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# 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.

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