1D

Question 1

How is the equilibrium of forces derived from an arbitrary cut?

Answer 1

Applied Force at left + Body Force in middle + Applied Force at right = 0

Question 2

What is the equilibrium of forces?

Answer 2

dp/dx + b(x) = 0

Question 3

What is strain as an equation?

Answer 3

du(x)/dx

Question 4

What is the governing equation for linear elasticity?

Answer 4

d/dx (AE du/dx) + b = 0

Question 5

What is AE du/dx equal to?

Answer 5

The internal force p

Question 6

What is the natural boundary condition for linear elasticity?

Answer 6

σn = E du/dx n = t on the natural boundary

Question 7

What is the natural boundary?

Answer 7

The end of the beam where traction is applied

Question 8

What is the essential boundary?

Answer 8

The point where displacement/temperature has a known constant value, usually zero

Question 9

What is the balance of energy equation?

Answer 9

d(qA)/dx = s(x)

Question 10

What is Fourier’s Law?

Answer 10

Heat flux q = -k dT/dx

Question 11

What is the governing equation for 1D heat conduction?

Answer 11

d/dx (Ak dT/dx) + s = 0

Question 12

How do you begin to derive the weak form in Linear Elasticity?

Answer 12

Multiply by an arbitrary test function δu and integrate

Question 13

What is the product rule for integrating?

Answer 13

Int δu (f)’ dx = Int (δuf)’ dx - Int (δu)’ f dx

Question 14

What is the other equation necessary for completing the weak form?

Answer 14

δuA(E du/dx n - t)|t = 0

Question 15

Apart from the terms used, how is the heat conduction weak form different to the linear elasticity one?

Answer 15

The |t term is negative

Question 16

What must be true of the approximations for u(x) and δu(x)?

Answer 16

They must be continuous

Question 17

How is the function across a linear element approximated?

Answer 17

By taking the values at either end of the element (each node)

Question 18

What is a shape function?

Answer 18

A function that has value 1 at the node it corresponds to and 0 at all other nodes

Question 19

What is matrix B?

Answer 19

The derivative of the shape function matrix N

Question 20

If a function F is defined as Nd, what is the derivative dF?

Answer 20

Bd

Question 21

What does the vector d contain?

Answer 21

The values of the function at each node (fx for each x)

Question 22

What is the scatter matrix (Le) for the first element of a 2 element system with 3 nodes?

Answer 22

Global Node: 1 2 3
Local Node 1: [ 1 0 0 ]
Local Node 2: [ 0 1 0 ]

Question 23

How do scatter matrices help to assemble the global matrix?

Answer 23

de = Le d, N = Σ Ne Le

Question 24

When solving the weak form with matrices, why are transposes of the test function used?

Answer 24

The test function has arbitrary value and using transposes allows use of the (Ab)T = bT AT relationship

Question 25

What is the element stiffness matrix Ke equal to?

Answer 25

Int (BeT Ae Ee Be) dx

Question 26

What is b and what are its units?

Answer 26

Body force per unit length, N/m

Question 27

What is the relationship between sigma and p?

Answer 27

Sigma = p/A

Question 28

What is p and what are its units?

Answer 28

Internal force, N

Question 29

If given an equation for stress, how can you construct the strong form?

Answer 29

Replace stress with p/A, rearrange for p, replace dp/dx in the balance of forces by d/dx of the rearranged equation

Question 30

Why is the weak form used instead of the strong form?

Answer 30

It is easier to solve
It can consider complex geometries and loading

Question 31

What is the equation form for solving a finite element problem?

Answer 31

Kd = f + r

Question 32

How is strain found after displacements have been solved?

Answer 32

Strain = Bd, where b is 1/L multiplied by the gradients of the shape functions

Question 33

What are two assumptions of linear elasticity?

Answer 33

Young’s Modulus is constant
Small deformations

Question 34

Why do all the terms in a stiffness matrix sum to zero?

Answer 34

To account for the lack of internal forces in the event of a rigid body movement where all displacement are equal