1D Flashcards
How is the equilibrium of forces derived from an arbitrary cut?
Applied Force at left + Body Force in middle + Applied Force at right = 0
What is the equilibrium of forces?
dp/dx + b(x) = 0
What is strain as an equation?
du(x)/dx
What is the governing equation for linear elasticity?
d/dx (AE du/dx) + b = 0
What is AE du/dx equal to?
The internal force p
What is the natural boundary condition for linear elasticity?
σn = E du/dx n = t on the natural boundary
What is the natural boundary?
The end of the beam where traction is applied
What is the essential boundary?
The point where displacement/temperature has a known constant value, usually zero
What is the balance of energy equation?
d(qA)/dx = s(x)
What is Fourier’s Law?
Heat flux q = -k dT/dx
What is the governing equation for 1D heat conduction?
d/dx (Ak dT/dx) + s = 0
How do you begin to derive the weak form in Linear Elasticity?
Multiply by an arbitrary test function δu and integrate
What is the product rule for integrating?
Int δu (f)’ dx = Int (δuf)’ dx - Int (δu)’ f dx
What is the other equation necessary for completing the weak form?
δuA(E du/dx n - t)|t = 0
Apart from the terms used, how is the heat conduction weak form different to the linear elasticity one?
The |t term is negative
What must be true of the approximations for u(x) and δu(x)?
They must be continuous
How is the function across a linear element approximated?
By taking the values at either end of the element (each node)
What is a shape function?
A function that has value 1 at the node it corresponds to and 0 at all other nodes
What is matrix B?
The derivative of the shape function matrix N
If a function F is defined as Nd, what is the derivative dF?
Bd
What does the vector d contain?
The values of the function at each node (fx for each x)
What is the scatter matrix (Le) for the first element of a 2 element system with 3 nodes?
Global Node: 1 2 3
Local Node 1: [ 1 0 0 ]
Local Node 2: [ 0 1 0 ]
How do scatter matrices help to assemble the global matrix?
de = Le d, N = Σ Ne Le
When solving the weak form with matrices, why are transposes of the test function used?
The test function has arbitrary value and using transposes allows use of the (Ab)T = bT AT relationship
What is the element stiffness matrix Ke equal to?
Int (BeT Ae Ee Be) dx
What is b and what are its units?
Body force per unit length, N/m
What is the relationship between sigma and p?
Sigma = p/A
What is p and what are its units?
Internal force, N
If given an equation for stress, how can you construct the strong form?
Replace stress with p/A, rearrange for p, replace dp/dx in the balance of forces by d/dx of the rearranged equation
Why is the weak form used instead of the strong form?
It is easier to solve
It can consider complex geometries and loading
What is the equation form for solving a finite element problem?
Kd = f + r
How is strain found after displacements have been solved?
Strain = Bd, where b is 1/L multiplied by the gradients of the shape functions
What are two assumptions of linear elasticity?
Young’s Modulus is constant
Small deformations
Why do all the terms in a stiffness matrix sum to zero?
To account for the lack of internal forces in the event of a rigid body movement where all displacement are equal
