General Spring Elements (TYPE8)

General spring elements are defined as TYPE8 element property. They are mathematical elements, which have 6 DOF, three translational displacements and three rotational degrees of freedom. Each DOF is completely independent from the others. Spring displacements refer to either spring extension or compression. The stiffness is associated to each DOF. Directions can either be global or local. Local directions are defined with a fixed or moving skew frame. Global force equilibrium is respected, but without global moment equilibrium. Therefore, this type of spring is connected to the laboratory that applies the missing moments, unless the two defining nodes are not coincident.

Time Step

The time step calculation for general spring elements is the same as the calculation of the equivalent TYPE4 spring (Time Step).

Linear Spring

See Linear Spring; the explanation is the same as for spring TYPE4.

Nonlinear Elastic Spring

See Nonlinear Elastic Spring; the explanation is the same as for spring TYPE4.

Nonlinear Elasto-plastic Spring: Isotropic Hardening

See Nonlinear Elasto-plastic Spring: Isotropic Hardening; the explanation is the same as for spring TYPE 4.

Nonlinear Elasto-plastic Spring: Decoupled Hardening

See Nonlinear Elasto-plastic Spring: Decoupled Hardening; the explanation is the same as for spring TYPE4.

Nonlinear Elasto-plastic Spring: Kinematic Hardening

See Nonlinear Elasto-plastic Spring: Kinematic Hardening; the explanation is the same as for spring TYPE4.

Nonlinear Elasto-plastic Spring: Nonlinear Unloading

See Nonlinear Elasto-plastic Spring: Nonlinear Unloading; the explanation is the same as for spring TYPE4.

Nonlinear Dashpot

See Nonlinear Dashpot; the explanation is the same as for spring TYPE4.

Nonlinear Viscoelastic Spring

See Nonlinear Viscoelastic Spring; the explanation is the same as for spring TYPE4.

Skew Frame Properties

To help understand the use of skew frames, the deformation in the local x direction of the spring will be considered. If the skew frame is fixed, deformation in the local X direction is shown in Figure 1:

The same local x direction deformation, with a moving skew frame, can be seen in Figure 2.

In both cases, the forces are in equilibrium, but the moments are not. If the first two nodes defining the moving skew system are the same nodes as the two spring element nodes, the behavior becomes exactly the same as that of a TYPE4 spring element. In this case the momentum equilibrium is respected and local Y and Z deformations are always equal to zero.

Fixed Nodes

If one of the two nodes is completely fixed, the momentum equilibrium problem disappears. For example, if node 1 is fixed, the force computation at node 2 is not dependent on the location of node 1. The spring then becomes a spring between node 1 and the laboratory, as shown in Figure 3.

It is generally recommended that a general spring element (TYPE8) be used only if one node is fixed in all directions or if the two nodes are coincident. If the two nodes are coincident, the translational stiffness' have to be large enough to ensure that the nodes remain near coincident during the simulation.

Deformation Sign Convention

Positive and negative spring deformations are not defined with the variation of initial length. The initial length can be equal to zero for all or a given direction. Therefore, it is not possible to define the deformation sign with length variation.

The sign convention used is the following. A deformation is positive if displacement (or rotation) of node 2 minus the displacement of node 1 is positive. The same sign convention is used for all 6 degrees of freedom.(1)

u_{i} = u_{i 2} - u_{i 1}

(2)

θ_{i} = θ_{i 2} - θ_{i 1}

Translational Forces

The translational forces that can be applied to a general spring element can be seen in Figure 4. For each DOF (i = x, y, z), the force is calculated by:(3)

F_{i} = f_{_{i}} (u_{i}) + C_{i} {\dot{u}}_{i}

Where,

$C$: Equivalent viscous damping coefficient
$f_{i} (u_{i})$: Force function related to spring displacement

The value of the displacement function depends on the type of general spring being modeled.

Linear Spring

If a linear general spring is being modeled, the translation forces are given by:(4)

F_{i} = K_{i} u_{i} + C_{i} {\dot{u}}_{i}

Where,

$K$: Stiffness or unloading stiffness (for elasto-plastic spring)

Nonlinear Spring

If a nonlinear general spring is being modeled, the translation forces are given by:(5)

F_{i} = f_{_{i}} (u_{i}) (A + B \ln (| \frac{{\dot{u}}_{i}}{D} |) + g ({\dot{u}}_{i})) + C_{i} {\dot{u}}_{i}

Where,

$f (u_{i})$: Function defining the change in force with spring displacement
$g ({\dot{u}}_{i})$: Function defining the change in force with spring displacement rate
$A$: Coefficient; Default = 1
$B$: Coefficient
$D$: Coefficient; Default = 1

Moments

Moments can be applied to a general spring element, as shown in Figure 5. For each DOF (i = x, y, z), the moment is calculated by:(6)

M_{i} = f_{_{i}} (θ_{i}) + C_{i} {\dot{θ}}_{i}

Where,

$C$: Equivalent viscous damping coefficient
$f_{i} (θ_{i})$: Force function related to spring rotation

The value of the rotation function depends on the type of general spring being modeled. Not all functions and coefficients defining moments and rotations are of the same value as that used in the translational force calculation.

Linear Spring

If a linear general spring is being modeled, the translation forces are given by:(7)

M_{i} = K_{i} θ_{i} + C_{i} {\dot{θ}}_{i}

Where,

$K$: Stiffness or unloading stiffness (for elasto-plastic spring)

Nonlinear Spring

If a nonlinear general spring is being modeled, the translation forces are given by:(8)

M_{i} = f_{} (θ_{i}) (A + B \ln (| \frac{{\dot{θ}}_{i}}{D} |) + g ({\dot{θ}}_{i})) + C_{i} {\dot{θ}}_{i}

Where,

$f (θ_{i})$: Function defining the change in force with spring displacement
$g (θ_{i})$: Function defining the change in force with spring displacement rate
$A$: Coefficient; Default = 1
$B$: Coefficient
$D$: Coefficient; Default = 1

Multidirectional Failure Criteria

Flag for rupture criteria: I_fail

I_fail=1

The rupture criteria flag is set to 1 in this case:(9)

F^{2} = D_{x}^{2} + D_{y}^{2} + D_{z}^{2} + D_{x x}^{2} + D_{y y}^{2} + D_{z z}^{2}

Where,

$D_{x} = D_{x p}$: The rupture displacement in positive x direction if $u_{x} > 0$
$D_{x} = D_{x n}$: The rupture displacement in negative x direction if $u_{x} > 0$

Graphs of this rupture criterion can be seen in Figure 6.