*The assessment of floor vibration concerns how the mass of its supporting structure moves when subjected to an imposed load. This is determined by ascertaining the natural frequency of the supporting structure to the floor, which is stated in Hz (cycles per second).*

*The assessment of floor vibration concerns how the mass of its supporting structure moves when subjected to an imposed load. This is determined by ascertaining the natural frequency of the supporting structure to the floor, which is stated in Hz (cycles per second).*

Since the invention of lighter structures that have longer spanning elements within them, the built in dampening effect on buildings have historically become less pronounced. However, in spite of this, floor vibration is one area of the serviceability limit state that can be, sometimes overlooked during the design process. Where floor vibration is onerous and has not been considered during design, it often leads to expensive remedial works being carried out on structures after they have been built, as occupants will soon begin to complain of discomfort due to excessive movements and vibrations.

While floor vibration is a serviceability limit state i.e., it relates to the service condition of a structure, it can in some instances have a detrimental effect on the integrity of the structure. This usually occurs when there is a continuous long term vibration inducing load, such as a piece of machinery that is causing a dynamic response within the structure. This can then lead to a fatigue-based failure of the supporting structure. This scenario however is rare as vibrations from such machinery would have to match the harmonic of the supporting structure, which is unlikely. Also, it is usually the building’s occupants that generate vibrations and as their sense of comfort deteriorates, the cyclical movements within the structure drop to the point where fatigue induced failure cannot occur.

The assessment of floor vibration concerns how the mass of its supporting structure moves when subjected to an imposed load. This is determined by ascertaining the natural frequency of the supporting structure to the floor, which is stated in Hz (cycles per second). Once that is calculated it is then possible to assess how the structure will move when subjected to various types of imposed loading

** Floor Vibration Assessment **

First of all, there are no established requirements for floor vibration indicated in EN 1990:2002 (Eurocode 0). Instead, it suggests that each project be assessed individually and that any criteria be decided with the customer. This open mindset is the result of the various vibration assessment techniques that may be used. Because each has particular advantages and disadvantages, floor vibration standards are developed early on in a project’s existence. It is impossible to overstate how important these criteria are since they frequently have a big influence on how the structure is designed. There are several restrictions provided by the UK National Annex to Eurocode, which designers can address with their clients.

In addressing floor vibration in structures, the floor elements are designed with a natural frequency’s minimal limit in mind. This is evaluated in relation to the structure’s stiffness in relation to the total amount of self-weight and superimposed dead load elements, as well as 10% of the anticipated applied load. This proportion permits inactive installations like furniture as well as semi-permanent installations. It is advised that this restriction be set at a least of 4 Hz for office buildings, and that this minimum be increased to 8.4 Hz for stages and dance floors due to the intense activity those surfaces would see. Low frequency floors are those with a fundamental natural frequency of 3 to 10 Hz, and high frequency floors are those with a fundamental natural frequency more than that. It is advised that low frequency floors are designed for dynamic loads due to the increase in imposed loading from the relatively lively floor structure.

This foretasted limit only applies to natural frequencies, it assesses the possibility of the structure’s base movement or harmonic matching the frequency range of an adult’s footfall, which is between 1.5 to 2.5 Hz. However, the base harmonic for staircases is substantially higher. It fluctuates between 3 and 4 Hz because of the way people ascend stairs. As individuals move through them, they exert more force at a higher frequency, which can cause the harmonic to grow.

Where a structure’s lowest natural frequency is more than double the footfall frequency, it will not display substantial vibrations when subjected to applied stress from human occupation. However, there is still that chance that severe floor vibrations will occur despite achieving this natural frequency threshold. This is due to the fact that the amount of floor reaction may still be unsatisfactory. As a result, excitation-based response requirements must be satisfied in order for a floor structure to be considered adequate.

**Assessing the Natural Frequency** **of** **Elements **

Finite element analysis approaches are preferred for estimating the natural frequency of structures because they offer more precise findings. Nonetheless, the natural frequency of basic structural element such as a simply supported beam or a grillage of primary and secondary beams supporting a floor may can be estimated.

The base equation for assessing the natural frequency of structures is given as:

f_n=\frac{\chi_n}{2\pi}\sqrt{\frac{EI}{ml^4}}

Where: f_{n} is the dynamic natural frequency of the beam; EI is the flexural rigidity of the member; m is the unfactored effective mass of the structure and the elements it’s supporting; l is the span of the beam; and κ_{n} is a constant that represents the support conditions of the beam for the nth node of vibration.

If it is assumed that only a uniformly distributed load is applied on a beam, that it is simply supported, and that the value of n is taken to be 2. Following these assumptions, the resultant equation is identical to that used to calculate the mid span deflection of a simply supported beam. Thus, if the maximum deflection at midspan of simply supported beam is given as:

\delta=\frac{5mgl^4}{384EI}

Where: δ is the maximum deflection due to the applied load and self-weight in mm; g is acceleration due the gravity, which equals 9.81 m/s2

Therefore, the natural frequency of the beam can be derived using the base equation and gotten as:

f=\frac{48}{\sqrt\delta}

**Assessing the Natural Frequency** **of** **Structures **

Floor vibration assessment using hand calculation can only be carried out for very simple elements, such as beams and floor joist. When there is need to carry out a holistic vibration analysis of a complete structure as would often be the case, computer analysis technologies are utilized to identify the natural frequencies. The process for developing FE models for natural frequency analysis differs slightly from that for stress and serviceability and may be summarized as follows:

- All aspects of the structure are subdivided into several elements. The more elements there are, the more accurate the analysis
^{1}. - All connections between elements are modelled as continuous/fixed
^{1}. - Vertical elements should be split at the point of contraflexure with a pinned connection; typically, this is at the mid-span. This comprises 10% of the imposed loads depending on fixtures
^{1}. - It is recommended that at least ten mode shapes be analyzed in order to obtain reliable findings
^{1}. - The material qualities of any concrete used in the building should allow for long-term impacts
^{1}.

#### Response Factor (R)

As previously stated, EN 1990:2002 (Eurocode 0) provides no guidance on the allowable limits for floor vibration. That obligation is shared by both the client and the structural engineer.

In certain cases, clients provide design parameters for floor vibrations in terms of a response factor (R). The response factor is derived from the acceleration of a structure as it moves during vibrations. It is generally advised that the response factor not exceed the values in *Table 1* in order for the floor structure to be acceptable.

Room Time | Time of Day | Limiting Response Factor |

Workshop | Day | 8 |

Night | 4 | |

Office | Day | 4 |

Night | 4 | |

Residential | Day | 2-4 |

Night | 1.4 | |

Hospital & | Day | 1 |

Night | 1 |

**Table 1: Limiting Values of Response Factor vs Time of Day vs Room Type**

### Remedial Measures of Addressing floor vibration

An evaluation of the structure is necessary when a building’s usage changes to the point where a rhythmic activity is almost certain. In order to ascertain what changes must be made for the structure to work as intended, the structure is usually tested and modelled.

In many instances, changes would be made. These changes typically concentrate on strengthening the structure without significantly increasing its bulk. They may involve shortening the major beams’ span by adding more supports to the building, or by stiffening beams by changing their sectional profile, and strengthening the support connections.

### Worked Example

A simply supported beam is required to span 8m in a proposed office development, if the service load on this beam is 15kN/m, and the beam is to have a natural frequency, greater than 5Hz to satisfy vibration requirement, verify the adequacy of a 457 x 191 x 74 UKB section.

N:B = I_{yy} = 33300cm^{3}

#### Deflection

\delta=\frac{5wl^4}{384EI}

=\frac{5\times15\times8000^4}{384\times205\times10^3\times33300\times10^4}=11.72mm

#### Natural Frequency

f=\frac{18}{\sqrt\delta}=\frac{48}{\sqrt(11.72)}= 5.3Hz > 5Hz \quad o.k

**457 x 191 x 74 UKB section is satisfacto****ry. **

Also see: **A Background to Extreme Loading Conditions**

**Sources & Citation**

- Institution of Structural Engineers Technical Guidance Note (Level 1) – Floor Vibration
- SCI (2009) P354 Design of Floors for Vibration: New Approach Ascot, Berkshire: Steel Construction Institute