated, and the principle stresses/maximum shear stress can be found.
6. Note that a rotation within the circle is twice that of what would be necessary for the coordinate system.

Question 9

How is it possible to find the principle stresses of a stress state algebraically?

Answer 9

Using the characteristic polynomial and invariant equations.

Question 10

How are the principle directions found from the principle stresses?

Answer 10

Using the eigenvector equation using the original stress state.

Question 11

How is it possible to construct a principle coordinate system from the principle directions x₁, x₂, x₃?

Answer 11

By concatenating their transposes in the row-wise direction:

Q = [x₁^T x₂^T x₃^T]^T

Question 12

What is the convention for ordering principle values?

Answer 12

ɛ₁ > ɛ₂ > ɛ₃

or

λ₁ > λ₂ > λ₃

Question 13

What strategy should be used when attempting to orient a coordinate system relative to its predecessor?

Answer 13

Use the rotation matrix’s rows: each row is a basis vector, and each element of the row gives the cosine of the angle between the new basis and the original axis. For example, x₂₃ would give the angle between the original axis 3 and the new basis vector 2.

Question 14

How is it possible to find the displacement vector from the motion of a body?

Answer 14

Subtract the original displacement component from their respective motion function then form the vector. I.e.

u₁ = x₁ - X₁

u₂ = x₂ - X₂

u₃ = x₃ - X₃

Question 15

What is the formula for finding the small strain tensor from the displacment vector?

Answer 15

ɛ = 0.5(u_i,j + u_j,i)

Question 16

What does I^ɛ = 0 say about the deformation of a body?

Answer 16

That it is incompressible

Question 17

For what strain magnitudes is the small strain approximation valid?

Answer 17

ɛ < 0.01

Question 18

Give ten words that could be used to describe a deformation state.

Answer 18

Uniaxial

Incompressible

Biaxial

Pure shear

Triaxial

Equi-

Expansive

Compressive

Question 19

What is the magnitude of a principle direction?

Answer 19

1

Question 20

When asked to draw the planes with maximum stress, what should you do?

Answer 20

Draw a cube of elementary material, then depict the planes within that using the directions of maximum stress vectors

Question 21

What does the Mohr’s circle look like in the principle coordinate system?

Answer 21

A dot on the normal axis

Question 22

How is it possible to prove that two vectors are orthogonal?

Answer 22

By taking their dot product - a zero result indicates orthogonality.

Alternatively: x₁ ・ x₂^T =