Matrix Method

Question 1

The displacement method in matrix form

Fundamental laws

Answer 1

Kinematic compatibility

Constitutive relationship (Material law)

Static equilibrium

Question 2

Assumptions, linear theory

Answer 2

Small displacements

Linear elastic material

Principle of superposition

Question 3

r = FR

Explain the letters

Answer 3

r = nodal degrees of freedom

F = flexibility matrix

R = nodal forces

Question 4

The stiffness matrix, K, is the inverse of

Answer 4

The flexibility matrix, F

Question 5

The stiffness component Kij may be defined as

Answer 5

Kij equals the force/moment in nodal degree of freedom i, due to a unit displacement/rotation in nodal degree of freedom j

Question 6

S^e = k^e * v^e

Explain the letters

Answer 6

S^e = element nodal forces

k^e = element stiffness matrix

v^e = element nodal displacement

Question 7

What is the dimension of the stiffness matrix for a 2 dof bar element

Answer 7

2 x 2

Question 8

What is the dimension of the stiffness matrix for a 4 dof beam element?

Answer 8

4 x 4

Question 9

What is the dimension of the stiffness matrix for a 6 dof frame/beam element

Answer 9

6 x 6

Question 10

Nodal point compatibility

Answer 10

Nodal displacement and rotation of elements that share a node is equal in the respective node

Question 11

What is the a-matrix

Answer 11

Called the connectivity or topology matrix. Number of rows equals the number of element degree degrees of freedom in v, while the number of columns equals the number of nodal degrees of freedom

Consists of zeros, and 1 were it correspond to r

Question 12

What is S^0

Answer 12

Consistent element load vector, also called fixed end forced

Guarantees equilibrium at the element

Question 13

General expression for system stiffness matrix

Answer 13

K = (a^e)^T * k^e * a^e

Question 14

Equation for S^e

Answer 14

S^e = k^e * a^e * r * S^oe

Question 15

Procedure to find S^e

Answer 15

1. establish S^o —> R^o
2. obtain complementary solution r
   - R —> K —> K^-1 —> r
3. Obtain total solution

Question 16

Difference between S and S^o

Answer 16

S = nodal forces

S^o = consistent element nodal load vector (fixed end forces)

Question 17

What is T^o

Answer 17

Transformation matrix
It consists of a combination of sin, cos, 0 and 1
It is orthogonal (inverse + transposed)

Question 18

How many rigid body modes do a 6 dof frame element have?

Answer 18

3

Two translations and one rotation

Question 19

How many deformation modes do a 6 dof frame element have?

Answer 19

3

Elongation, symmetric bending and asymmetric bending

Question 20

What does the rank of the stiffness matrix rank(k) tell you?

Answer 20

Tells you the number of independent deformation modes.

Question 21

What is the consequence of a stiffness matrix rank lower than the size of the stiffness matrix?

Answer 21

The mateix is singular. I.e., det(k)=0