Stiffness Matrix Method

Question 1

In the stiffness matrix method, What form is are the equilibrium equations written in?

Answer 1

Force {F} = Stiffness {K} * Displacement {u}

Question 2

In terms of u1 and u2 in a 1D bar, what is strain (€) equal to?

How is strain related to stress?

Answer 2

ε = (u2 - u1) / L

σ = ε * E

Question 3

What does the tensile force in a 1D bar equal?

Answer 3

N = (EA)/L * (u2 - u1)

Question 4

What is the stiffness matrix of a 1D bar?

Answer 4

[Ke] = EA/L * [1,-1 ; -1,1]

Question 5

Outline the procedure for solving equilibrium equations for a system of 1D bars.

Answer 5

1) Write out equilibrium equations for internal forces and local stiffness matrix for each bar.
2) Set up a global stiffness matrix using these.
3) Apply the boundary conditions to the equations.
4) Find the reduced system equilibrium equation
5) Solve these equations to find the unknown displacements and forces
6) Use these values to find the last unknown.
7) Substitute these values back into the initial equations to find internal forces.

Question 6

How is the inclination angle α found in terms of Sin (S) and Cos (C)?

What is the sign convention?

Answer 6

sin(α) = s = (y2 - y1) / L
cos(α) = c = (x2 - x1)/ L

Anti-clockwise is Positive.

Question 7

What is GT equal to?

Answer 7

= [c,0 ; s,0 ; 0,c ; 0,s]

Question 8

Outline the procedure for solving a 2D bar system using the stiffness matrix method.

Answer 8

1) Find the inclination angle (α) for each bar, making sure anti-clockwise is positive.
2) Find S and C for each bar.
3) Set up a local stiffness matrix for each bar then combine to find the global stiffness matrix.
4) Apply boundary conditions to the equilibrium equitation.
5) Form the reduced system equations to find the unknown displacements.
6) Use these to find the other unknowns in the equations.
7) Find the internal forces in each bar.