Springs

The relative displacement across a SPRING1 element is the ith component of displacement of the spring's node:

where i is defined as described below and can be in a local direction (see Defining the direction of action for SPRING1 and SPRING2 elements).

The relative displacement across a SPRING2 element is the difference between the ith component of displacement of the spring's first node and the jth component of displacement of the spring's second node:

where i and j are defined as described below and can be in local directions (see Defining the direction of action for SPRING1 and SPRING2 elements).

It is important to understand how the SPRING2 element will behave according to the above relative displacement equation since the element can produce counterintuitive results. For example, a SPRING2 element set up in the following way will be a “compressive” spring:

If the nodes displace so that $u_i^1=1$ and $u_j^2=0$, the spring appears to be in compression, while the force in the SPRING2 element is positive. To obtain a “tensile” spring, the SPRING2 element should be set up in the following way:

For geometrically linear analysis the relative displacement is measured along the direction of the SPRINGA element in the reference configuration:

where ${\mathbf{X}}^1$ is the reference position of the first node of the spring and ${\mathbf{X}}^2$ is the reference position of its second node.

For geometrically nonlinear analysis the relative displacement across a SPRINGA element is the change in length in the spring between the initial and the current configuration:

where $l=\sqrt{\left({\mathbf{x}}^1-{\mathbf{x}}^2\right)\cdot \left({\mathbf{x}}^1-{\mathbf{x}}^2\right)}$ is the current length of the spring and $l_0$ is the value of l in the initial configuration. Here ${\mathbf{x}}^1$ and ${\mathbf{x}}^2$ are the current positions of the nodes of the spring.

In either case the force in a SPRINGA element is positive in tension.

You define linear spring behavior by specifying a constant spring stiffness (force per relative displacement).

The spring stiffness can depend on temperature and field variables. See Input Syntax Rules for further information about defining data as functions of temperature and independent field variables.

For direct-solution steady-state dynamic analysis the spring stiffness can depend on frequency, as well as on temperature and field variables. If a frequency-dependent spring stiffness is specified for any other analysis procedure in Abaqus/Standard, the data for the lowest frequency given will be used.

SPRING, DEPENDENCIES=n
first data line
spring stiffness, frequency, temperature, field variable 1, etc.
...

Property or Interaction module: SpecialSprings/DashpotsCreate: select connectivity type: select points: Property: Spring stiffness:
spring stiffness

You define nonlinear spring behavior by giving pairs of force–relative displacement values. These values should be given in ascending order of relative displacement and should be provided over a sufficiently wide range of relative displacement values so that the behavior is defined correctly. Abaqus assumes that the force remains constant (which results in zero stiffness) outside the range given (see Figure 1).

Figure 1. Nonlinear spring force–relative displacement relationship.

Initial forces in nonlinear springs should be defined as part of the $F\left(u\right)$ relationship by giving a nonzero force, $F\left(0\right)$, at zero relative displacement.

The spring stiffness can depend on temperature and field variables. See Input Syntax Rules for further information about defining data as functions of temperature and independent field variables.

Abaqus/Explicit will regularize the data into tables that are defined in terms of even intervals of the independent variables. In some cases where the force is defined at uneven intervals of the independent variable (relative displacement) and the range of the independent variable is large compared to the smallest interval, Abaqus/Explicit may fail to obtain an accurate regularization of your data in a reasonable number of intervals. In this case the program will stop after all data are processed with an error message that you must redefine the material data. See Material data definition for a more detailed discussion of data regularization.

SPRING, NONLINEAR, DEPENDENCIES=n
first data line
force, relative displacement, temperature, field variable 1, etc.
...