Example: how are relative deformations calculated?

This worked out example explains both use cases for calculating the relative deformations namely the simply supported span (both ends fixed) use case as well as the cantilever span (one end fixed) use case.

Structure configuration

An IPE270 beam with a total length of 6m is supported on its end nodes with hinged supports which are flexible in the Z-direction. The beam is loaded by means of two line loads: qz = -20,0kN/m and qy = 2kN/m.

The beam is supported on both its ends via flexible nodal supports as described in the picture below:

The deformations will be evaluated in the middle of the span at position dx = 3000mm.

Buckling group configuration

The buckling system is configured as follows.

• For deflection z: ‘simply supported span' (both ends fixed)
  

• For deflection y: 'cantilever span' (one end fixed)
  

Calculation of relative deformation u_z

Absolute deformations u_z can be seen in the picture below. Because the buckling system for deflection z has two fixed buckling supports. We can thus construct the reference line as the line between these two nodes.

The reference line can be calculated if the absolute deformations of the fixed (buckling)supports are known. In node N1 -> uz = -12,0mm and in node N2 -> uz = -6,0mm.

The function which describes this reference line can then be obtained:

Let’s calculate the value of the reference line in the middle (pos.x = 3,00m) by using the above mentioned formula:

The relative deformation is then the subtraction between the absolute deformation at position x and the the u_z-value of the reference line.

This value corresponds with the calculated value in SCIA Engineerwhen executing the Steel SLS-check for the value u_z,max:

Calculation of relative deformation u_y

Absolute deformations u_y can be seen in the picture below. Because the buckling system for deflection y has only one fixed buckling support. The reference line should be constructed starting from the fixed support and parallel with the member system line.

The reference line can be calculated if the absolute deformations of the fixed (buckling)support is known. In node N1 -> u_y = 2,4mm. An additional step needs to be done for cantilever configurations namely: apply the absolute deformation of the fixed node to all the other free nodes (i.e. nodes without buckling supports) of that buckling system configuration deflection y in this case. This means the following for this example:

Now we can re-use this formula for the reference line u_y:

Let’s calculate the value of the reference line u_y in the middle (pos. x = 3,00m) by using the above mentioned formula:

The relative deformation is then the subtraction between the absolute deformation at position x and the u_y-value of the reference line.

This value corresponds with the calculated value in SCIA Engineerwhen executing the Steel SLS-check for the value u_y,max: