Spring element functions¶

The spring element, shown below, can be used for the analysis of one-dimensional spring systems and for a variety of analogous physical problems.

Quantities corresponding to the variables of the spring are listed in Table 1.

Analogous quantities¶
Problem type	Spring stiffness	Nodal displacement	Element force	Spring force
Spring	\(k\)	\(u\)	\(P\)	\(N\)
Bar	\(\frac{EA}{L}\)	\(u\)	\(P\)	\(N\)
Thermal conduction	\(\frac{\lambda A}{L}\)	\(T\)	\(\bar{H}\)	\(H\)
Diffusion	\(\frac{D A}{L}\)	\(c\)	\(\bar{H}\)	\(H\)
Electrical circuit	\(\frac{1}{R}\)	\(U\)	\(\bar{I}\)	\(I\)
Groundwater flow	\(\frac{kA}{L}\)	\(\phi\)	\(\bar{H}\)	\(H\)
Pipe network	\(\frac{\pi D^4}{128{\mu}L}\)	\(p\)	\(\bar{H}\)	\(H\)

Quantities used in different types of problems¶
Problem type	Designations	Description
Spring	\(k\), \(u\), \(P\), \(N\)	spring stiffness, displacement, element force, spring force
Bar	\(L\), \(E\), \(A\), \(u\), \(P\), \(N\)	length, modulus of elasticity, area of cross section, displacement, element force, normal force
Thermal conduction	\(L\), \(\lambda\), \(T\), \(\bar{H}\), \(H\)	length, thermal conductivity, temperature, element heat flow, internal heat flow
Diffusion	\(L\), \(D\), \(c\), \(\bar{H}\), \(H\)	length, diffusivity, nodal concentration, nodal mass flow, element mass flow
Electrical circuit	\(R\), \(U\), \(\bar{I}\), \(I\)	resistance, potential, element current, internal current
Groundwater flow	\(L\), \(k\), \(\phi\), \(\bar{H}\), \(H\)	length, permeability, piezometric head, element water flow, internal water flow
Pipe network (laminar flow)	\(L\), \(D\), \(\mu\), \(p\), \(\bar{H}\), \(H\)	length, pipe diameter, viscosity, pressure, element fluid flow, internal fluid flow

The following functions are available for the spring element:

Spring functions¶
spring1e	Compute element matrix
spring1s	Compute spring force

spring1e¶

Purpose:

Compute element stiffness matrix for a spring element.

Syntax:

Ke = spring1e(ep)

Description:

spring1e provides the element stiffness matrix \(\bar{\mathbf{K}}^e\) for a spring element.

The input variable

ep\(= [k]\)

supplies the spring stiffness \(k\) or the analog quantity defined in Table Analogous quantities.

Theory:

The element stiffness matrix \(\mathbf{K}^e\), stored in Ke, is computed according to

\[\begin{split}\mathbf{K}^e = \begin{bmatrix} k & -k \\ -k & k \end{bmatrix}\end{split}\]

where \(k\) is defined by ep.

Purpose:

Compute spring force in a spring element.

Syntax:

es = spring1s(ep, ed)

Description:

spring1s computes the spring force in the spring element spring1e.

The input variable ep is defined in spring1e and the element nodal displacements ed are obtained by the function extract_ed.

The output variable

es\(= [N]\)

contains the spring force, or the analog quantity.

Theory:

The spring force \(N\), or analog quantity, is computed according to

\[N = k \left(u_2 - u_1\right)\]