Based on classic vibrational bending theory on beams, this paper provides comprehensive analytical formulae for dynamic characteristics of two equal span continuous timber flooring systems, including frequency equations, modal frequencies, and modal shapes. Four practical boundary conditions are considered for end supports, including free, sliding, pinned, and fixed boundaries, and a total of sixteen combinations of flooring systems are created. The deductions of analytical formulae are also expanded to two unequal span continuous flooring systems with pinned end supports, and empirical equations for obtaining the fundamental frequency are proposed. The acquired analytical equations for vibrational characteristics can be applied for practical design of two-span continuous flooring systems. Two practical design examples are provided as well.
Vibrational serviceability performance of timber floors has become an important issue in the world due to their resonance frequencies and low material masses. In Europe, Eurocode 5 [
Human beings are considered as precarious sensors of vibrations, and their distress to timber floor vibrations concern many researchers, and human activities and machine-induced vibrations can cause distress. Human sensitivity and perception are basically related to structural vibrations.
Over past decades, extensive investigations have been conducted on evaluating the dynamic performance of timber floors and human vibrational perception in many European countries, Canada, Australia, and Japan. Ohlsson [
Recently, considerable attention has been paid to the dynamic responses of timber floors constructed with LVL (laminated veneer lumber) beams, glulam beams and CLT (cross-laminated timber) panels, and TCC (timber-concrete composite) floors. Basaglia et al. [
EN 1995-1-1 [
The majority of European countries have directly adopted equation (
Similarly, Finland [
Both Austria and Finland specify that the floor mass
Spain [
With the development of engineered timber products, floor joist sizes can be manufactured much larger for longer floor spans or multispans. Some research work on structural dynamic characteristics of general continuous beams has been reported [
For two-span continuous timber flooring systems with equal joist spacing but various end supports, they can be treated as two-span beams and analysed on a single two-span continuous timber floor joist (see Figure
A typical two-span continuous timber floor.
For a two-span continuous Bernoulli–Euler beam with a uniform cross-sectional area, mass density, and flexural stiffness, the equation of motion for each beam span, i.e., transverse displacement
Substituting equation (
The natural frequencies and coefficients
Four typical boundary conditions for two-span continuous beams are considered with various end supports, e.g., free, sliding, pinned, and fixed boundaries (Table
Boundary conditions for two-span continuous beams.
Boundary conditions | Control equations |
---|---|
Free end |
|
Sliding end |
|
Pinned end |
|
Fixed end |
|
Pinned middle support |
|
|
For continuous timber flooring systems with two equal spans,
To establish the frequency equations of two equal span flooring systems and to determine the modal frequencies and mode shapes, the displacement equation (
A two equal span continuous timber floor beam with left end fixed and right end simply supported.
For spans 1 and 2, equation (
Differentiating equations (
Thus, equation (
Solving equations ( Thus, equation (
Combining equations ( Combining equations ( Combining equations ( Combining equations (
A nontrivial solution for equations (
Thus, the frequency equation can be obtained as
Similarly, based on equation (
Frequency equations for two-span beams with various end support conditions.
Boundary conditions | Control equations | |
---|---|---|
Free-free |
|
|
Free-sliding |
|
|
Free-pinned |
|
|
Free-fixed |
|
|
Sliding-free |
|
|
Sliding-sliding |
|
|
Sliding-pinned |
|
|
Sliding-fixed |
|
|
Pinned-free |
|
|
Pinned-sliding |
|
|
Pinned-pinned |
|
|
Pinned-fixed |
|
|
Fixed-free |
|
|
Fixed-sliding |
|
|
Fixed-pinned |
|
|
Fixed-fixed |
|
|
From the frequency equations, the modal frequencies for two equal span timber flooring systems can be obtained numerically. Here commercial software MathCAD is used for such purpose, and Table
Frequency parameters for two-span beams with various end support conditions.
Boundary conditions |
|
|
|
|
---|---|---|---|---|
Free-free | 0.0000 |
1.8751 | 3.9266 | 4.6941 |
Free-sliding | 1.1705 | 2.1695 | 4.1798 | 5.2329 |
Free-pinned | 1.5059 | 3.4131 | 4.4373 | 6.5446 |
Free-fixed | 1.5708 | 3.9266 | 4.7124 | 7.0686 |
Sliding-free | 1.1705 | 2.1695 | 4.1798 | 5.2329 |
Sliding-sliding | 1.5708 | 2.3650 | 4.7124 | 5.4978 |
Sliding-pinned | 1.9633 | 3.5343 | 5.1051 | 6.6759 |
Sliding-fixed | 2.0295 | 4.1973 | 5.2391 | 7.3300 |
Pinned-free | 1.5059 | 3.4131 | 4.4373 | 6.5446 |
Pinned-sliding | 1.9633 | 3.5343 | 5.1051 | 6.6759 |
Pinned-pinned | 3.1416 | 3.9266 | 6.2832 | 7.0686 |
Pinned-fixed | 3.3932 | 4.4633 | 6.5454 | 7.5916 |
Fixed-free | 1.5708 | 3.9266 | 4.7124 | 7.0686 |
Fixed-sliding | 2.0295 | 4.1973 | 5.2391 | 7.3300 |
Fixed-pinned | 3.3932 | 4.4633 | 6.5454 | 7.5916 |
Fixed-fixed | 3.9266 | 4.7300 | 7.0686 | 7.8532 |
The vibrational mode shapes only for the first modes of two equal span continuous floor beams with various boundary conditions are illustrated in Figures
First vibrational modes for two equal span continuous timber floor beams with free left ends and various boundary conditions for right ends.
First vibrational modes for two equal span continuous timber floor beams with sliding left ends and various boundary conditions for right ends.
First vibrational modes for two equal span continuous timber floor beams with pinned left ends and various boundary conditions for right ends.
First vibrational modes for two equal span continuous timber floor beams with fixed left ends and various boundary conditions for right ends.
Figure
Figure
Finally, Figure
The shapes for higher modes are more complex and will not be discussed here further.
For continuous floors with two unequal spans (
A continuous timber floor beam with two unequal spans and pinned-pinned end supports.
Solving equations ( Thus, equation (
Solving equations ( Thus, equation (
Combining equations ( Combining equations ( Combining equations ( Combining equations (
A nontrivial solution for equations (
Thus, the frequency equation can be obtained as
From the frequency equations (
Frequency parameters for continuous beams with two unequal spans and pinned-pinned end supports.
|
|
|
|
|
---|---|---|---|---|
0.00001 | 3.9266 | 7.0686 | 10.2101 | 13.3517 |
0.01 | 3.9137 | 7.0456 | 10.1774 | 13.3093 |
0.10 | 3.8143 | 6.8839 | 9.9647 | 13.0537 |
0.20 | 3.7298 | 6.7630 | 9.8118 | 12.8402 |
0.25 | 3.6947 | 6.7132 | 9.7294 | 12.5664 |
0.40 | 3.6070 | 6.5443 | 8.6517 | 10.1140 |
0.50 | 3.5564 | 6.2832 | 7.4295 | 9.8488 |
0.60 | 3.5060 | 5.6983 | 6.9143 | 9.6741 |
0.75 | 3.4167 | 4.7872 | 6.6908 | 8.7697 |
0.80 | 3.3785 | 4.5499 | 6.6399 | 8.3275 |
1.00 | 3.1416 | 3.9266 | 6.2832 | 7.0686 |
First vibrational modes for two unequal span continuous timber floor beams with pinned ends.
When
Figure
Relationships between
This equation can be used to determine the fundamental vibrational frequency,
Sometimes, a linear relationship between
A two-span timber floor is designed for a domestic timber frame building. It is constructed with continuous solid timber joists (Figure
A two-span floor constructed from solid timber joists with fixed left end and pinned right end. (a) A two-span floor constructed with solid timber joists. (b) A two-span floor with fixed left end and pinned right end.
From equation (
A two-span timber floor is designed for an office timber frame building. It has a width
A two unequal span floor constructed from
From equation (
Based on classic vibrational bending theory on beams, comprehensive analytical formulae for dynamic characteristics of two-span continuous timber flooring systems have been established, including frequency equations, modal frequencies, and mode shapes. Four practical boundary conditions are considered for end supports, including free, sliding, pinned, and fixed supports, and a total of sixteen combinations of flooring systems are created.
The characteristic equations for modal frequencies of two equal span continuous beams with various boundary conditions have been deduced, and four sets of mode shapes have been illustrated. A rigid mode exists for free-free end boundary conditions. A full example for fixed-pinned boundary conditions has been presented to show the deduction procedure.
The characteristic equations for modal frequencies of two unequal span continuous beams with pinned-pinned end conditions have been deduced, and one set of the corresponding mode shapes for various span ratios has been illustrated. A full example for pinned-pinned boundary conditions with a general span ratio has been presented to show the deduction procedure. Also, two empirical equations, cubic and linear, for determining the modal frequency parameters with respect to varying span ratios have been proposed and can be used directly for practical design.
Finally, two practical design examples for determining fundamental modal frequencies have been presented, one for a two equal span continuous floor constructed with solid timber joists and fixed-pinned ends and the other for a two unequal span continuous floor constructed with JJI 400D
The data used in this study will be deposited in a repository.
Part of the work was presented at World Conference on Timber Engineering (WCTE 2016), 22–25 August 2016 in Vienna, Austria.
The authors declare that they have no conflicts of interest.
The assistance of Dr. Martin Cullen toward this paper is highly appreciated.