Springs & Connectors¶

Springs in OASIS model mechanical connections between two Boundary Condition Points (BCPs). They support linear and nonlinear constitutive models, damping, and stick-slip friction.

Overview¶

Each spring connects two BCPs and computes forces in 6 degrees of freedom (3 translational + 3 rotational) using local coordinate frames defined by orientation vectors.

graph LR
    B1[BCP 1] --- S((Spring 6-DOF)) --- B2[BCP 2]

Spring Properties¶

Parameter	Type	Description
`stressModelFlag`	int	1 = linear (matrix), 2 = nonlinear (stress-strain curves)
`dampingFlag`	int	0 = no damping, 1 = viscous damping
`frictionFlag`	int	0 = none, 1 = velocity-based, 2 = stick-slip
`frameFlag`	int	0 = average frame, 1 = BCP1 frame, 2 = BCP2 frame
`BCP_1`, `BCP_2`	int	Global BCP indices (1-based in input, converted to 0-based)
`BCP_1_type`, `BCP_2_type`	int	0 = fixed point, 1 = point on line, 2 = point on plane

Orientation Vectors¶

Six unit vectors define the local coordinate frames at each BCP — three at BCP1 (normal, tangent-y, tangent-z) and three at BCP2. These vectors must form orthonormal bases.

Forces are computed in the local frame defined by frameFlag:

0 — average of both BCP frames
1 — BCP1 frame
2 — BCP2 frame

Stiffness Models¶

Linear (`stressModelFlag = 1`)¶

A 6×6 stiffness matrix \(\mathbf{K}\) relates displacements to forces:

\[\mathbf{F} = \mathbf{K} \cdot \boldsymbol{\delta}\]

Nonlinear (`stressModelFlag = 2`)¶

Each DOF has an independent stress-strain curve defined as a piecewise-linear lookup table:

Displacements: vector of \(N\) displacement values
Forces: \(6 \times N\) matrix — one row per output DOF at each displacement point

This allows cross-coupling: displacement in DOF 1 can produce forces in DOFs 1–6.

Damping¶

When dampingFlag = 1, a 6×1 damping vector \(\mathbf{D}\) provides viscous damping per DOF:

\[\mathbf{F}_d = \mathbf{D} \cdot \dot{\boldsymbol{\delta}}\]

Friction¶

Velocity-based (`frictionFlag = 1`)¶

A smooth transition between static and dynamic friction using cubic polynomial interpolation.

Stick-slip (`frictionFlag = 2`)¶

Parameter	Symbol	Description
`mu_d`	\(\mu_d\)	Dynamic friction coefficient
`mu_s`	\(\mu_s\)	Static friction coefficient
`vt`	\(v_t\)	Transition velocity [m/s]
`Dt`	\(D_t\)	Maximum displacement for stick phase [m]

A 6×6 friction matrix \(\mathbf{M}\) scales the friction forces. The spring transitions between stick (elastic deformation below \(D_t\)) and slip (kinetic friction at \(\mu_d\)).

Input Format¶

ASCIIYAML

In dataSprings.dat:

1               // Number of springs
//////////////////
////////////////// New spring [1]
//////////////////
2               // stressModelFlag (nonlinear)
1               // dampingFlag
1               // frictionFlag
1               // frameFlag (BCP1 frame)
1               // BCP_1 (1-based)
2               // BCP_2 (1-based)
1               // BCP_1_type (on line)
0               // BCP_2_type (fixed)
1.0 0.0 0.0     // normal vector BCP1
0.0 1.0 0.0     // tangent-y BCP1
0.0 0.0 1.0     // tangent-z BCP1
1.0 0.0 0.0     // normal vector BCP2
0.0 1.0 0.0     // tangent-y BCP2
0.0 0.0 1.0     // tangent-z BCP2
// 6x6 stiffness matrix (36 values)
...
// friction coefficients
0.02            // mu_d
0.20            // mu_s
0.001           // vt
0.01            // Dt
// 6x6 friction matrix
...
// 6x1 damping vector
200000.0
200000.0
0.0
0.0
0.0
0.0
// stress-strain curves (6 DOFs)
5               // n points for DOF 1
-100 -1.0 0.0 1.0 100        // displacements
-1e12 -4.5e6 0.0 15e6 1e12   // forces DOF 1
0.0 0.0 0.0 0.0 0.0          // forces DOF 2
...                           // forces DOF 3-6
// repeat for DOF 2-6

In dataProblem.yaml:

springs:
  - stress_model_flag: 2
    damping_flag: 1
    friction_flag: 1
    frame_flag: 1
    BCP_1: 1
    BCP_2: 2
    BCP_1_type: 1
    BCP_2_type: 0
    vectors:
      - [1.0, 0.0, 0.0]
      - [0.0, 1.0, 0.0]
      - [0.0, 0.0, 1.0]
      - [1.0, 0.0, 0.0]
      - [0.0, 1.0, 0.0]
      - [0.0, 0.0, 1.0]
    stiffness_matrix:
      - [0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
      # ... 6 rows
    mu_d: 0.02
    mu_s: 0.20
    vt: 0.001
    Dt: 0.01
    mass_matrix:
      - [0.0, 1.0, 0.0, 0.0, 0.0, 0.0]
      # ... 6 rows
    damping_vector: [200000.0, 200000.0, 0.0, 0.0, 0.0, 0.0]
    stress_strain:
      - displacements: [-100, -1.0, 0.0, 1.0, 100]
        forces:
          - [-1e12, -4.5e6, 0.0, 15e6, 1e12]
          - [0.0, 0.0, 0.0, 0.0, 0.0]
          # ... 6 rows
      # repeat for DOF 2-6

Example	Feature
`springs/pile_connector`	Pile-to-anchor spring with friction
`springs/neoprene_connector`	Body-to-body neoprene+wire connection