^{Max IV may be affected by vibrations from many sources, e.g. traffic on E22, wind load at the façade, people walking at the floors in the facility, and other internal loads as fans and pumps (illustration: NGI)Illustration above: Architects Snøhetta (Oslo) and Fojab (Malmö).}

Strict stability tolerances were therefore enforced, and a dynamic group was appointed to handle the structural and soil dynamic Challenges.

The dynamic group performed measurements of current vibration levels at the site, prediction of future vibration levels at the facility through calculations, and described cost effective counter measures to reduce the vibration levels. NGI was appointed as partner to the dynamic group by the contractor, Peab, in 2011.

The unique performance of MAX IV gives the opportunity to focus the electron beam on a very small target area, resulting in an extremely good resolution. However, it also makes the facility sensitive to vibrations that can cause disturbance of the electron beam.

As a target value and starting position for the design, the vibration limit value was set to RMS = 20-30 nm in the frequency range from 5 Hz to 100 Hz. The vibration target value applies for the beam lines and storage rings. 20-30 nm is roughly the size of a virus and about 4000 times smaller than the width of a single strand of human hair.

A number of vibration sources may affect the facility. Possible sources were identified by the dynamic group, and the ones estimated to have greatest impact on the facility and to be most expensive to handle in future were chosen to be subject for analysis:

- Traffic on E22, which passes the facility at about 100m distance.
- Wind load at the facade.
- People walking at the floors in the facility.
- Other internal loads as fans and pumps (handled as unity loads)

### NGI's approach

For safe design of dynamically loaded systems, properly accounting for their coupling to the ground is crucial. Modeling dynamic interaction between a structure and its soil foundation is in general a demanding task. For large and complicated structures, and particularly when the structures are also embedded into the soil, complexity, size of problem and computational efforts may be excessive.

The approach NGI uses makes it possible to model very large structures with surrounding soil in 3D with manageable model size and computer time, and without influence of reflections from the computational boundaries.

NGI's approach implies representing the soil surrounding the facility through stiffness matrices describing the dynamic properties of the soil. To determine the stiffness matrices the ground is modelled as a horizontally layered unbounded medium, and rigorous solutions of the wave equations are obtaining for each layer. The dynamic parameters that constitute input data to the calculations are determined from geotechnical investigations and geophysical measurements at the site.

The resulting frequency dependent stiffness matrices are coupled to the nodes of the FE-model of the structure at the interface between the soil and the structure. Since the MAX IV structure is partly buried in the ground, the approach was further developed in the project to also accommodate buried structures.

### FE-model

Part of the soil just beneath the structure was included in this structure model in order to be able to evaluate the effect of soil stabilization.

NGI modelled the surrounding soil as complex dynamic stiffness matrices. Since the matrices are frequency dependent, separate matrices were computed for each excitation frequency. The matrices were coupled to the FE-model of the structure and the total system was solved.

The analysis was performed in the frequency domain by applying harmonic unity loads (1 N) in X-, Y- Z-directions in selected positions. The analysis was carried out in the frequency range from 1 Hz to 30 Hz with 1 Hz resolution.

The computation time with the method described was about four hours per frequency. If the surrounding soil had to be modelled by the conventional FE (including absorbing boundary domains), the model would be enormous and the computation time would be unaffordable.

### Calculated vibration values

The load case of traffic on E22 was applied by scaling calculation results for a model without the structure with free field measurement data collected before construction work started at the site.

The results show that traffic on E22 lead to about the same vibration values in the construction as on the ground without construction (free field values).

Further, the results show good effect of the soil stabilization below the construction. For a case with four meters of soil stabilization the vibration target value for the design will be exceeded in some parts of the construction.

However, the frequency range of the design target value has been under discussion, and the calculations show that the design target value may be met if the lower frequency of the design target is changed to 10 Hz.