Statics Flashcards
What is the layout of a 2D rotation matrix?
[cosϕ, sinϕ]
[-sinϕ, cosϕ]
How do you convert a vector between the original and rotated coordinate systems?
[v’] = [R][v]
[v] = [R]T[v’]
How do you convert a matrix between the original and rotated coordinate systems?
[A’] = [R][A][R]T
[A] = [R]T[A’][R]
What does the ‘prime’ indicator show of a vector or matrix?
Aligned with the bar axis
What is the Euler-Bernoulli displacement matrix equation?
[x’] = [x+u(x)] + y[-sin𝛙(x)]
[y’] [ w(x) ] [ cos𝛙(x)]
What is assumed by the Euler-Bernoulli model?
Zero shear
Planes are consistent
Planes remain orthogonal to the neutral axis
What is the Neutral Axis displacement angle according to the small angle assumption?
𝛙 = dw/dx
What is εyy for Euler-Bernoulli bars?
εyy = 0
What is εxx for Euler-Bernoulli bars?
εxx = -y(d2w/dx2)
What does the curvature χ symbolise?
χ = d2w/dx2
What is the simplified equation for the z moment?
Mz = EIχ
What is the function, f, for the Euler-Bernoulli strong form?
f = EI(d4w/dx4)
How is shear force Q derived from the moment?
Q = dM/dx
Which essential boundary conditions cannot coexist at the same point?
ΓQ & Γw or ΓM & Γθ
What must be true for the Euler-Bernoulli weak form to work?
Test function δw must be C1 continuous
What does C1 continuity mean?
The function and its first derivative are both continuous
How is a physical domain (x1 - x2) translated to a parent domain (-1 < ξ < 1)
ξ = (2(x-x1)/le) - 1
Why is a parent domain used?
Shape functions for a parent domain are easier to find
What is the gradient of the w shape functions at the extents?
Gradient of zero at -1 and 1
What are the gradients of the θ shape functions at the extents?
For Nθ1, the gradient is 1 at -1 and 0 at 1
For Nθ2, it is 0 at -1 and 1 at 1
How do the shear and moment matrices affect the deflections?
Q (shear) matrix affects w (vertical deflection), M (moment) matrix affects θ (angular deflection)
How are the matrix properties of a frame created?
A combination of bar and beam properties, where bar parts constitute ux and beam parts contribute uy and θ
What are γ and β in Timoshenko Beam Theory?
γ is shear angle
β is rotation of fibre
What is the curvature χ for Timoshenko Theory?
dβ/dx
What is warping?
When plane sections do not remain planar
What is shear force Q in Timoshenko Theory?
Q = GAγ
How are slender and stocky beams defined?
L/t >= 20, Slender
L/t < 10, Stocky
For what kind of beams do Timoshenko and Euler-Bernoulli diverge significantly?
Stocky beams
Why is the correction factor κ introduced for Timoshenko?
Because γ is overcorrected by the method
What is the value of κ for rectangular beams?
κ = 5/6
What is the benefit of using beam theory over 3D elastic theory
The reduced beam theory is far simpler than the full theory and allows for efficient numerical implementation.
How do you begin the derivation of the horizontal translation u(x,y)?
u(x,y) = -y sinθ
How do you find the shear strain εxy?
2εxy = du/dy + dw/dx
How do you find the moment on an area A, caused by stress σ at distance y?
M = -∫ yσ dA
Simplify the balance of forces equation Q(x + Δx) - Q(x) + f(x + Δx/2)Δx = 0
dQ + fdx = 0
Therefore dQ/dx = -f
Why is C1 continuity important?
f and δf and their derivatives have to be continuous for the integral in the weak form to work
How are the element stiffness matrix Ke and the shape factor vector Be related?
Ke = ∫ Ee Ie BeT Be dx
What is the vector Be?
The vector of second derivatives of the shape functions Ne
For Euler-Bernoulli theory, what are the shape functions Ne functions of?
Ne(w) are functions of ξ
Ne(θ) are functions of ξ and Le
What are the 3 steps for finding the Euler-Bernoulli weak form?
- Multiply (S) with an arbitrary test function δw.
- Integrate the result over the length of the bar.
- Perform an integration by parts twice.
What is assumed by Timoshenko Theory?
A plane section normal to the centroid remains plane but undergoes shear deformation
What is the required continuity for w and β in the Timoshenko weak form?
C0 continuity
What is dw/dx - β representative of in the Timoshenko weak form?
γ
