Statics

Question 1

What is the layout of a 2D rotation matrix?

Answer 1

[cosϕ, sinϕ]
[-sinϕ, cosϕ]

Question 2

How do you convert a vector between the original and rotated coordinate systems?

Answer 2

[v’] = [R][v]
[v] = [R]T[v’]

Question 3

How do you convert a matrix between the original and rotated coordinate systems?

Answer 3

[A’] = [R][A][R]T
[A] = [R]T[A’][R]

Question 4

What does the ‘prime’ indicator show of a vector or matrix?

Answer 4

Aligned with the bar axis

Question 5

What is the Euler-Bernoulli displacement matrix equation?

Answer 5

[x’] = [x+u(x)] + y[-sin𝛙(x)]
[y’] [ w(x) ] [ cos𝛙(x)]

Question 6

What is assumed by the Euler-Bernoulli model?

Answer 6

Zero shear
Planes are consistent
Planes remain orthogonal to the neutral axis

Question 7

What is the Neutral Axis displacement angle according to the small angle assumption?

Answer 7

𝛙 = dw/dx

Question 8

What is εyy for Euler-Bernoulli bars?

Answer 8

εyy = 0

Question 9

What is εxx for Euler-Bernoulli bars?

Answer 9

εxx = -y(d2w/dx2)

Question 10

What does the curvature χ symbolise?

Answer 10

χ = d2w/dx2

Question 10

What is the simplified equation for the z moment?

Answer 10

Mz = EIχ

Question 11

What is the function, f, for the Euler-Bernoulli strong form?

Answer 11

f = EI(d4w/dx4)

Question 12

How is shear force Q derived from the moment?

Answer 12

Q = dM/dx

Question 13

Which essential boundary conditions cannot coexist at the same point?

Answer 13

ΓQ & Γw or ΓM & Γθ

Question 14

What must be true for the Euler-Bernoulli weak form to work?

Answer 14

Test function δw must be C1 continuous

Question 15

What does C1 continuity mean?

Answer 15

The function and its first derivative are both continuous

Question 16

How is a physical domain (x1 - x2) translated to a parent domain (-1 < ξ < 1)

Answer 16

ξ = (2(x-x1)/le) - 1

Question 17

Why is a parent domain used?

Answer 17

Shape functions for a parent domain are easier to find

Question 18

What is the gradient of the w shape functions at the extents?

Answer 18

Gradient of zero at -1 and 1

Question 19

What are the gradients of the θ shape functions at the extents?

Answer 19

For Nθ1, the gradient is 1 at -1 and 0 at 1
For Nθ2, it is 0 at -1 and 1 at 1

Question 20

How do the shear and moment matrices affect the deflections?

Answer 20

Q (shear) matrix affects w (vertical deflection), M (moment) matrix affects θ (angular deflection)

Question 21

How are the matrix properties of a frame created?

Answer 21

A combination of bar and beam properties, where bar parts constitute ux and beam parts contribute uy and θ

Question 22

What are γ and β in Timoshenko Beam Theory?

Answer 22

γ is shear angle
β is rotation of fibre

Question 23

What is the curvature χ for Timoshenko Theory?

Answer 23

dβ/dx

Question 24

What is warping?

Answer 24

When plane sections do not remain planar

Question 25

What is shear force Q in Timoshenko Theory?

Answer 25

Q = GAγ

Question 26

How are slender and stocky beams defined?

Answer 26

L/t >= 20, Slender
L/t < 10, Stocky

Question 27

For what kind of beams do Timoshenko and Euler-Bernoulli diverge significantly?

Answer 27

Stocky beams

Question 28

Why is the correction factor κ introduced for Timoshenko?

Answer 28

Because γ is overcorrected by the method

Question 29

What is the value of κ for rectangular beams?

Answer 29

κ = 5/6

Question 30

What is the benefit of using beam theory over 3D elastic theory

Answer 30

The reduced beam theory is far simpler than the full theory and allows for efficient numerical implementation.

Question 31

How do you begin the derivation of the horizontal translation u(x,y)?

Answer 31

u(x,y) = -y sinθ

Question 32

How do you find the shear strain εxy?

Answer 32

2εxy = du/dy + dw/dx

Question 33

How do you find the moment on an area A, caused by stress σ at distance y?

Answer 33

M = -∫ yσ dA

Question 34

Simplify the balance of forces equation Q(x + Δx) - Q(x) + f(x + Δx/2)Δx = 0

Answer 34

dQ + fdx = 0
Therefore dQ/dx = -f

Question 35

Why is C1 continuity important?

Answer 35

f and δf and their derivatives have to be continuous for the integral in the weak form to work

Question 36

How are the element stiffness matrix Ke and the shape factor vector Be related?

Answer 36

Ke = ∫ Ee Ie BeT Be dx

Question 37

What is the vector Be?

Answer 37

The vector of second derivatives of the shape functions Ne

Question 38

For Euler-Bernoulli theory, what are the shape functions Ne functions of?

Answer 38

Ne(w) are functions of ξ
Ne(θ) are functions of ξ and Le

Question 39

What are the 3 steps for finding the Euler-Bernoulli weak form?

Answer 39

• Multiply (S) with an arbitrary test function δw.
• Integrate the result over the length of the bar.
• Perform an integration by parts twice.

Question 40

What is assumed by Timoshenko Theory?

Answer 40

A plane section normal to the centroid remains plane but undergoes shear deformation

Question 41

What is the required continuity for w and β in the Timoshenko weak form?

Answer 41

C0 continuity

Question 42

What is dw/dx - β representative of in the Timoshenko weak form?

Answer 42

γ