Element

Each instance represents one truss member

Element class methods

method	input	returns	description
__init__(nd0, nd1, material)	two Node() objects, one Material() object.		constructor.
getForce()		list of (1d) np.array objects	A list of nodal forces. Each nodal force shall be represented as a 1d np.array with two components of the force.
getStiffness()		2-by-2 list of 2-by-2 np.array	A list of lists (matrix) containing nodal tangent matrices as np.array([[.,.],[.,.]])

Note: a Node() object may have changed its state between calls, so you need to recompute every time!

Element class variables

name	type	description
nodes	List of Node() instances	representing the two nodes at either end of a truss.
material	Material()	pointer to an instance of Material(). Needed to compute stress and tangent modulus.
force	2-list of (1d) np.array objects.	holding nodal forces P0 and P1
Kt	2-by-2 array of 2-by-2 np.array objects.	tangent stiffness matrix. Representing all nodal stiffness matrices.

Equations

1. \({\bf L} = {\bf X}_1 - {\bf X}_0\)

2. \(\ell = ||{\bf L}||\)

3. \({\bf n} = \frac{1}{\ell} \, {\bf L}\)

4. Strain: \(\varepsilon = \frac{1}{\ell} \, {\bf n}\cdot( {\bf U}_1 - {\bf U}_0)\)

5. Force: \(f = \sigma(\varepsilon) A\) using material.setStrain(eps) and material.getStress().

6. Nodal force vector: \({\bf P}^e = f \, {\bf n}\)

7. \({\bf P}_0 = -{\bf P}^e ~~~~~~ {\bf P}_1 = {\bf P}^e\)

8. Nodal stiffness matrix: \({\bf k}^e = \frac{E_t(\varepsilon)\,A}{\ell}\, {\bf n}\otimes{\bf n}\) using material.getStiffness() to find \(E_t\).

9. \({\bf k}_{00} = {\bf k}_{11} = {\bf k}^e ~~~~~~~~ {\bf k}_{01} = {\bf k}_{10} = -{\bf k}^e\)