MasterSeries Blog

Footfall analysis is essential in structural engineering because it assesses how a building or structure responds to vibrations caused by people walking or moving across floors, walkways, or other structural components. This is particularly critical in buildings with large open spaces, such as offices, shopping centres, stadiums, and footbridges, where the impact of foot traffic can lead to noticeable vibrations.

Determining the structure's mode shapes and frequencies is one key component in analyzing and designing floor structures for footfall vibration. While the published guidances for footfall induced vibration, published by the Concrete Centre and the Steel Construction Institute (SCI), make numerous references to mode shapes, neither gives a particularly clear explanation of what mode shapes are or their significance.

What is a Mode Shape?

Mode shapes are a fundamental property of the structure. They represent special types of vibration or time-dependent deflections. Each mode shape is a form of vibration when the whole of the structure is displacing at a single specific frequency and where the phase relationship between any two parts of the structure remains constant. The mode shapes for any structure will depend on the mass, support and/or boundary conditions and stiffness of the structural elements. While the mode shapes are dependent on the mass of the structure, they are independent of the time-dependent force that is applied to the structure. In other words, the mode shapes are determined independent of any force being applied to the structure.

Since mode shapes are independent of load, they are reported in Masterframe with normalized deflections and rotations at each node, but there do not represent the deflection of the structure under the applied dynamic load. The results describe the shape of the particular mode at a point in time, with the point of maximum amplitude given a value of 1.

Properties of Mode Shapes

Each mode shape and its corresponding frequency are unique for a given structure. In other words, no two modes can share the exact same shape and frequency. In applied mathematics, this property is termed orthogonality. This is a generalization of the concept whereby, in a coordinate system, it is possible to move in the x-axis, say, without changing coordinates in the z and y-axes. This means that any one mode shape cannot be formed by combining the other mode shapes.

The orthogonality of the mode shapes means they form a basis for any periodic vibration of the structure. This means that any general vibration of the structure can be constructed from a linear combination of the mode shapes. To find the displacement of the structure at any given point in time, a specific combination of the mode shapes is used. This is the principle at the heart of the footfall vibration analysis method.

What Range of Mode Shapes are Required?

When finding the displacement of a structure under a time-varying dynamic load, while the behaviour of the structure can be found by creating a linear combination of the mode shapes, not all mode shapes contribute equally. For example, for vertical vibrations, purely horizontal mode shapes will not contribute significantly. Also, mode shapes with natural frequencies significantly far from that of the forcing function will be less significant.

Since the final solution depends on the sum of mode shapes, it is important to include enough mode shapes such that sufficient accuracy is obtained in the footfall vibration results, yet avoiding summing too many modes shapes where the mode shapes are not significant, thereby increasing the analysis time for no added accuracy in the final result.

The range of mode shapes to be considered is given in the design guidance produced by the SCI or the Concrete Centre. The range of mode shape frequencies to be included depends on whether a steady state or transient analysis is to be carried out.

SCI P354 - Design of Floors for Vibration: A New Approach, recommends;

• For the steady state response, that mode shapes up to 2Hz higher than the cut off frequencies (low to high frequency) are used
• For floors, this ranges up to around 12 Hz
• For stairs, this would be mode shapes up to around 14 Hz
• For a transient analysis, SCI P354 recommends that all modes with a frequency up to twice the fundamental (first mode) frequency are used.

The Concrete Centre publication (CCIP-016: A Design Guide for Footfall Induced Vibration of Structures) states that mode shapes up to 15 Hz can be significant for a resonant type response. This applies to the steady state analysis. For the impulse (transient) analysis, CCIP-016 states that mode shapes up to twice the fundamental (first) mode shapes are used.

Higher-frequency mode shapes are less important in the steady-state analysis since they are far from the frequencies of the harmonics of the walking force, contributing little to the results. However, this is not the case for the transient analysis, and higher-frequency mode shapes can return large accelerations. Therefore, for transient analysis, care should be taken not to include mode shapes with too high a frequency.

Which Mode Shapes are the Most Significant?

In the linear combination of mode shapes, some mode shapes have a larger weighting than others since some mode shapes contribute more to the final result than others. Often, therefore, the vibration response of a structure will be driven by a relatively small number of mode shapes.

Within the Vibration Design tab, the relative weighting column indicates the percentage contribution of each mode shape to the final result. Selecting the R weight column will sort the results to help list the mode shapes in order of relative weighting.

In the following example, just 3 mode shapes contribute to almost 70% of the vibration design result.

In some structures, the vibration design is spread over several modes, while in others, the response may be dominated by a single mode shape.

Using Relative Weighting to Target Changes to a Structure

Since the relative weighting identifies the mode shapes which are most significant when determining the results of a vibration design when a structure does not meet the required response factor, modifying the structure to modify the most significant mode shapes will be the most effective approach.

Once the most critical mode shapes have been identified, viewing the animation deflected shape for each mode shape can help to identify which structure elements contribute to the vibration result. This can indicate if the mode shape being considered is being driven by specific elements such as primary or secondary beams, or does the mode shape affects the structure more generally. In the first case, altering specific beams may assist in meeting the required response factor, whereas in the second case, modifying the slab depth may be required to affect the vibration design result.

Conclusion

Mode shapes are a vital component of determining the vibration design when using a linear elastic method such as that given in the vibration design guidance from either the Steel Construction Institute (SCI) or the Concrete Centre. However, not all mode shapes contribute equally, and being able to identify the key mode shapes driving the result can give additional information as to which parts of the structure are key to driving the final result.

Insights from the IStructE Northern Ireland Hub, QUB Liasion Officer

Dr. Tony Martin of Queens University Belfast and the IStructE Northern Ireland Hub, provided some valuable industry insight and practical perspectives with some advice aimed at junior engineers or those unfamiliar with vibration design, explaining why it’s a critical area to master and offering tips for incorporating vibration analysis into design work:

"For junior engineers, gaining an understanding of vibration design early in your career is essential, especially when working on projects with long spans or sensitive installations. Mastering tools like mode shape analysis will help you ensure that your designs are not only structurally sound but also comfortable for occupants and compliant with vibration performance criteria. Investing time in learning these techniques will significantly enhance your ability to tackle complex design challenges."

MasterFrame: Dynamic Analysis add-on

The MasterFrame: Dynamic Analysis add-on analyses and designs structural models for the effects of dynamic behaviour. The MasterFrame: Dynamic Analysis module uses the MasterFrame model to assess the structure's natural frequencies and then evaluates the structure in accordance with either the Steel Construction Institute or Concrete Society design guidance.

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