3.3.3: Spring Cross Sections

Springs allow you to directly define the stiffness relation between two nodes via spring constants. Each node has six degrees of freedom (DOFs): three translations and three rotations. Using the “Cross Sections” multi-component with “Cross Section” set to “Spring” lets one couple these DOFs by means of six spring-constants. A relative movement $u_{i,rel}$ between two nodes thus leads to a spring force $F_i = c_i \cdot u_{i,rel}$. In this equation $u_{i,rel}$ stands for a relative translation or rotation in any of the three possible directions x, y, z, $c_i$ is the spring stiffness. In Karamba3D the latter has the meaning of kilo Newton per meter $kN/m$ in case of translations and kilo Newton meter per radiant $kNm/rad$ in case of rotations. The input-plugs “Ct” and “Cr” expect to receive vectors with translational and rotational stiffness constants respectively. Their orientation corresponds to the local beam coordinate system to which they apply. In case of zero-length springs this defaults to the global coordinate system but can be changed with the “OrientateBeam”-component.

In case one wants to realize a rigid connection between two nodes the question arises as to which spring stiffness should be selected. A value too high makes the global stiffness matrix badly conditioned and can lead to a numerically singular stiffness matrix. A value too low results in unwanted relative displacements. So you have to find out by trial and error which value gives acceptable results.

Figure 3.3.3.1: Spring fixed at one end and loaded by a point load on the other

Fig. 3.3.3.1 shows a peculiarity one has to take into account when using springs: They are unaware of the relative position of their endpoints. This is why the load on the right end of the spring does not evoke a moment at the left, fixed end of the spring.