Vibration analysis falls under the serviceability limit state design criteria. It is often an overlooked criteria which engineers neglect to check. This is especially the case for concrete slabs. However, ignore vibration for a concrete slab at your own peril! Even concrete slabs can perform poorly for vibration. Luckily Bentley’s RAM Concept has a built in vibration analysis module to help. So lets take a deeper look at vibration analysis of floor slabs using RAM Concept.

The general assumption is usually **“concrete slabs are fine for vibration”**. There is some element of truth to this assumption. It’s large mass does help it perform well under vibration however at longer spans (particularly cantilever spans) concrete can start to fail for general vibration criteria.

Lets first take a look at what vibration is all about, then explore vibration analysis of slabs using RAM Concept and its built-in vibration module. If you know all about vibration already you can cut straight to the chase by clicking HERE to start seeing how to analyse vibration using RAM Concept…

## What is Vibration

In its simplest form, vibration is a periodic back and form (cyclical) motion of a particle or solid body. Vibration commonly results when an elastic system is displaced then released; allowing it to return to its state of equilibrium. The elastic nature of the system provides a restorative force which allows its return to the state of equilibrium after multiple oscillations.

Vibration falls into one of two catagories:

**Free Vibration:**A system which is momentarily displaced (or excited) then allowed to freely move.**Forced Vibration:**A system which is continuously excited through outside rhythmic forces.

Lets take a look at the elements which effect the vibration of a system with reference specifically to floor slabs…

### Damping

In Vibration terms, damping is the restraint of vibration movement.

Damping can be best visualised in respect to your cars shock absorber. The shock absorber contains a cylinder, piston and a spring.

The cylinder is filled with viscous oil (thicker than water). As the piston travels up and down within the oil filled cylinder the oil passes through holes in the piston head. Friction is generated as the oil passes through these small holes. The friction generates heat which helps to dissipate the vibrating energy caused when traveling over a bump in the road. This allows the vibration to eventually stop…

The thicker the oil is, the higher the damping and therefore the quicker the vibration stops. The thinner the oil, the less the damping which means the vibration continues for longer.

In Vibration Analysis, a damping ratio is a measure of how quickly the amplitude decays in an oscillating (vibrating) system. there are three levels of damping ratio:

**Under Damped:**Systems with a damping ratio less than one are said to be under damped because they experience one or more oscillation cycles before returning to equilibrium.**Over Damped:**Systems with a damping ratio greater than 1 are said to be over-damped because the systems oscillation stops before one cycle is completed.**Critically Damped:**Systems with a damping ratio of exactly 1 are said to be critically damped because they return to equilibrium as fast as possible.

Here is what different levels of damping ratio look like graphically…

In a floor slab, supported elements usually provide damping to the system. These are generally non-structural items which deform and restrain the slab, restricting its motion during vibration. These elements include:

- Non-loadbearing Partitions
- Furniture
- Façade elements
- Ceiling and floor finishes.

The table below gives some guidance on different damping ratios for reinforced concrete floor slabs:

Floor Type | Damping Ratio |

Open Floor Plan (office with little to no partitions or walls) | 0.02 |

Partially Closed Floor Plan (Office spaces with partial height partitions) | 0.03 |

Closed Floor Plan (apartment or hotel floor with multiple full height partitions) | 0.05 |

### Vibrating Forces on Floor Slabs

Most vibration in floor slabs is generated through foot fall induced motion caused by occupants and users of the floor. Vibration may also be generated by slab mounted plant equipment and other specialised elements. The forces and frequencies imparted by these elements are usually covered by the manufacturers specifications.

There are a number of design guides which cover the effects of human footfall vibrations in slabs.

The vibration generated through foot fall traffic is dependant upon:

- The walkers weight (the higher the weight, the higher the force of excitation on the slab)
- The frequency of the footsteps (how many steps per second)

### Frequency and Resonance

In vibration terms, the frequency is the amount of cycles (or oscillations) the system experiences per second.

When a slab vibrates, it does so at different frequencies. Each frequency is called a “mode” (mode of vibration or frequency mode). As vibration occurs these different modes are superimposed on one another to give the overall response of the system.

Generally the first 3 or 4 modes of vibration are considered for slabs, since the higher modes are quickly decayed by damping.

The different responses of a system may be represented by “mode shapes”, which show the deflections of the deformed shapes of the system during vibration. The fundamental (or first mode) frequency always corresponds to the mode shape with the lowest frequency.

Mode shapes of a slab can be compared to that of a vibrating guitar string. The first three mode shapes of a simply supported slab are shown in the figure below…

Frequency is very important when discussing resonance…

**Resonance describes the scenario in which an excitation force is applied to a system with a frequency which matches the systems fundamental frequency. This phenomenon can result in the system displacing with significant amplitude compared to when the same force is applied at ****different non-resonant frequencies.**

Lets take a look at a couple of practical examples of this. If you have ever pushed a child on a swing, this is resonance at play. To push the child higher and higher, the frequency of the excitation force you apply to the child (you pushing on their back) is perfectly timed to the swing systems fundamental frequency (which includes the child, the seat, chains and swing frame).

The rhythmic pushing, in step with the swings natural frequency allows the displacement to be amplified significantly (resulting in a fun time for you and the child!). If you were to push the child with a frequency out of step with the swings natural frequency the child’s displacement would not be very high at all by comparison (resulting in an upset child!).

Another fun example is the opera singers voice frequency matching that of a wine glass. If the frequency is just right, this can result in the wine glass vibrating with enough amplitude to shatter itself apart…

Taking this phenomenon and applying it to concrete floor slabs we can now asses the frequency of an excitation force (a person walking for example), against the resonant frequency or fundamental frequency of the slab.

The table below illustrates a range of footfall frequencies for individuals walking. The information is taken from a data set average:

Floor Activity | Frequency |

Walk/Stroll | 1.5 – 2.5hz |

Brisk Walk | 2.0 – 3.0hz |

Jog/Run | 2.5 – 3.5hz |

Most general use floor slabs are designed to cater for walking/strolling traffic. These may include offices, apartment buildings and hotels. For this reason it is best practice to ensure that these floor slabs have a natural frequency of greater than **3.0hz**.

For rhythmic activity such as gyms and aerobics halls, it becomes a little trickier. The range of frequencies for the excitation force can be quite large (depending on the activity). In addition to this, the excitation force is usually much higher as multiple individuals are providing the excitation force at the same time compared to one or two walkers at a given time on a floor slab.

## Analysis of Vibration in Floor Slabs using RAM Concept

RAM Concept is a powerful 3D modelling software for the design of conventionally reinforced and post-tensioned concrete slabs. (for an in-depth article on what a post-tensioned slab is refer to THIS article)

For this example, we will assume that you already know how to model and design your floor slab for strength and deflection.

The steps for analysis of vibration in floor slabs using RAM concept are as follows:

- Set up your model including support structure, slabs, beams openings, set-downs etc.
- Specify your calculation criteria for vibration within RAM Concept
- Specify the modal mass of the slab system
- Specify the excitation area on the floor slab
- Run the vibration analysis
- Check the natural frequency of your floor slab
- Check the response factors against a pre-determined maximum limit.

We will be focusing on steps **2** through to **7** as these are specifically associated with analysing vibration in RAM Concept.

## Setting Criteria for Analysis of Vibration in Slabs within RAM Concept

The most critical component in setting up your Vibration analysis of floor slabs in RAM concept is the design criteria.

The design criteria is located by navigating to the following location…

**Criteria >> Calc. Options>> Vibrations tab…**

you now should see the following dialogue box…

Lets take a look at the individual criteria and discuss the best approach for each..

### Number of modes to Consider for Vibration Analysis in RAM Concept

**Explanation**

This option within RAM Concept outlines the quantum of modes to consider during the vibration analysis. Higher modes are usually less effective in representing the slabs overall vibration response. The most important modes being 1 through to 4 generally, with each subsequent mode decaying in importance (with modes going up to infinity).

**Ranges of Values**

The range to use is generally dependant on computing power. With the power of today’s computers a mode range of 50-100 will still be readily solved pretty quick.

**Affects on Analysis for Vibration**

A lower number of modes will generally result in a less accurate result, while higher number of modes will gain more accuracy. The return on accuracy for higher numbers of modes however is an exponentially decaying relationship.

### Dynamic Concrete Modulus Factor

**Explanation**

The elastic modulus of concrete (or Young’s Modulus, “E”) is the elastic stiffness property of the material. In mathematic terms, modulus is a resultant strain (change in over its original length) under a given stress (force/area)…

At very small displacements (such as when a concrete slab vibrates) the elastic modulus is found to be enhanced by a certain percentage. this percentage is dependant on the concrete mix and the amplitude of the displacement applied to the slab.

**Ranges of Values**

Tests have shown that under vibration conditions the elastic modulus of concrete can be enhanced anywhere from 10 to 20%. This would mean that a factor of 1.1 to 1.2 can be adopted for this criteria in the vibration analysis within RAM Concept. You should consult with your concrete supplier to confirm this exact value for the mix design on your project.

**Affects on Analysis for Vibration**

A stiffer slab and therefore higher elastic modulus will give better vibration performance whereas conversely a less stiff slab and therefore a lower elastic modulus will give poorer vibration performance.

### Setting Footfall Frequency for Vibration Analysis of Slabs in RAM Concept

**Explanation**

As discussed earlier in this article, a floor slabs vibration responses is dependant on its resonant frequency and the applied force frequency.

This section within RAM concept allows the user to specify the minimum and maximum footfall frequency to be considered in the analysis (how slow or fast the hypothetical walker steps while moving)

**Ranges of Values**

A range of footfall frequencies were explored earlier in t his article. For those that skipped straight to this section, here is a summary table re-produced for your convenience…

Floor Activity | Frequency |

Walk/Stroll | 1.5 – 2.5hz |

Brisk Walk | 2.0 – 3.0hz |

Jog/Run | 2.5 – 3.5hz |

For general floor slabs including offices, hotels and apartment buildings, a range of **1.5-2.5**Hz is usually a good range to adopt. This would correspond to the minimum and maximum footstep frequencies within RAM Concept.

If your floors natural frequency is close to the applied footstep frequency, a wider range should be used in order to check sensitivity against poor performance under vibration.

**Affects on Analysis for Vibration**

The affects of footstep frequency on your vibration analysis will be dependant on your particular floor slabs natural frequency…

- The closer the forcing frequency is to the slabs natural frequency, the higher the risk of a resonant response resulting.

For this reason, it is good practice to ensure that your floor slab is proportioned to have a fundamental frequency higher than **3.0Hz** (we will explore how to check this later on in this article)

### Setting Damping Ratio for Vibration Analysis in RAM Concept

**Explanation**

As outlined earlier in this article, the damping ratio of a floor system is its ability to restrain (or reduce) motion due to vibration over time.

While some inherent damping exists within the structure itself, the majority of damping is provided by non-structural elements attached to the floor such as partitions, facades and finishes etc.

**Ranges of Values**

A range of damping ratios were explored earlier in his article. For those that skipped straight to this section, here is a summary table re-produced for your convenience…

Floor Type | Damping Ratio |

Open Floor Plan (office with little to no partitions or walls) | 0.02 |

Partially Closed Floor Plan (Office spaces with partial height partitions) | 0.03 |

Closed Floor Plan (apartment or hotel floor with multiple full height partitions) | 0.05 |

**Affects on Analysis for Vibration**

A higher damping ratio will give better performance of the floor for vibration, while a lower damping ratio will yield a poorer performance. Therefore if there is uncertainty about your floor arrangement it is better to err on the conservatism and use a slightly lower damping ratio for the vibration analysis of floor slabs using RAM Concept.

## Setting Response Type in RAM Concept

**Explanation**

One or both options can be selected within RAM Concept for response type…

**Resonant Response:**A resonant response tends to build up over time, and is generally most critical for lower frequency modes less than about 4 times the footstep frequency.**Impulsive Response:**An impulsive response tends to dissipate before the next footstep, and is generally most critical for higher frequency modes.

**Ranges of Values**

Both response options should generally be checked for vibration analysis of floor slabs in RAM Concept. It is important to capture the effects of vibration both at lower frequencies and higher frequencies.

Using both responses does not come at the expense of too much computing power and therefore analysis time, so generally it is best to check both options.

**Affects on Analysis for Vibration**

Checking resonant response captures the performance of the slab at lower frequencies while impulsive response captures the slabs behaviour at higher frequencies.

### Resonant Response Options

**Explanation**

Within the resonant response options are several selection and factor criteria. The table below provides explanations for each…

Factor | Explanation |

Simplified Calculation vs Modal Analysis | The simplified calculation is generally used for day-to-ay analysis checks of vibration, where only response factor is required. The modal analysis type should be used for highly vibration sensitive floors which require calculation of RMS velocity values. |

Duration | Defines the duration over which the time increments are applied (see immediately below) |

Time Increment | Defines the number of time points that are used to calculate the modal analysis (greyed out for Simplified Calculation). |

Weight of Person | The static weight of the individual walking. |

Max Natural Frequency | Defines the maximum natural frequency that is used in the dynamic analysis for the resonant response. |

**Ranges of Values**

The following table outlines guidance on the selection and factors criteria…

Factor | Value |

Simplified Calculation vs Modal Analysis | Use simplified calculation for quick analysis of simple floors for apartment, office and hotel floors. Use Modal analysis for hospital wards, and floor supporting sensitive equipment such as MRI units and other applications where the RMS velocity value is required. |

Duration | The duration should be set to capture at least 30 cycles of forcing (footsteps). |

Time Increment | The time increment should be set to at least 10 times shorter than the 4th harmonic of the fastest walking frequency expected in seconds. |

Weight of Person | A representative statistical weight should be used or a range of weights between 65kg and 85kg (143 – 187 lbs) |

Max Natural Frequency | Maximum natural frequency should be set anywhere from 8 to 15Hz |

**Affects on Analysis for Vibration**

The following table provides commentary around how each factor may influence your vibration analysis in RAM concept…

Factor | Value |

Simplified Calculation vs Modal Analysis | The modal analysis gives better more accurate results. However this is at the expense of computing power and therefore analysis time. |

Duration | If the duration is too low (less than 30 cycles of forcing for example) the higher mode effects may not be adequately captured and the results under-conservative. |

Time Increment | Shorter time increments adds more data points to the analysis and therefore more accuracy. |

Weight of Person | A lighter person will give lower amplitude vibration and therefore better performance, a heavier person will give higher amplitude vibration and therefore worse performance. |

Max Natural Frequency | Capping maximum natural frequency to a low value may erroneously cut off the higher mode effects in the analysis. |

## Specifying the Modal Mass for Vibration Analysis in RAM Concept

The in-built vibration analysis module within RAM Concept provides flexibility to modify the modal mass of the slab system.

By default, the self weight of the slab system is accounted for when RAM Concept runs the vibration check. The self-weight alone however may produce results which are over-conservative. The mass of attached elements such as partitions, ceilings and floor finishes will contribute to the mass which is excited during vibration (the modal mass).

The full design load for strength should not be adopted because this will then produce very under-conservative vibration results. Instead, the Structural Engineer should specify a realistic load which represents the likely loading scenario under service conditions.

This specified loading within RAM Concept is only applicable to the vibration analysis and can be navigated via…

**Layers>>>Vibrations>>>Additional Mass Loading>>>All loading..**.

In this layer, the user has flexibility to specify point, line and area loading for the vibration case (with interface the same as for design load input).

## Specifying the Excitation area for slabs in RAM Concept

To further refine the analysis, the user can specify the excitation area for the slab. This is simply the area where you will be anticipating the footfall traffic to exist.

This can be located by navigating via…

**Layers>>>Vibrations>>>Vibration Analysis>>>Excitation Areas Plan..**.

The snapshot above illustrates a screen grab for an example office floor slab. The hatched area denotes the designated are where the majority of foot traffic is expected. The floor plan is relatively open requiring a large degree of flexibility.

## Run the Vibration Analysis of the Floor in RAM Concept

After all the calc. options are set-up within RAM concept, you are ready to run the vibration analysis.

This is a straightforward process, the vibration analysis option is located next to the “Calc Load history” option in the top toolbar…

## Determine Natural Frequency of Floor for Vibration Analysis in RAM Concept

Mode shape number 1 corresponds with the natural frequency (fundamental or harmonic frequency) of the floor slab. This can be found in one of the layers within RAM Concept by navigating to…

**Layers>>>Vibration Analysis>>>Mode Shape 1 Perspective**

The frequency for mode 1 is indicated at the top left of screen. Here is an example office slab I have been working on. It has quite large cantilevers making it susceptible to vibration particularly at the tips of the cantilevers. Note that RAM Concept also shows a deflected mode shape which is an exaggerated view for visual representation purposes.

You can see that the fundamental frequency (mode 1) for this slab is **5.742**Hz. This gives us good separation from **3.0**Hz which we discussed earlier. This reduces the likelihood of resonances occurring in the slab system.

Another great tool in vibration analysis of floors in RAM Concept is the animated view of the mode shapes. TO view these, first you need to select the plot settings from the top toolbar then in the dialogue box on the bottom left enable the check box for “Enable animation”…

You can then choose the number of frames of the animation then press OK. Once you are ready to view the animation, you simply press the play button inthe main toolbar…

The slab movement in the animation above is exaggerated for viewing purposes (although it may look it, the slab is not about to fly away). The behaviour looks very similar to the vibrating guitar strings we saw earlier in this article.

## Check Response Factors in RAM Concept

The final step in Vibration analysis of floor slabs using RAM Concept is to check that the vibrations are acceptable.

A simple way of doing this is viewing the response factors in the response factor layer then comparing the plot against your chosen criteria. This can be found by navigating to…

**Layers>>>Vibration>>>Vibration Analysis>>>Max Response Factor Plan…**

The response factors can be checked against acceptable criteria depending on the floors usage. A response factor of 1 represents the threshold of human perception of the vibration movement. Values higher than one are multiples of human perception and values below 1 are theoretically not perceptible to humans.

The table below provides guidance for indicative limits for some common floor usages…

Floor Usage | Response Factor Limit | Description of Use |

Workshops, Office | 8 – 10 | Perceptible vibration, suitable for non-sensitive areas. |

Residential | 4 – 8 | Possible perceptible vibration, suitable for sleep areas in most cases |

Operating Rooms | 1- 4 | Near threshold perception, suitable for sensitive sleep areas. |

Sensitive Equipment Rooms | 0.0625 – 1 | Suitable for sensitive equipment (consult manufacturers specifications) |

Comparing the response factor plot with the upper bound response factor limit for an office (10), we can see that the slab is compliant for the chosen vibration criteria.

Did you know that RAM Concept also designs mat foundations and isolated footings? Take a look at a step-by-step walk through of how to do this via the article HERE.