1

SERVICEABILITY LIMIT STATES OF TIMBER FLOORS

Vibrations and Comfort

Margarida Maria Bebiano Coutinho Winck Cruz

margaridawinck@tecnico.ulisboa.pt

Instituto Superior Técnico, University of Lisbon, Lisbon, Portugal

October 2013

Abstract: The goal of the current work is to study and evaluate the empirical formulas presented in

Eurocode 5 in order to verify timber floors to the serviceability limit state of vibration. Hence,

numerical models using the finite element method were developed to analyse those formulas on

different floors. Using these models and the Eurocode 5 expressions, a parametric study was

conducted to determine the factors affecting the vibration characteristics of the floors. At last, the

static and dynamic requirements used to verify the serviceability limit state of vibration were applied

to the studied floors.

Keywords: Timber floors, serviceability limit state of vibration, finite element modelling.

INTRODUCTION

Timber is one of the most traditional materials

used in buildings construction around the

world. Applied on floors, walls and roof

systems, timber has the great advantage of

being a renewable natural resource and having

a high strength-weight relation. However,

because of its sensibility to water and

biological attacks and its natural imperfections,

timber must be used carefully. These

limitations led to the development of wood-

based materials with higher strength and less

imperfections; however the use of timber in

structures became less common.

Despite the decay of its use in most countries

in the last century, being replaced by concrete

and steel, timber structures are still important,

mostly in rehabilitation constructions. Floors

are the main structural timber system in old

residential buildings, so they need to be

preserved and properly designed.

Being light and flexible structures makes

vibration a significant problem for timber floors.

Vibration is a source of discomfort in the use of

this floor systems and its major cause are

dynamic movements produced by human

activities, such as walking.

In this paper the principal factors affecting

vibration and the design method for its

limitation, referred in Eurocode 5 as the

serviceability limit state of vibration, are

analysed. Numerical models for the timber

floors were developed for a proper study of the

empirical formulas presented in the regulation.

NUMERICAL MODELS

Numerical analysis was carried out using the

finite element method. The numerical models

pretend to simulate a basic timber floor system

with the main beams and the wood sheathing.

Three different models were developed using

2

the finite element program ADINA (ADINA

R&D, 2001). A detail of each model is

represented in Figure 1.

i. Frame elements – the beams and the

sheathing were modelled as frame

elements. These elements have three

degrees of freedom per node: x, x and y.

ii. Frame and shell elements – the beams

were modelled as frame elements and the

sheathing as a continuous shell element.

The given elements also have three

degrees of freedom per node: x, x and y.

iii. Frame and solid elements – the beams

were modelled as frame elements and the

sheathing as a continuous solid element.

This model has six degrees of freedom per

node: x, y, z, x, y and z.

The value of the fundamental frequency of the

structure and the results obtained for the

maximum moment and displacement at the

centre of the conditional beam due to an

uniformly distributed load allowed the

comparison of the three models. As those

results are similar, it can be considered that

the models are equivalent (Cruz, 2013).

(i) (ii) (iii)

Figure 1 – Detail of each model

The model adopted to run the numerical tests

in this paper is the one with the frame

elements, since it is the model associated to

the lowest computational effort.

In this paper three different floors (5,0x5,0 m2,

4,0x5,0 m2 and 3,0x3,0 m

2) are analysed, each

one with three different beam sections

(0,075x0,15 m2, 0,10x0,20 m

2 and 0,15x0,25

m2). The beams must have an equivalent T-

section so that the sheathing is considered in

the beam properties (Figure 2). Table 1 lists

the properties of each beam element of the

model (beam and sheathing).

The distance between the edges of the beams

is 0,30 m and the sheathing is constituted by

boards with a 0,10x0,02 m2 section.

The material considered in this study is a

timber with E0,05=6,0 GPa and mean=380 kg/m3,

which corresponds to a strength class of C18.

Figure 2 – T-section beams

Table 1 – Properties of the frame elements in the finite element model

Section [m

2] Ix [m

4] Iy [m

4] A [m

2] J [m

4]

Beam (a) 0,075x0,15 5,39 x 10-5

9,32 x 10-5

0,0150 8,49 x 10-5

Beam (b) 0,10x0,20 1,36 x 10-4

1,23 x 10-4

0,0240 2,67 x 10-4

Beam (c) 0,15x0,25 3,28 x 10-4

2,22 x 10-4

0,0420 7,82 x 10-4

Sheathing 0,10x0,02 6,94 x 10-8

1,67 x 10-6

0,0010 1,33 x 10-7

(a) (b) (c)

3

SERVICEABILITY LIMIT STATE OF

VIBRATION

The first and only code that specifies rules for

timber structures design to be used in Portugal

is Eurocode 5 (EC5) (EN 1995-1-1:2004). The

section 7 of this code is devoted to the

verification of the serviceability limit states,

where the vibration problem is included.

The rules presented in EC5 are applied to

residential floors with fundamental frequency

greater than 8 Hz. This limit was defined after

several researches where it was concluded

that floors with natural frequency with a lower

value have a higher risk of resonance effects

caused by walking, so they should the studied

in a special investigation.

The method defined by EC5 to verify the

serviceability limit state of vibration consists in

satisfying two requirements. The first

requirement (1) is related to the displacement

caused by a static point load and should be

limited by a parameter a, so that movements

due to low-frequency components (f<8Hz),

caused by walking, are supressed. Since the

floors are considered to have natural

frequencies higher than 8 Hz, these

movements are semi-static in nature; hence

the static criterion is adequate. Hence, the

quotient between the maximum displacement

(w), measured in mm, and the vertical point

load that causes it (F), applied at any point of

the floor and measured in kN, should be lower

than the value of a parameter a.

(1)

The second requirement (2) limits the

magnitude of the transient response due to the

heel impact of a footstep. This impact excites

higher frequency components and the timber

floor response is governed by its stiffness,

mass and damping. This dynamic criterion is

translated to the limitation of the maximum

initial value of the vertical floor vibration

velocity (v), measured in m/s, caused by an

ideal unit impulse (1 Ns) applied at the point of

the floor giving maximum response by the

combination between a parameter b, the floor

fundamental frequency (f1), in Hz, and its

modal damping ratio ().

(2)

These requirements should be applied

assuming that the floor is unloaded, i.e., only

the mass corresponding to the self-weight of

the floor and other permanent actions should

be considered.

The values for the parameters a and b are not

specified in EC5. It is only presented a graphic

with the recommended range of limiting values

and the recommended relationship between

the parameters (Figure 3). It is also pointed out

that more information about this parameter

choice should be included in the National

Annex.

Figure 3 – Recommended range of and

relationship between a and b (EN 1995-1-1:2004)

Each variable defined in criteria (1) and (2) will

be studied through the formulas presented in

EC5 and through the numerical model to

4

determine the factors that affect them the

most. Then, a comparison of the analytical and

numerical results will be presented.

As previously stated, the dimensions l x b

(Figure 4) of the floors studied in this paper are

5,0x5,0 m2, 4,0x5,0 m

2 and 3,0x3,0 m

2, being

the beams span (l) always the smallest length.

The section of the analysed beams are

0,075x0,15 m2, 0,10x0,20 m

2 and 0,15x0,25

m2 (Figure 2) and the values of the properties

used in the EC5 formulas are displayed in

Table 2. In (Cruz, 2013) more types of floors

with other dimensions and beam sections were

analysed. It should be noted that all formulas

consider rectangular floors simply supported

along all four edges.

Figure 4 – Timber floor

Table 2 – Properties of the floor elements

Beams sections [m

2]

0,075x0,15 0,10x0,20 0,15x0,25

Ix [m4] 5,39 x 10

-5 1,36 x 10

-4 3,28 x 10

-4

A [m2] 0,01875 0,02800 0,04650

m [kg/m2] 19,00 26,60 39,27

(EI)l [Nm2/m] 8,62 x 10

5 2,04 x 10

6 4,37 x 10

6

Sheathing

Ix [m

4] 3,33 x 10

-6

(EI)b [Nm

2/m] 4,00 x 10

3

Frequencies and vibration modes

The fundamental or natural frequency of a

structure is the frequency of the first vibration

mode. The natural frequency is the most

important characteristic in the study of the

structure response to a dynamic action. The

higher the stiffness of the structure is, the

higher its fundamental frequency is and the

lower its vibration magnitude is. An undamped

free vibration system is considered to compute

the frequencies and vibration modes of a

structure.

In Eurocode 5 the fundamental frequency (f1)

of a timber floor is given by the formula (3). Its

value depends of the dimension of the beams

span (l), in meters, the mass per unit area, in

kg/m2, and the equivalent plate bending

stiffness of the floor about an axis

perpendicular to the beam direction ((EI)l), in

Nm2/m.

√

(3)

Through the analysis of the expression it can

be verified that, since it only depends of the

dimension of the beams span, the value of the

natural frequency is independent of the

dimension b of the floor. So, all the floors with

the same beam length and section have equal

frequencies. The results of the formula for the

defined floors are presented in Table 3.

Table 3 – Fundamental frequencies, in Hz, obtained through the formula (3)

Beams sections [m

2]

l x b [m2] 0,075x0,15 0,10x0,20 0,15x0,25

3,0 x 3,0 37,17 48,36 58,24

4,0 x 5,0 20,91 27,20 32,76

5,0 x 5,0 13,38 17,41 20,97

Comparing the results it can be verified that

the value of the frequency decreases with the

increasing of the beams span, effect that can

be justified by the decreasing stiffness of the

floor. The frequency is also lower for smaller

beam sections, meaning that the decrease of

5

the floors’ stiffness is bigger than the increase

of its mass.

The analytical values for the natural frequency

are now compared with the results obtained

with the numerical models. For each floor two

models were developed, based on the model

with frame elements previously described, one

with all four edges simply supported and the

other with only two edges simply supported.

The ends of the beams are the two supported

edges of the second model, being the l edges

of the Figure 4 not supported.

Figure 5 represents the two firsts vibration

modes of the floors with all four edges

supported (a) and the floors with only two of

the edges supported (b). These vibration

modes are representative for all the floors

studied, because all 1st vibration modes are

similar, the same happening with the 2nd

vibration modes.

1st mode 1

st mode

2nd

mode 2nd

mode

(a) (b)

Figure 5 – Vibration modes of the floors

As previously stated, the 1st vibration mode is

the deformed configuration of the floor with the

lower stiffness and it also is the one that

mobilizes more mass. It is then simple to

conclude that the 1st vibration mode, which is

linked to the fundamental frequency, is the one

with the lowest frequency value of the

structure.

Table 4 and Table 5 display the values for the

natural frequencies obtained with the

numerical models for floors with four supported

edges and floors with only two supported

edges, respectively.

Table 4 – Natural frequencies, in Hz, from the numerical model supported along four edges

Beams sections [m

2]

l x b [m2] 0,075x0,15 0,10x0,20 0,15x0,25

3,0 x 3,0 41,86 51,61 59,36

4,0 x 5,0 23,42 29,01 33,61

5,0 x 5,0 16,40 19,91 22,29

Table 5 – Natural frequencies, in Hz, from the numerical model supported along two edges

Beams sections [m

2]

l x b [m2] 0,075x0,15 0,10x0,20 0,15x0,25

3,0 x 3,0 37,51 48,59 58,29

4,0 x 5,0 21,04 27,28 32,81

5,0 x 5,0 13,47 17,48 21,02

With the analysis of results listed above, the

conclusion previously made, about the

increase of the value of the frequency with the

increase of the beams cross section

dimensions and the decrease of the beams

span, is strengthened. The factor that affects

frequency the most is the length of the floors’

beams, being the second factor the section of

the beams.

From the comparison of the formula results

with the numerical results it is possible to verify

that they are quite similar for the model with

two supported edges. In fact, since the formula

(3) only considers the equivalent bending

stiffness for the floor’s beams, its results

represent a floor with cylindrical bending,

despite the fact that EC5 states that the

6

formula is applied to floors simply supported

along all four edges. The frequency values

obtained with the model with four supported

edges are slightly higher than the others due to

the higher stiffness introduced by the two

additional supports. Therefore, it can be

considered that the formula is on the safety

side since it gives lower frequency values for

floors with all supported edges than it was

expected.

Static displacement

The Eurocode 5 does not define how to

determine the displacement due to a static

point load. Hence, the displacement was

computed using the formula defined for simply

supported beams, for a point load (F) applied

at the centre of the beam, in N, and an

uniformly distributed load (p) equivalent to the

beam weight per unit length of the floor, in

N/m2. The value of the displacement also

depends of the beams span (l), in meters, and

the equivalent plate bending stiffness of the

floor about an axis perpendicular to the beam

direction ((EI)l), in Nm2/m, as showed in the

expression (4).

(4)

The displacement caused by a static load of

700 N applied at the middle of the floor is

presented in Table 6 for several test cases.

Table 6 – Displacement, in mm, obtained using formula (4)

Beams sections [m

2]

l x b [m2] 0,075x0,15 0,10x0,20 0,15x0,25

3,0 x 3,0 0,457 0,193 0,090

4,0 x 5,0 1,084 0,457 0,214

5,0 x 5,0 2,117 0,894 0,418

The displacement values are higher for floors

with bigger beam spans and smaller beam

sections. This was expected and is easily

deduced from the analysis of the formula.

The displacements were also determined by

the numerical models for the same static point

load and for the models supported along four

or only two edges. The corresponding results

are presented in Table 7 and Table 8,

respectively.

Table 7 – Displacements, in mm, obtained using the numerical model with four supported edges

Beams sections [m

2]

l x b [m2] 0,075x0,15 0,10x0,20 0,15x0,25

3,0 x 3,0 0,437 0,256 0,154

4,0 x 5,0 0,716 0,416 0,274

5,0 x 5,0 1,051 0,602 0,411

Table 8 – Displacements, in mm, obtained using the numerical model with two supported edges

Beams sections [m

2]

l x b [m2] 0,075x0,15 0,10x0,20 0,15x0,25

3,0 x 3,0 0,443 0,257 0,154

4,0 x 5,0 0,728 0,417 0,274

5,0 x 5,0 1,129 0,621 0,413

The difference between the values obtained

with both models is small. The highest

differences appear for the more flexible floors

and decrease with the stiffness increase. This

same tendency is verified in the difference

between the analytical and the numerical

values. The displacement values given by the

formula are higher than the values obtained

with the numerical models as the adjacent

beams considered in the model increase the

stiffness of the floor. Therefore, it can be

considered that the formula is from the safety

side from the structural point of view.

7

Unit impulse velocity response

An impulse not only represents the variation of

a force in a period of time but it can also be

related to the linear momentum variation. For a

system initially at rest, the impulse is equal to

the linear momentum, which is the product of

the system mass and its velocity at the mass

centre. Hence, for the same impulse value, the

increase of the system mass leads to the

decrease of its velocity.

The unit impulse velocity response (v), in m/s,

is determined in EC5 through the formula (5),

which depends of the mass of the entire floor,

considered by the product between its

dimensions l and b, in meters, and its mass per

unit area, in kg/m2, and also its number of

modes with natural frequencies up to 40Hz

(n40). The floors referred in the code are light-

weight floors, which mean that the presence of

a human occupant modifies their modal

properties. Therefore, an additional mass of

50kg at the middle of the floor is considered to

simulate the partial mass of an occupant,

translated into the expression (5) by the 200/4

ratio.

(5)

The restriction of the peak velocity response

value due to an unit impulse has the purpose

of limiting the dynamic effects caused by the

heel impact of a footstep, as previously stated.

Depending on the intervals between

successive impacts and damping of the

vibration, adjacent transient vibration response

may interact with each other (Hu, et al., 2001).

This interaction is denoted by the ratio of the

across-joist direction stiffness ((EI)b) and

along-joist direction stiffness ((EI)l), which

controls the spacing of two adjacent natural

frequencies, that increases with increasing

(EI)b. This concept is introduced in EC5 by the

number of first-order modes with natural

frequencies up to 40Hz (n40), given in

expression (6), which depends of the floor’s

dimensions l and b, in meters, its fundamental

frequency (f1), in Hz, and its along and across

beam stiffness, (EI)l and (EI)b in Nm2/m.

{[(

)

] (

)

}

(6)

The results of the formulas (5) and (6) applied

to the defined floors are presented on Table 9.

The formulas only apply to floors with values of

natural frequency bellow 40Hz, reason why the

table is not completely filled.

Table 9 – Results from the formulas (5) and (6) for v, in m/s, and n40

Beams sections [m

2]

l x b [m2] 0,075x0,15 0,10x0,20 0,15x0,25

3,0 x 3,0 n40 2,42 - -

v 0,01994 - -

4,0 x 5,0 n40 6,12 6,17 6,02

v 0,02807 0,02241 0,01628

5,0 x 5,0 n40 6,43 6,84 7,33

v 0,02523 0,02082 0,01624

In order to obtain the same results with the

numerical models (Table 10 and Table 11) an

unit impulse was applied at the middle of each

model, with a time function of magnitude

1000N and duration of 0,001s, and a mass of

50kg. The Rayleigh damping (Clough, et al.,

1995) was applied to the models with the

modal damping ratio defined in EC 5 as 1%.

To compute the value of the velocity response

to the impulse, an implicit dynamic analysis

was performed using the Newmark’s method

(Clough, et al., 1995). The number of first-

8

order modes with natural frequencies up to

40Hz was determined using the previous

defined models considering undamped free

vibration properties.

Table 10 – Results obtained with the model for four supported edges (v, in m/s, and n40)

Beams sections [m

2]

l x b [m2]

0,075x0,15 0,10x0,20 0,15x0,25

3,0 x 3,0 n40 0 0 0

v 0,02290 0,02015 0,01623

4,0 x 5,0 n40 3 2 3

v 0,02287 0,01979 0,01540

5,0 x 5,0 n40 3 3 4

v 0,02277 0,01987 0,01547

Table 11 – Results obtained with the model for two supported edges (v, in m/s, and n40)

Beams sections [m

2]

l x b [m2]

0,075x0,15 0,10x0,20 0,15x0,25

3,0 x 3,0 n40 1 0 0

v 0,02289 0,02016 0,01623

4,0 x 5,0 n40 4 4 4

v 0,02285 0,01978 0,01541

5,0 x 5,0 n40 5 5 6

v 0,02293 0,01980 0,01546

The results obtained with the finite element

models are very similar for the models with

four and two supported edges. This means that

the support conditions of the floor are not

important when determining the value of the

velocity response to an impulse.

From the analysis of all results, it can be

concluded that the decrease of the floors’

dimensions decreases the number of vibration

modes below 40Hz. This was expected as

smaller floors have higher frequency values, as

previously concluded in this paper.

The factor affecting the velocity value the most

is the dimension of the beams cross section.

Larger cross sections mean higher mass

which, by the linear momentum theory, leads

to lower velocity values. This tendency can be

verified either in the formulas’ results or in the

numerical results.

It was expected that the velocity value would

decrease with the increase of the floors

dimensions (l and b), due to their higher mass,

which was verified in some cases. In other

cases, however, the velocity value was higher

for larger floors in the analytical and the

numerical results. These results could be

explained by the effect of interaction of

adjacent transient vibration response.

Comparing the results obtained using the

formula (5) with the ones obtained with the

numerical models, it can be noticed that the

firsts are lower than the seconds for smaller

floor dimensions. The opposite situation

happens for floors with bigger dimensions.

Verification of the serviceability

limit state of vibration

Based on the values presented in the previous

sections, the formulas (1) and (2) were applied

to the studied floors in order to determine the

values of the parameters a and b and to

perform the verification of the serviceability

limit state of vibration. Table 12, Table 13 and

Table 14 present the values for parameters a

and b computed using the EC5 expressions

and considering the numerical model with four

supported edges and the numerical model with

two supported edges, respectively.

Analysing the values of the parameters and

having as only reference the graph in Figure 3,

it can be concluded that all floors verify the

serviceability limit state of vibration, since there

is no value of a higher than 4,0 mm/kN and no

value of b lower than 50.

9

Table 12 – Parameters a and b determined with the results from the formulas of EC5

Beams sections [m

2]

l x b [m2]

0,075x0,15 0,10x0,20 0,15x0,25

3,0 x 3,0 a 0,65 0,28 0,13

b 508,3 - -

4,0 x 5,0 a 1,55 0,65 0,31

b 91,6 184,4 456,8

5,0 x 5,0 a 3,02 1,28 0,60

b 70,0 108,6 183,7

Table 13 – Parameters a and b determined with the model with all edges supported

Beams sections [m

2]

l x b [m2]

0,075x0,15 0,10x0,20 0,15x0,25

3,0 x 3,0 a 0,62 0,37 0,22

b 662,2 1031,7 25336,3

4,0 x 5,0 a 1,02 0,59 0,39

b 138,9 251,0 536,8

5,0 x 5,0 a 1,50 0,86 0,59

b 92,2 133,3 213,7

Table 14 – Parameters a and b determined with the model with two edges supported

Beams sections [m

2]

l x b [m2]

0,075x0,15 0,10x0,20 0,15x0,25

3,0 x 3,0 a 0,63 0,37 0,22

b 421,7 1986,8 19519,1

4,0 x 5,0 a 1,04 0,60 0,39

b 119,8 220,2 498,0

5,0 x 5,0 a 1,61 0,89 0,59

b 78,5 115,9 196,2

Figure 6 shows that all parameters a and b

respect the specified limits. These are the

values that appear with a grey shade in the

tables displayed above.

Figure 6 - Relationship between a and b for the studied floors

The limit of the static criterion (a) depends of

the stiffness of the floor, corresponding the

lower values of a to the floors with the higher

stiffness. The limit of the dynamic criterion (b)

depends of the floor’s mass and stiffness,

associating its higher values to the floors with

higher mass and higher stiffness.

From the comparison of the values obtained

using Eurocode 5 formulas and considering the

numerical models it is possible to conclude that

the EC5 equations are conservative. For the

same floor, the value of a is higher and the

value of b is lower when using the results from

the expressions present in the code. This

tendency is perceptible in the graph presented

in Figure 6 and is valid for the majority of the

floors, except for those with higher beam

sections, i.e., with higher stiffness, where the

parameter a is lower when the EC5 method is

used.

CONCLUSIONS

The goal of this paper was to study the

methodology defined by Eurocode 5 for the

verification of the serviceability limit state of

vibration and the main factors affecting it.

The finite element models used to obtain the

results displayed in this paper were developed

with frame elements simulating both the floor’s

beams and sheathing. An equivalent T-section

has been considered for the beams to ensure

the correct evaluation of the stiffness of the

beam and the sheathing. In spite of the

simplicity of the models, the results are similar

to the ones determined with more sophisticate

models using shell or solid elements and have

the advantage of requiring much less computer

effort. This model type is considered to give

5060708090

100110120130140150

0 1 2 3 4

b

a [mm/kN]

EC5

Model 4supports

Model 2supports

10

good results when developing dynamic

analysis of timber floors.

The formula present in EC5 to determine the

fundamental frequency is considered

appropriate for timber floors, since its results

are similar to the numerical results. It was

concluded that this formula was developed for

floors with cylindrical bending since it is

independent of the dimension of the floor in the

across-beam direction and its results are

closer to the ones obtained with numerical

models with two supported edges. The beams

length and section are the factors affecting

frequency value the most, being the first the

most conditioning.

The formula for the calculation of the unit

impulse velocity response is simple to use. It

was shown that the mass of the floor is the

most important factor affecting the velocity,

being its value lower in floors with higher mass.

It is then concluded that the floors with the

higher mass and the higher stiffness have the

best response to vibration problems. This

statement is confirmed by the values obtained

for parameters a and b. However, the

designers should be careful when designing a

timber floor so that the increase of the floor’s

mass does not lead to a decrease in its natural

frequency. Hence, the increase of the mass

value should always predict the increase of the

stiffness.

The application of Eurocode 5 techniques

presents some practical problems. One issue

is related to the determination of the static

point load displacement, because no guidance

is given about how to proceed. The formula (4)

used gave good results and should be

considered as an option for the displacement

value determination. Another difficulty is

related to the definition of the value of the

parameters a and b. These are the values that

limit the requirements that allow the verification

of the serviceability limit state of vibration and

the range of possible values is too big. The

information given by the EC5 is considered

insufficient. These difficulties should be studied

in order to make possible the presentation of

an unified procedure to assess the

serviceability limit state of vibration, which can

be included in the National Annex of this

building code.

REFERENCES ADINA R&D, Inc. 2001. ADINA User Interface. Report ARD 01-6. Watertown, MA, USA : s.n., 2001.

Clough, Ray W. and Penzien, Joseph. 1995. Dynamics of Structures. Computeres & Structures, Inc. Berkeley,

CA USA : s.n., 1995.

Cruz, Margarida W. 2013. Estados Limites de Utilização de Pavimentos de Madeira. Vibrações e Conforto.

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