# 3.2.2: Beam Loads

The group of loads described in this section works on beams and - with some limitations - on truss elements. The latter are considered as shear rigid with hinges at their ends: Beam-loads on trusses transform into transverse forces at their end-points. All beam load-components feature the "BeamID"- and "LCase"-input-plug:

"BeamId" determines the element on which to apply the load. A regular expression like '&"id1"|"id2"|...' selects multiple elements.

"LCase" specifies the name of the load's load-case.

The parameter "t" serves to specify the load-position along elements: t=0 refers to the starting point, t=1 to the end-point.

Karamba3D offers these types of beam-loads:

## Concentrated Load

Use the 'Concentrated'-option to specify forces and moments at arbitrary positions along the element. Vectors 'Force' and 'Moments' let one set the direction and size of the corresponding external loads. Fig 3.2.2.1 shows a cantilever with a span of 3m under a concentrated transverse load of 1kN. The load sits at a distance of 0.75 x 3.00 = 2.25m from the supports and causes a linear bending moment diagram.

The radio buttons in the sub-menu 'Orientation' determine the reference coordinate system of the 'Force'- and 'Moment'-vectors: either local to the element or global.

## Block Load

Fig. 3.2.2.2 shows the application of uniformly distributed transverse loads an on a cantilever beam. The span of the cantilever is two meters. The loads act between t0=0.25 and t1=0.75. With the default values t0=0 and t1=1 distributed block-loads cover the whole beam length. The graphical output in fig. 3.2.2.2 shows the support reactions well as the bending-moment-diagram (in orange) for My.

In combination with the radio buttons in sub-menu "Orientation" the vectors at the input-plugs "Force" and "Moment" sets the external loads to be applied - see fig. 3.1.2.8 for the meaning of the different types of orientation.

There are limitations for distributed rotational loads: In case they rotate about the element's local Y- or Z-direction they need to be specified over the whole length. This does not apply to distributed torsional moments when the Orientation is set to 'Local to element'.

The input-plug **“BeamId”** receives the identifier of the beam on which the load shall act. Multiple beams can be specified via a regular expression (e.g. '&"id1"|"id2"...'). See section 3.1.6 for how to attach identifiers to beams. By default beams are named after their index in the FE-model. There are three options for the orientation of the load: **“local to element”**, **“global”** and **“global proj.”**. Their meaning corresponds to the options available for mesh-loads (see fig. 3.2.1.8).

The input-plug **“LCase”** which designates the load case defaults to “0”.

## Gap Load

With Gap-loads it is possible to prescribe rotation or displacement discuntinuities at arbitrary element-positions. When using unit vectors are used, the resulting displacement shapes represent influence lines (see e.g. https://en.wikipedia.org/wiki/Influence_line).

Fig 3.2.2.3 shows a fully fixed beam under a translational gap load of 0.1m in local z-direction. The displaced shape shows, that in order to maximize the shear force Vz at the location of the gap-load one should place transverse external loads either to the left or right side of the gap only.

## Imperfection

There exists no such thing as an ideally straight column positioned perfectly vertical. The deviation of a real column from its ideal counterpart is called imperfection. This term comprises geometric and material imperfections.

The **“Imperfection”** variant of the **“Loads”** multi-component allows to specify geometric imperfections (see fig. 3.2.2.4). **“psi0”** takes the vector of the initial inclination of the beam axis about the axes of the local element coordinate system in radians. With **“kappa0”** one can specify the initial curvature. A positive component of curvature means that the rotation of the middle axis about the corresponding local coordinate axis increases when moving in longitudinal beam direction. Small inclinations and curvatures are assumed.

In fig. 3.2.2.4 one can see displacements and reaction forces of an initially straight beam with a second order theory normal force of $N^{II} = 10 kN$, an initial inclination of $0.1 rad$ about the local y-axis and an initial curvature of $0.1 rad/m$.

Imperfection loads do not add directly to the beam displacements. They act indirectly and only in the presence of a normal force $N^{II}$. An initial inclination $\psi_0$ causes transverse loads $\psi_0 \cdot N^{II}$ at the elements endpoints. An initial curvature $\kappa_0$ results in a uniformly distributed line load of magnitude $\kappa_0 \cdot N^{II}$ and transverse forces at the elements endpoints that make the overall resultant force zero. For details see e.g. [10].

## Polylinear Load

Distributed loads consisting of an arbitrary number of linear segments can be defined via the "Polylinear" option (see fig. 3.2.2.5). The input-plug "Dir" specifies the direction of the load. The vector supplied there gets scaled to unity and has no impact on the load magnitudes. The latter get set by lists of values connected to the "Force" or "Moment" inputs. For each value supplied there needs to be a corresponding location parameter "ts". In case there are more location parameters than force or moment-values the longest list principle applies like in fig. 3.2.2.5.

In case of subsequent identical t-values and different corresponding load-values a step in the distributed load results.

## Trapezoidal Load

The 'Trapezoidal'-option makes it more convenient to specify trapezoidal loads as compared to 'Polylinear'. Teh parameter input is simlar to that of block-loads. The four parameters t0, t1, t2 and t3 specify the trapezoids shape.

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