Introduction to vibration in buildings Handout 4

Guidance for modelling footfall-induced vibration in composite floors using GSA

1. Advanced features. In order to carry out footfall analysis in GSA, ensure that appropriate

advanced features are enabled. Go to Tools Preferences and click on the Advanced features

tab. The 2D element analysis, Modal dynamic analysis and Footfall induced vibration analysis

options must be selected.

2. Element types. Where possible, use parabolic (Quad8 and Tri6) elements for a 2D mesh. If linear

(Quad4 and Tri3) elements are used, ensure they have MITC, and not Mindlin, formulation. The

formulation can be set in the 2D Element Analysis tab of the Advanced Solver Settings on the last

screen of the Analysis Wizard. Linear elements will result in a smaller model size.

3. Mesh planning. If you take time to plan your mesh first, you will save yourself troubles later.

4. Mesh quality. Try to limit the aspect ratio of elements to 1:3. The internal element angles should

also be limited to >45 degrees and <135 degrees where possible.

5. Mesh building. Generating 2D element meshes is covered in the GSA help file with that name.

However, for areas that can be broken down into rectangular areas, a quick way to build 2D

element meshes is to add, for each rectangle, large Quad4 elements that are subdivided to get the

mesh density required. You can change Quad4/Tri3 elements to Quad8/Tri6 elements by

selecting all the 2D elements and using the Modify toolbar in GSA to change the 2D element type

to parabolic. The beam elements must also be subdivided to get mesh compatibility. 2D elements

with property n can be viewed by selecting PAn; beam elements with property n can be viewed

by selecting PBn. If there are dissimilar mesh densities either side of an interface, the two meshes

can be connected by a Tied interface, found under Constraints in the Gateway.

6. Mesh density. Try and use a mesh density of at least 8 elements along a beam’s length and about

3-4 elements between secondary beams in a floor bay. In general there should be at least 4

elements for each half sine wave of mode shape, including those for higher modes. How do you

know if your mesh is fine enough? If you do a natural frequency analysis, you need enough

nodes to define the mode shape correctly. If in doubt, subdivide the mesh by 2 and re-run.

Compare the natural frequencies and modal masses of the two models.

7. Composite Floors. There are several ways to model a composite floor. The easiest way is to

model the slab with 2D elements explicitly and the beams with offset beam elements. This will

result in axial forces in the slabs and beams, which is why flat shell elements must be used. Make

sure your beam mesh is compatible with your 2D element mesh. A quick way to create beam

elements which are compatible with the 2D element mesh is to select a line of nodes and use the

Add string of 1D elements command, found under Sculpt 1D element operations.

8. Element offsets. Element offsets are important when modelling a composite floorplate. The

beams can be offset downwards, or the slab can be offset upwards, or there can be a combination

of the two. What is important is that the total offset equals:

(Height of slab NA above steel beam + depth of centroid below top of beam)

The height of the slab neutral axis above the steel beam (the centroid of the decking and

uncracked concrete) is one of the parameters determined by the Footfall vibration spreadsheet

(see section 14 below).

If the level of the top of the beams is defined as a Grid plane, an easy way to model the beam

offsets is to Apply offsets. Having selected the beams, go to Tools Manipulate model Apply

offsets. Select Offset beams downwards to align top surfaces, Align with grid planes and OK. If

the slab is not at the level of its NA, the slab elements must be selected and the necessary offset

applied using Modify selected elements Offsets, select all eight nodes and enter the appropriate

offset under z. Note that the top of the beams will not generally be flush with the bottom of the

slab in the model, because the profiled decking and slab are modelled as an equivalent

rectangular section.

9. Concrete properties. The following concrete properties should be used when modelling dynamic

effects:

Normal weight concrete: Young’s Modulus: 38GPa

Density: 2400kg/m3

Lightweight concrete: Young’s Modulus: 22GPa

Density: 1800 kg/m3

In order to create this user-defined material:

Go to Properties Materials User defined in the gateway.

Click the top left cell in the table and click the Wizard button.

Select Concrete long term from the Copy standard material dropdown menu.

Name the material something such as ‘Concrete dynamic’.

After clicking Next, change the Elastic modulus and Density to the values above.

10. Web openings. If the steel beams have web openings, to compensate for the reduction in mass, a

reduced web thickness should be specified in the element properties. The web thickness should

be calculated such that the volume of steel in the web of the equivalent plain section is the same

as that in the web of the cellular section. A stiffness correction should also be applied if the steel

beams have web openings. In the absence of better information, the Kz section modifier should be

reduced to about 25% of the ratio of the solid web area (the web height being defined as the total

section depth minus the flange thicknesses for a welded section) to the total section area.

11. 2D element properties. Make sure you use Shell elements, which can be defined in the 2D

element properties area of GSA. Shell elements are formulated to take in-plane stresses as well as

bending stresses, while Flat plate elements can only take bending stresses: this is very important

for composite floors.

12. Concrete Cracking. The small amplitude of dynamic strains means that usually, where crack

widths are controlled, concrete cracking will have little impact on dynamic axial and bending

stiffnesses. Even where the concrete is cracked, the stiffness between cracks will approach the

uncracked value, resulting in only a small overall reduction. An example of a situation where

stiffness should be reduced is a composite slab, propped during construction, and reinforced with

nominal mesh, rather than designed reinforcement. On depropping and further loading, there will

be uncontrolled cracking over the supports, for which a reduced bending stiffness (of, say, 70%

of the uncracked value) should be used.

13. Mass. It is extremely important that the mass of the floor is not overestimated. Only mass that is

going to be on the floor should be included in the natural frequency analysis. Excess mass may

result in unconservative answers. As a guide, no more than 10% of live load should be included

as additional mass. Typically, the non-structural mass for an office floor can be taken as

100kg/m2. Do NOT apply the full live load as added mass. For stairs and bridges, NO live load

should be added.

There are four ways to add non-structural mass in GSA. The easiest way is to add Additional

mass in the 2D element properties table or the 2D property wizard. Alternatively, the

Mass/weight modifier in the same table can be adjusted; otherwise you can add mass elements, or

you can specify a load case to be converted to mass when setting up the natural frequency

analysis. To check if this has been done, scan the dynamic details in the output view where it

shows any load cases added as mass. Gravity, thermal and prestress loads are NOT converted to

masses, even when included in a load case with externally applied loads that are converted to

masses.

14. Trapezoidal decking. To model trapezoidal decking, appropriate properties must be calculated for

the 2D elements. However, until GSA has orthogonal elements, you must decide in which

direction the decking’s properties are more important. If the span direction properties are

dominant, a convenient way to obtain properties is to use the Footfall vibration spreadsheet from

the SSN site (Tools Spreadsheets General); the spreadsheet should be saved locally on

your machine. Once saved and opened, open the Response worksheet and click the Composite

beam floor radio button. Click on the Composite Beam Floor tab, which will now have appeared.

After opening this worksheet, enter the overall slab depth, enter the type of decking and select

normal weight or lightweight concrete. Make a note of the following four numbers from the

Decking table:

Axial factor

Bending factor

Mass factor

Height of slab neutral axis above steel beam (centroid of decking and uncracked concrete).

From these numbers, the Thickness, Bending, In-plane and Mass/weight thickness modifiers can

be calculated and entered into the 2D element properties table:

Thickness is the overall slab depth

Bending is the spreadsheet’s Bending factor

In-plane is the spreadsheet’s Axial factor

Mass/weight is the spreadsheet’s Mass factor.

It is also possible to specify Additional mass resulting from up to 10% of imposed load (kg/m²)

= up to 0.1 × imposed load (kPa) ×g

1000 (optional)

These section properties will not be constant for the whole slab if there are changes in the type of

decking, slab thickness, superimposed dead or imposed loads etc. If this is the case, different 2D

element properties should be assigned to different 2D elements.

15. Beam Connections. Normally, for strength or serviceability design, structural engineers assume

pinned end connections. For dynamics, however, assume all connections are fixed unless they

truly are pins. This is because pin connections do, in practice, provide a small moment

connection, which is significant when considering small dynamic rotations. The same applies

around concrete core walls.

16. Façade Supports. Similarly, a façade can usually be taken as offering vertical support to a floor

because the small dynamic displacements do not overcome friction.

17. Columns. The bending stiffnesses of columns help increase the natural frequency of a floor and

reduce the modal mass, so it is not clear whether their contribution is of benefit or not. Rather

than providing pin supports at column positions, the columns above and below the floor should

be modelled.

18. Over Restraining. If modelling a floor plate using 2D elements and offset beams, make sure that

the slab edges are not fully restrained against moving laterally (ie pinned). This will result in the

slab attracting membrane forces and will give a higher natural frequency than the floor actually

has.

19. Symmetry. If you use symmetry in your model the modal mass will be incorrect. You will not get

antisymmetric modes either. It is strongly recommended to use a full model.

20. Modal dynamic analysis. A Modal dynamic analysis must be carried out before a footfall analysis

can be run. You should determine all modes with frequencies up to the greater of 15Hz and twice

the natural frequency of the lowest mode that is active in the area being considered. As a starting

point, try around 40 modes.

21. Lumped mass or consistent mass matrix. Providing you have enough nodes the lumped mass

option in the analysis wizard will give a good result with far less computational effort. Mass will

be lumped at the nodes and the mass matrix for each element will be diagonal. The consistent

mass matrix formulation, which calculates the mass from the element shape functions, will result

in a slightly better mass distribution but involves far more computations to get the mass matrix,

which will be fully populated and based on the same element shape functions as used to derive

the element stiffness matrix.

22. Footfall analysis. Following a modal analysis, the Footfall analysis can be run. The excitation

method should be set to Full excitation and the Use search technique box selected. The damping

ratio depends on the structure being analysed. For an open-plan, lightly furnished office, the

damping ratio may be lower than 2% of critical damping. For a fully partitioned, heavily

furnished office, the damping ratio may be up to 4.5%. For a composite floor with no furniture,

partitions or raised floors (eg: airport terminal) damping may be as low as 1.0% of critical

damping. The response factor is quite sensitive to the damping ratio, and care should be taken not

to select too high a level of damping. The default value of 3% should be used with caution, as it

is likely to result in a non-conservative design for most offices. The default Number of footfalls

of 100 is likely to be a conservative starting point for an office floor plate. After running the

footfall analysis, the initial results can be analysed and a more appropriate number of footfalls

chosen. The default values for the other options should be acceptable.

23. Results. After running the Footfall analysis, look at the results by selecting your footfall analysis

case, clicking on the Contour settings button and selecting Nodal results (or Nodal results on

elements if you want the results interpolated along elements) Footfall induced vibration

summary Overall maximum response factor.

24. Refining number of footfalls. In order to more accurately determine the correct number of

footfalls to input into the analysis, the areas with peak response factor should be investigated. For

each area, the following method should be employed:

Find the node number of the peak node.

Go to View New chart view Footfall analysis and select the Participation factor v.

modes: resonant chart from the Footfall induced vibration - resonant analysis group.

Enter the number of the peak node into both the Response node and Excitation node boxes

and click on OK.

This chart shows which mode or modes make a significant contribution to this resonance.

Go back to the footfall analysis chart options and, this time, select the Response factor v.

walking frequency chart from the resonant analysis group, again entering the peak node

number in both boxes.

Consider the peak or peaks of this graph. For each peak, multiply the harmonic number by

the walking frequency. This should give approximately the same resulting frequency for

each peak. Check that this corresponds to the frequency of the mode identified from the

previous chart.

Examine the deflected shape of this mode at the area in question. Measure the dimensions

of the deflected zone. Take the largest dimension and divide it by the length of a pace, say

0.7m. This gives the maximum number of footfalls that could excite this area.

Having determined the appropriate number of footfalls for each area of peak response, different

footfall analyses should be run with the different numbers of footfalls. The results from an

analysis run with a particular number of footfalls only apply to the region(s) for which that

number of footfalls is appropriate. It is, therefore, not possible to produce a single output in GSA

that gives a completely accurate representation of the response factor over the whole slab without

generating user modules, combining them in a spreadsheet and creating a new user module. One

could, however, take the highest number of footfalls and produce a single output which is

accurate over the area for which that number of footfalls was calculated and conservative over

the rest of the slab.

June 2012