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On the determination of the modulus of elasticity of plasterboard plates
Construction and Building Materials 187 (2018) 923–930
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Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat
On the determination of the modulus of elasticity of plasterboard plates M. Chimeno Manguán a,⇑, E. Roibás Millán c, F. Simón Hidalgo b a
Universidad Politécnica de Madrid, ETSIAE, 28040 Madrid, Spain Universidad Politécnica de Madrid, IDR/UPM, 28040 Madrid, Spain c Consejo Superior de Investigaciones Científicas, ITEFI, 28006 Madrid, Spain b
h i g h l i g h t s Study of standard ISO 16940 and alternative methods for determining the Young’s modulus. Determination of the modulus of elasticity by mobility resonances instead of anti-resonances allows increasing accuracy. Simpler method that allows the better accuracy and uncertainty than ISO 16940.
a r t i c l e
i n f o
Article history: Received 6 October 2017 Received in revised form 13 February 2018 Accepted 1 August 2018 Available online 9 August 2018 Keywords: Modulus of elasticity Young’s modulus Resonances Anti-resonances
a b s t r a c t The importance of an accurate determination of the modulus of elasticity has increased with the development, characterization and manufacturing processes of more complex materials as plasterboards or layered materials. This fact has led to the need of methods beyond the classical stress-strain slope. Several standards can be found describing such procedures in order to determine the Young’s modulus. This work presents an analysis of different methods, including the standard ISO 16940. Several of the methods presented are based on the response of the system to a harmonic force as this standard, but this work also presents a method based on the free response of the tested sample to an initial loading condition. This method involves a significant reduction of the experimental procedure and equipment investment. The methods formulation is presented and illustrated with a numerical case based on a finite element model. The methods presented are also used on an actual case: the determination of the modulus of elasticity of a plasterboard. From both cases, conclusions on the accuracy and precision of the methods are stated. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction Sound transmission of plain building elements depends, among other things, on its bending stiffness, which, in turn, depends on the Young’s modulus (or modulus of elasticity). Traditionally, general tabulated values of this last parameter have been used for ordinary building materials such as brick, concrete, plasterboard or glass. An accurate characterization of the elastic properties of these materials is increasingly important due to the necessity to develop materials that are stronger, stiffer, lighter and even more resistant to extreme environmental conditions in order to assure their performances as well as to guide further developments. This is the case for new products like laminated glasses or some types of plasterboard plates. Nowadays, the use of plasterboards for both structural and nonstructural elements in buildings construction has ⇑ Corresponding author. E-mail address: [email protected] (M. Chimeno Manguán). https://doi.org/10.1016/j.conbuildmat.2018.08.001 0950-0618/Ó 2018 Elsevier Ltd. All rights reserved.
been increased since the properties of these materials, particularly the gypsum, can be modified to meet specific requirements such as impact, fire or humidity resistance [1,2]. A significant effort has been made in the last years to improve specific characteristics, as thermal storage by means of incorporating phase change materials to the plasterboards [3], or mechanical properties using reinforcement materials as fibres of different materials [4,1] (glass, carbon, polypropylene and expanded vermiculite, between others). The accurate determination of the mechanical properties (i.e, Young’s modulus or tensile/compressive strength) of the plasterboards plays a key role in the whole performance of the structure or building, as well as in the previous mechanical studies. For example, numerical models [5] of building components which include plasterboard elements require the definition of plasterboards properties with a confidence level good enough. In addition, their mechanical properties also influence the acoustic performance and modelling of plasterboard components [6] such as internal partitions.
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For plasterboards, there are procedures for determining their mechanical properties [7,8] or particular performances [9] that concern both the plasterboard panels [1] or partitions [10,11]. Nevertheless, very limited studies are available in literature for these type of materials, despite their increasing importance in different areas of civil engineering. Some test procedures are included in references [3,12], however a lack of comprehensive test campaign on plasterboards in the current literature is noted. Numerical studies that are currently available employ values resulting from a small number of tests and the uncertainty related to this material is not usually taken into account. Significant advances have been made recently in the use of computer techniques to model new products manufacturing processes. The continuous development of methods to asses the elastic parameters denotes an actual effort to consider the mechanical behaviour of these materials [13,14]. Traditional static procedures for the determination of the modulus of elasticity [15] require specific equipment not usually found in acoustic laboratories. These methods are based on direct measurements of stresses and strains during mechanical tests, and Young’s moduli are determined from the slope of the linear region of the stress-strain curve. These techniques suffer from certain disadvantages, they are time consuming, expensive (because of their destructive nature) and considerable sample preparation is required [16,2]. On the other hand, the dynamical methods have the advantage of being non-destructive in nature, presenting an ease specimen preparation and being able to be often performed more rapidly, particularly with the availability of modern electronic test equipment. Nevertheless, they usually require sample sizes not big enough to be representative of the bulk building materials. In the last decade progress on digital processing has allowed developing dynamical procedures based on the flexural response of the specimen. As stated above, for acoustical applications, the Young’s modulus arises in the equation of the flexural stiffness modulus that, in turn, is a key quantity to determine the critical frequency by which the profile of the sound insulation curve of building elements is strongly influenced [17,18]. It also appears, for instance, in the equations of the vibration reduction indices of some types of wall junctions [19]. Thus, in the last decade, there have been published some measurement standards that, specifically, include guidelines for the determination of this quantity for building elements, as in [20] or [21, Appendix F]. In particular, for non isotropic materials as plasterboards, these standards establish also a procedure based on the average of the modulus of elasticity for several samples obtained in the longitudinal and transversal directions of a plate, thus obtaining an equivalent single value. The aim of these standards is to determine the modulus of elasticity analysing the response of the system in the same location where a harmonic force is applied to the sample. Such response can be studied in terms of mobility (vibration velocity divided by force) or its inverse, the impedance (force divided by vibration velocity). The characteristic values of such responses are their maximum and minimum values that can be theoretically related to the modulus of elasticity and so be used to its determination. The terms resonance and anti-resonance frequencies are usually used to name the mobility maxima and minima respectively, as it will be used in this work. Nevertheless, it is worth to highlight that mobility resonances are the impedance anti-resonances and that the mobility anti-resonances are the impedance resonances. The purpose of this study is to present a discussion about different methods for dynamic elastic properties determination of plasterboard elements taking as a reference the one included in [20] (together with [21]) emphasizing aspects such as its accuracy, simplicity, or equipment needed. The study is based on the determination of the modulus of elasticity from two different
sources: fundamental bending resonances or anti-resonances of the response to a harmonic excitation and the free response to an initial loading condition. Section 2 describes the fundamentals of the methods presented along with a description of [20]. A numerical application is studied in Section 3 to show the differences between the methods in terms of accuracy and precision. In Section 4 an experimental application with plasterboard specimens is considered. The conclusions from the work are summarised in Section 5. 2. Methods This section presents several methods for the determination of the bending modulus of materials. First, the ISO 16940 standard is summarised as this is one of the methodologies more widely accepted for the determination of the bending modulus. Later, three alternative methods are presented: two of them are based on the stationary response of a sample to a harmonic force (the same type of response studied in the ISO 16940) and the last one is based on the free response of the specimen. 2.1. The ISO 16940 procedure The specific aim of the ISO 16940 standard is the determination of the loss factor and bending stiffness modulus of glass beams. In laminated glass the interlayer plays a main role in the mechanical properties of the glazing and, as a consequence, in its acoustic properties. This way different interlayers can be compared, enabling the development of glazings with better sound insulation performances. Bending stiffness determination is based on the impedance response of a free-free beam excited by a driven force in its midpoint, where the vibration velocity is also measured. The standard establishes a relationship between each impedance resonance freISO
quency f ai and the bending stiffness as
ðkISO ai Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EI
2p
mðL=2Þ4
2
ISO f ai
¼
i ¼ 1; 2; 3 . . .
ð1Þ
where E is the modulus of elasticity, I is the area moment of the cross-section about the neutral axis, m is the mass per unit length and kISO ai is a parameter whose values are listed in Table 1. Solving for Ei :
0 12 m @2pðL=2Þ2 ISO A Ei ¼ f ai 2 I ðkISO Þ
ð2Þ
ai
The standard recommends using the first three resonance frequencies to obtain a mean E value. The values of parameter kISO ai shown in Table 1 come from an assumption in the standard that establishes that the vibration modes of the sample loaded in its middle point correspond to the bending modes of two ‘‘clamped-free” half length beams. Therefore it is established in the standard that impedance resonance frequencies—the mobility anti-resonances—of the free-free
Table 1 Values of the parameter kISO ai . i
kISO ai
1 2 3 4
1.87510 4.69410 7.85476 10.99554
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beam match the eigenfrequencies of a ‘‘clamped-free” half length beam.
Table 2 Values of parameter kFF i for eigenfrequencies of a free-free beam.
2.2. Anti-resonances of a free-free beam method This method is straightforwardly derived from the standard ISO 16940 [20] but unlike this one (that assimilates the response to that one of a half length ‘‘clamped-free” beam) it is derived from the analytical anti-resonances of the free-free beam. The displacement of a beam with free-free conditions between x ¼ 0 and x ¼ L by means of a continuous model can be found in textbooks as [22] and it can be expressed by modal superposition as
qðx; tÞ ¼
1 X
gi ðtÞ/i ðxÞ
kFF i
1 2 3 4 5
4.73004 7.85321 10.99561 14.13717 17.27876
Table 3 Values of the parameter kFF ar i for the anti-resonances of a free-free beam.
ð3Þ
i¼0
where gi ðtÞ are the modal coordinates and /i ðxÞ the normalized eigenmodes, considering i ¼ 0 as the rigid-body motion of the beam and modes i > 0 the flexural modes of the beam. The first flexural mode shapes (with unitary maximum) are illustrated in Fig. 1. The beam bending eigenfrequencies can be expressed as
sffiffiffiffiffiffiffiffiffi 2 xi ðkFF Þ EI fi ¼ ¼ i 2p 2p mL4
i
N
i
kFF ari
1
1
3.77003
6
1 2 3 4
3.75038 9.39740 15.73438 22.05536
7
1 2 3 4
3.75030 9.39504 15.72371 22.02538
ð4Þ
where parameter kFF i satisfies
cos kFF cosh kFF ¼1 i i
ð5Þ
First solutions are shown in Table 2. The response of the system to a harmonic load of intensity p0 and circular frequency X applied at x0 ; pðx; tÞ ¼ p0 dðx x0 ÞeiXt is determined by the response of the beam in normalized modal coordinates
g€ i þ x2i gi ¼ p0 /i ðx0 Þ
ð6Þ
For the case under analysis the load is applied in the centre of the beam (x0 ¼ L=2). As shown in Fig. 1, /i ðx0 Þ ¼ 0 for even i as they are antisymmetric functions respect to x ¼ L=2. Therefore the response of the system is
qðx; tÞ ¼ p0
" 1 X /2i1 ðx0 Þ/2i1 ðxÞ i¼1
x22i1 X2
1 mLX2
2 2 i¼1 x2i1 X
1 mLX2
f ari ¼
eiXt
ð7Þ
¼0
ðkFF ar i Þ
sffiffiffiffiffiffiffiffiffi EI
2
2p
mL4
ð9Þ
values of N. Then, from the experimentally determined anti-resonances of a beam at its middle point, the modulus of elasticity can be calculated from each anti-resonance as
EFF ar i
2 32 m4 2pL2 5 f ari ¼ 2 I ðkFF Þ
ð10Þ
ar i
ð8Þ
2.3. Resonances of a free – free beam method As shown in the previous section, the value of the numerical anti-resonance frequencies are dependant on the number of modes considered in Eq. (8). To avoid this dependency, a method based on the resonance frequencies of the response (that depends only on the corresponding eigenfrequency) is considered. Under the same conditions as above—free-free beam with harmonic load applied on the middle point—the response of the beam at the centre is due to odd i modes as even i modal shapes are null at that point as they are antisymmetric functions respect to that point as illustrated in Fig. 1. Therefore, the response of the beam at the centre will show resonance frequencies corresponding only to the odd i modes. Hence, the resonance frequencies are 2
FF f ri
Fig. 1. First four flexural mode shapes normalised to unitary maximum.
i ¼ 1; 2; 3 . . .
for parameter kFF ar i whose values are shown in Table 3 for different
#
where the term corresponding to the rigid-body motion of the beam has been excluded from the summation. Anti-resonances in the response of the system at the point of excitation are the solutions of qðx0 ; tÞ ¼ 0. Then, anti-resonances of the beam at its middle point are the circular frequencies X that satisfy 1 X /2i1 ðx0 Þ2
This equation can be solved for a given number of bending modes, N, determining a certain number of anti-resonances, f ari , that can be expressed in the same form as the eigenfrequencies
ðkFF Þ ¼ 2i1 2p
sffiffiffiffiffiffiffiffiffi EI mL4
i ¼ 1; 2; 3 . . .
ð11Þ
for the values of the parameter kFF i shown in Table 2. Then, from the experimentally determined resonances of the beam at its middle point, the modulus of elasticity can be calculated from the resonance frequencies as
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EFF ri
i
kCF i
1 2 3 4
1.87510 4.69409 7.85476 10.99554
" #2 m 2pL2 f ¼ I ri ðkFF Þ2 2i1
i ¼ 1; 2; 3 . . .
ECF i ¼
ð12Þ
Previous methods imply to perform a dynamic test on an actual specimen, that is, to apply a harmonic force on the specimen (usually through an electromagnetic shaker) and to measure both such force and the specimen response. Free response of a cantilever beam method is conceived to avoid the need of an externally applied force and it is based on the free response of a cantilever beam to a static load on its free end followed by a quick release. The displacement of a cantilever beam of mass per unit length m, modulus of elasticity, E, area moment of inertia of the crosssection about the neutral axis I and length L can be expressed by ~ superposition of the normalized modes /ðxÞ as 1 X
gi ðtÞ/~ i ðxÞ
ð13Þ
i¼1
The eigenfrequencies of this case can be expressed also in terms of a parameter kCF i that satisfies
cos kCF cosh kCF ¼ 1 i i
ð14Þ
whose first solutions are shown in Table 4. Modal coordinates are gi ðtÞ ¼ Ai cosð2pf i tÞ þ Bi sinð2pf i tÞ where constants Ai and Bi are determined by the initial conditions in dis_ 0Þ respectively. For an static placement qðx; 0Þ and velocity qðx; load of unitary amplitude applied on the free end of the beam fol_ 0Þ ¼ 0, then Bi ¼ 0. Ai can be defined lowed by a quick release, qðx; from the static response of the cantilever beam [23]
qðx; 0Þ ¼
1 Lx2 x3 EI 2 6
ð16Þ
3. Numerical application
2.4. Free response of a cantilever beam method
qðx; tÞ ¼
" #2 m 2pL2 fi 2 I ðkCF i Þ
!
The methods proposed are applied on a numerical case along with the methodology proposed in the standard ISO 16940 [20]. The different tests associated to each method are simulated numerically with a one-dimensional FE model solved with MSC Nastran [24] using beam elements (CBAR). Two beams are considered: a free-free beam of length 0.5 m and a cantilever beam of length 0.25 m, both 0.06 m wide and 0.005 m thick. They both are made up of aluminum alloy 5754 with a modulus of elasticity of 68 GPa, Poisson’s ratio 0.33 and a density of 2660 kg/m3. A modal damping of 5% of the critical one is assumed. The mesh size is set to ensure 10 elements per wavelength up to 4500 Hz. For the mechanical and area properties stated, the bending wavelength at said frequency is 0.1 m, therefore the element size is set to 0.01 m. 3.1. Simulated test results The mobility of the free-free beam in the middle point to a unitary force is calculated considering a frequency resolution of 0.1 Hz from 0 Hz to 4000 Hz as depicted in Fig. 2. The first resonance frequencies are 103.9 Hz, 561.0 Hz and 1382.3 Hz while the first antiresonance frequencies are 65.3 Hz, 408.8 Hz and 1141.9 Hz. The transitory response of the cantilever beam for the application of method described in Section 2.4 is also simulated assuming a perfect clamped boundary condition. The time response of the free end of the beam is shown in Fig. 3. The discrete Fourier transform [25] is calculated with a frequency resolution of 0.1 Hz and its root-mean-square values are shown in Fig. 4 whose first resonance peak corresponds to 65.3 Hz. As described in Section 2.4 and observed in Fig. 4 only one resonance frequency can be determined adequately. 3.2. Determination of the modulus of elasticity From the resonance and anti-resonance values obtained with the FE model, the methods proposed in Section 2 are applied along with the method described in standard ISO 16940 [20] as reference.
ð15Þ
Coefficients Ai decrease with i as Table 5 shows for the first coefficients (normalized to the one corresponding to the fundamental harmonic) that shows that the response of the system is mainly due to the fundamental mode as the contribution of the rest of modes is at least two orders of magnitude lower. For an arbitrary initial displacement (large enough to measure the response of the system but within the linear response regime), the Fourier Transform of the response can be used to determine the first eigenfrequencies of the beam, f i , and the corresponding modulus of elasticity.
Table 5 Coefficients of the modal displacements normalised with the first one. i
j Ai =A1 j
1 2 3 4 5
1 0.025461 0.003249 0.000834 0.000310
Fig. 2. FEM mobility of the middle point of the free-free beam to a unitary load in the same location.
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Table 6 Value of the modulus of elasticity obtained from each resonance or anti-resonance, mean value and standard deviation through each method: ISO 16940, anti-resonances of a free-free beam method, resonances of free-free beam method and free response of a cantilever beam method. i
EISO (GPa)
EFF ar (GPa)
EFF r (GPa)
ECF (GPa)
1 2 3
67.916 67.773 67.448
67.909 67.576 67.205
67.941 67.828 67.533
67.916
l r
67.712 0.240
67.563 0.352
67.768 0.210
67.916
Table 7 Error in percentage of the values of the modulus of elasticity obtained from each resonance or anti-resonance for each method: ISO 16940, anti-resonances of a freefree beam method, resonances of free-free beam method and free response of a cantilever beam method.
Fig. 3. FEM displacement of the cantilever beam free end after a static load followed by a quick release.
i
EISO
EFF ar
EFF r
ECF
1 2 3
0.12% 0.33% 0.81%
0.13% 0.62% 1.17%
0.09% 0.25% 0.69%
0.12%
l
0.42%
0.64%
0.34%
0.12%
Table 8 Values of the parameter kISO ai referred to the whole length of the free-free beam compared to parameter kFF ar i . kISO ai 2
kFF ar i
Diff. (%)
3.75020 9.38818 15.70952 21.99100
3.75030 9.39504 15.72371 22.02538
0.003 0.073 0.090 0.156
and anti-resonance frequencies of the simulated tests and therefore of the modulus of elasticity obtained from them. Standard ISO 16940 is based on considering equal the response of the tested free-free beam to the response of a half-length clamped-free beam. In fact, it approximates the anti-resonances of the free-free beam by the resonances of the clamped-free beam. This approximation can be analysed taking the Standard values kISO ai Fig. 4. Discrete Fourier transform of the FEM displacement of the cantilever beam free end.
Standard ISO 16940 and the method described in Section 2.2 take into account the mobility anti-resonances while method described in Section 2.3 takes into account the mobility resonance frequencies. Method described in Section 2.4 takes into account the eigenfrequency determined by means of the DFT of the free response. The value of the modulus of elasticity obtained by means of each method is shown in Table 6 for each resonance or antiresonance considered as well as its mean and standard deviation except for the free response of a cantilever beam as it has only one resonance, then it is not necessary to calculate the mean and standard deviation values. The error of the results respect to the nominal value (68 GPa) is shown in Table 7. 3.3. Discussion Results show that all methods provide values of the modulus of elasticity close to the actual value (maximum error of 1:17%) although a bias to lower values is found. This is due to not considering any damping in the methods formulation unlike in the numerical model. This fact leads to lower values of the resonance
(Table 1) that are defined for a beam of half length of the tested one and refer them to the full length (in practice it implies that the values are doubled). Now, these values can be compared to the parameter values of the anti-resonances determined in Section 2.2, kFF ari . Both magnitudes and their difference are shown in Table 8. The comparison shows that the assumption made in Standard ISO 16940 is adequate although it adds a bias to lower values in addition to the one related to the damping as stated above. Both the ISO 16940 standard and the method described in Section 2.2—anti-resonances of a free-free beam method—calculate the modulus of elasticity from the anti-resonance of the response and as discussed in Section 2.2, its formulation depends on the number of modes considered to contribute to the system response. Unlike these methods, method described in Section 2.3—resonances of a free-free beam method—determines the resonance frequency only from the eigenfrequencies of the system, providing a much lower error compared to both former ones as shown in Table 7. The method based on the free response of the clamped-free beam, Section 2.4, provides a lower error than the methods based on the anti-resonances of the response. This is especially important if the actual application of the methods is taken into account as the testing procedure and requirements of this method are much less cumbersome than the other ones. For instance, it does not require
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Table 9 Dimensions and density of the samples. Sample
Length (m)
Width (m)
Thickness (m)
Density (kg/m3)
1 2 3 4 5 6
0.599 0.599 0.599 0.600 0.600 0.599
0.0600 0.0609 0.0605 0.0613 0.0600 0.0600
0.0124 0.0125 0.0124 0.0124 0.0125 0.0126
646.2 626.3 640.6 654.8 652.4 651.1
to apply any load to the system so no electromagnetic shaker is needed, leading to a much lower cost in terms of equipment. 4. Experimental application Standard ISO 16940 [20] and the three methods proposed are applied to determine the modulus of elasticity of a plasterboard. As it is an anisotropic material, several samples in longitudinal and transversal directions of a plate are considered [21]: three samples in the longitudinal direction (samples 1–3) and three in the transversal direction (samples 4–6). The dimensions and density of the six samples are listed in Table 9. 4.1. Determination of resonances and anti-resonances A test campaign is performed following the measurement setup described in Standard ISO 16940, Annex A [20] to determine the resonance and anti-resonances of the free-free beam. A harmonic force is applied at the centre of the sample as shown in Fig. 5. The force applied and the acceleration at the excitation point are measured to determine the sample mobility. The load is applied by means of a LDS V406 shaker while the force and acceleration are measured through a PCB 208C02 force sensor and a PCB 352C33 accelerometer respectively. Transducer measurements are acquired through an OROS OR36 analyzer/recorder. The results obtained with the set of samples are depicted in Fig. 6, from which the first resonances and anti-resonances can be determined. Their values are shown in Tables 10 and 11 including their means by material direction, overall mean and standard uncertainty (u) [26]. The test required for the method proposed in Section 2.4, based on the transient response of the beams, is performed six times per each one of the six samples considered clamping one end as depicted in Fig. 7. The response at the free end is measured with a PCB 352C33 accelerometer acquired with an OROS OR36 analyzer/recorder. The test consists of shifting and then releasing the free end of the sample. The first resonance of each cantilever beam are determined. They are listed in Table 12.
Fig. 6. Mobility measured in the centre of the six samples of plasterboard.
Table 10 First three resonance frequencies determined from the mobility measured in the centre of the samples, overall and by material direction means and standard uncertainty. Sample
f r1 (Hz)
f r2 (Hz)
f r3 (Hz)
1 2 3
70.3 65.6 71.1 69.0
370.3 350.8 376.6 365.9
909 864 926 900
71.1 61.7 63.3 65.4
376.6 330.5 333.6 346.9
928 809 826 854
67.2 1.7
356.4 8.6
877 21
lL 4 5 6
lT l u
Table 11 First three anti-resonance frequencies determined from the mobility measured in the centre of the samples, overall and by material direction means and standard uncertainty.
1 2 3
lL 4 5 6
lT l u
Sample
f ar1 (Hz)
f ar2 (Hz)
41.4 38.3 42.2 40.6
289.8 271.1 293.0 284.6
806 754 810 790
42.2 35.2 36.7 38.0
293.8 257.8 264.1 271.9
816 714 734 755
39.3 1.2
278.3 6.5
772 18
4.2. Determination of the modulus of elasticity
Fig. 5. Measurement set up for measuring the mobilitiy in the center of the plasterboard samples.
The three methods proposed are applied considering antiresonances (ISO 1940 and the method described in Section 2.2), resonances (method described in Sections 2.3) and first eigenfrequency (method described in Section 2.4). Tables 13–16 list for each method the value of the modulus of elasticity determined from each frequency value Ei , along with the mean value per eigenfrequency and its standard uncertainty.
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Table 14 Modulus of elasticity determined by the anti-resonances of a free-free beam method for each anti-resonance, overall and by material direction means and standard uncertainty. Sample
E1 (GPa)
E2 (GPa)
E3 (GPa)
1 2 3
2.22 1.81 2.29 2.11 2.35 1.60 1.71 1.89 2.00 0.13
2.76 2.31 2.80 2.62 2.90 2.19 2.24 2.44 2.53 0.13
2.73 2.27 2.73 2.58 2.85 2.14 2.21 2.40 2.49 0.13
lL 4 5 6
lT l u
Table 15 Modulus of elasticity determined by the resonances of a free-free beam method for each resonance, overall and by material direction means and standard uncertainty.
Fig. 7. Measurement set up for measuring the free response of samples to a static load followed by a quick release.
Sample
E1 (GPa)
E2 (GPa)
E3 (GPa)
1 2 3
2.53 2.10 2.57 2.40 2.64 1.95 2.00 2.20 2.30 0.13
2.41 2.06 2.47 2.31 2.54 1.92 1.90 2.12 2.21 0.12
2.38 2.05 2.44 2.29 2.53 1.88 1.91 2.11 2.20 0.12
lL 4 5 6
Table 12 First eigenfrequencies determined from the discrete Fourier transform of the response to a static load followed by a quick release on the free end with the other end clamped, overall and by material direction means and standard uncertainty. Sample
f1
1 2 3
10.57 10.76 10.91 10.75
lL 4 5 6
lT l u
Table 16 Modulus of elasticity determined by the free response of a cantilever beam method, overall and by material direction means and standard uncertainty.
10.09 9.35 10.24 9.89
lT l
10.32 0.23
u
E1 (GPa)
E2 (GPa)
E3 (GPa)
1 2 3
2.22 1.81 2.29 2.11
2.77 2.31 2.81 2.63
2.74 2.28 2.74 2.59
2.35 1.60 1.71 1.89
2.91 2.19 2.25 2.45
2.86 2.15 2.22 2.41
2.00 0.13
2.54 0.13
2.50 0.13
lT l u
1 2 3
2.317 2.288 2.445 2.350 2.155 1.813 2.120 2.029 2.190 0.089
4 5 6
Sample
4 5 6
E1 (GPa)
lL
lT l
Table 13 Modulus of elasticity determined applying standard ISO 16940 for each antiresonance, overall and by material direction means and standard uncertainty.
lL
Sample
The final value of the modulus of elasticity determined by each method is the average of the value for each sample and eigenfrequency following [21]. These values and their standard uncertainty are listed in Table 17. 4.3. Discussion Experimental results obtained from the plasterboard show a clear dependency on the material direction as expected in this type
u
Table 17 Modulus of elasticity determined by means of each method: overall mean value and standard uncertainty. Method
E (GPa)
u (GPa)
Standard ISO 16940 (Section 2.1) Anti-resonances of a free-free beam method (Section 2.2) Resonances of a free-free beam method (Section 2.3) Free response of a cantilever beam method (Section 2.4)
2.34 2.34 2.24 2.190
0.13 0.13 0.12 0.089
of materials. Higher value of the modulus of elasticity is found for samples taken in the longitudinal direction (samples 1–3) than those for samples in the transversal direction (4–6). For these materials, usually, the overall mean is considered the modulus of elasticity of the material [21] although, due to their orthotropic and heterogeneous nature considering statistical properties [2] should be considered. Also, for final construction elements, as partitions, the directional values should be taken into account when considering the mounting direction and mounting conditions [10,11].
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For methods based in anti-resonances (Tables 13 and 14) a consistent lower value is found for the results related to the first anti-resonance. To analyse this result, the modal damping for each sample has been determined [27] from the mobility measured depicted in Fig. 6. For all the samples, a much higher modal damping is found for the first mode in free-free conditions. For instance, for sample 1, first mode damping is 15% of the critical damping compared to the second and third ones that are 3% and 2% of the critical damping, respectively. The high damping of the first mode has the highest influence on the first anti-resonance and the first resonance of the free-free response [28]. Given that the methods studied in this work are undamped, determination of the modulus of elasticity from highly damped modes should be done with caution, as they lead to underestimated values. Nevertheless, regarding the results between the different methods studied, values of the Young’s modulus shown in Tables 13–17 show the same trends identified in the results from the numerical case discussed in Section 3.3. Regarding the differences between methods based on anti-resonances and resonances, for this case it is also found that the method based on the resonances of a free-free beam method is characterised by a higher precision than those ones based on anti-resonances (ISO 16940 and the method based on the anti-resonances of the free-free beam). Since the accuracy of the method based on the free response of the clamped-free beam is similar to the ones of the other methods, it can be concluded that it is an efficient method to determine the modulus of elasticity without the need to have an electromagnetic shaker available or another means of applying a harmonic force as standard ISO 16940 requires. 5. Conclusions In this paper a study on methods to determine the Young’s modulus of materials has been presented. Methods are based on different approaches: standard ISO 16940 and some of the methods presented are based on the mobility of a specimen under a harmonic load and one of the methods proposed is based on the free response of the sample to an initial loading condition. Within the first group, the study shows that determining the modulus of elasticity from the resonance frequencies of the response leads to a higher accuracy and lower spread as it is shown by the study of a numerical and an experimental case. Also, it has been shown that analysing the mobility anti-resonances of a freefree beam by means of the eigenfrequencies of a half-length clamped-free beam adds a small bias to the determination of the modulus of elasticity towards higher values. In addition to methods based on the response to a harmonic load, that require in practice a device to apply such a force, a method based on the free response of the sample has been presented. The results of the numerical application show that this method allows determining the Young’s modulus with the same level of accuracy that those methods based on anti-resonances with a much simpler and affordable testing procedure. Regarding its uncertainty, results of the experimental application show that it is lower that the other methods one. Finally, it is worth mentioning that, according to these results, when the sample material has a high damping value, it would be advantageous to drop the first anti-resonance from the determination after procedures based on anti-resonances as the ISO procedure and the method based on the anti-resonances of a free-free beam.
Acknowledgements This work as been supported by the Spanish Agencia Estatal de Investigación under grant DPI2016-79559-R (AEI/FEDER, UE) References [1] O. Gencel, J.J. del Coz Díaz, M. Sutcu, F. Koksal, F.A. Rabanal, G. MartínezBarrera, W. Brostow, Properties of gypsum composites containing vermiculite and polypropylene fibers: numerical and experimental results, Energy Build. 70 (2014) 135–144. [2] C. Petrone, G. Magliulo, G. Manfredi, Mechanical properties of plasterboards: experimental tests and statistical analysis, J. Mater. Civil Eng. 28 (11) (2016) 135–144. [3] A. Oliver-Ramírez, A. García-Santos, F.J. Neila-González, Physical and mechanical characterization of gypsum boards containing phase change materials for latent heat storage, Mater. Constr. 61 (303) (2011) 465–484. [4] A.H.H. Al-Masoodi, A. Kawan, M. Kasmuri, R. Hamid, M.N.N. Khan, Static and dynamic properties of concrete with different types and shapes of fibrous reinforcement, Constr. Build. Mater. 104 (2016) 247–262. [5] A. Asiz, Y.H. Chui, G. Doudak, C. Ni, M. Mohammad, Contribution of plasterboard finishes to structural performance of multi-story light wood frame buildings, Proc. Eng. 14 (1) (2011) 1572–1581. [6] A. Arjunan, C.J. Wang, K. Yahiaoui, D.J. Mynors, T. Morgan, M. English, Finite element acoustic analysis of a steel stud based double-leaf wall, Build. Environ. 67 (1) (2013) 202–210. [7] British Standard, EN 520: Gypsum Plasterboards: Definitions, Requirements and Test Methods, 2004. [8] EN 13279-2 Gypsum Binders and Gypsum Plasters – Part 2: Test Methods, 2014. [9] FEMA 461 Interim Testing Protocols for Determining the Seismic Performance 479 Characteristics of Structural and Nonstructural Components, 2007. [10] C. Petrone, G. Magliulo, P. Lopez, G. Manfredi, Seismic fragility of plasterboard partitions via inplane quasistatic tests, Earthq. Eng. Struct. Dyn. 44 (14) (2015) 2589–2606. [11] C. Petrone, G. Magliulo, P. Lopez, G. Manfredi, Out-of-plane seismic performance of plasterboard partitions via quasi-static tests, Bull. New Zealand Soc. Earthq. Eng. 49 (1) (2016) 125–137. [12] A. Bouterf, S. Roux, F. Hild, G. Vivier, X. Brajer, E. Maire, S. Meille, Damage law identification from full field displacement measurement: application to fourpoint bending test for plasterboard, Eur. J. Mech. A/Solids 49 (1) (2015) 60–66. [13] M.Z. Hossain, A.S.M.A. Awal, Experimental validation of a theoretical model for flexural modulus of elasticity of thin cement composite, Constr. Build. Mater. 25 (3) (2011) 1460–1465. [14] Z. Wang, Z. Gao, Y. Wang, Y. Cao, G. Wang, B. Liu, Z. Wang, A new dynamic testing method for elastic, shear modulus and Poisson’s ratio of concrete, Constr. Build. Mater. 100 (Suppl. C) (2015) 129–135. [15] CEN (European Comitee for Standarization), EN 789, Determination of mechanical properties of wood-based panels, Brussels, Belgium, 2004. [16] A. Wolfender (Ed.), Dynamic Elastic Modulus Measurements in Materials, ASTM, Philadelphia, PA, 1990. [17] F.J. Fahy, P. Gardonio, Sound and Structural Vibration Radiation, Transmission and Response, second ed., Academic Press, 2007. [18] C. Martins, P. Santos, P. Almeida, L. Godinho, A. Dias, Acoustic performance of timber and timber-concrete floors, Constr. Build. Mater. 101 (Part 1) (2015) 684–691. [19] ISO 12354-1 Building Acoustics—Estimation of Acoustic Performance of Buildings From the Performance of Elements—Part 1: Airborne Sound Insulation Between Rooms, 2017. [20] ISO 16940 Glass in Building—Glazing and Airborne Sound Insulation— Measurement of the Mechanical Impedance of Laminated Glass, 2008. [21] EN 16703 Acoustics—Test Code for Drywall Systems of Plasterboard with Steel Studs—Airborne Sound Insulation, 2015. [22] A.A. Shabana, Theory of Vibration. Vol. II. Discrete and Continous Systems, Springer-Verlag, New York, 1991. [23] S. Timoshenko, Strength of Materials, Vol. I: Elementary theory and Problems, CBS Publishers & Distributors, 2004. [24] Nastran, 2014. MSChttp://www.mscsoftware.com/product/msc-nastran. [25] G. Heinzel, A. Rüdiger, R. Schilling, Spectrum and spectral density estimation by the Discrete Fourier transform (DFT), including a comprehensive list of window functions and some new flat-top windows, 2002. [26] ISO/IEC Guide 98-3:2008, Uncertainty of Measurement – Part 3: Guide to the Expression of Uncertainty in Measurement (GUM:1995). [27] D.K. Anthony, F. Simon, J. Juan, Determining material damping type by comparing modal frequency estimators, J. Acoust. Soc. Am. 126 (3) (2009) EL86–EL92. [28] D.K. Anthony, F. Simon, Accuracy and robustness of four basic single degree of freedom methods for determining the modal parameters of non-lightly damped systems, J. Sound Vib. 331 (2012) 5191–5208.
Contents lists available at ScienceDirect
Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat
On the determination of the modulus of elasticity of plasterboard plates M. Chimeno Manguán a,⇑, E. Roibás Millán c, F. Simón Hidalgo b a
Universidad Politécnica de Madrid, ETSIAE, 28040 Madrid, Spain Universidad Politécnica de Madrid, IDR/UPM, 28040 Madrid, Spain c Consejo Superior de Investigaciones Científicas, ITEFI, 28006 Madrid, Spain b
h i g h l i g h t s Study of standard ISO 16940 and alternative methods for determining the Young’s modulus. Determination of the modulus of elasticity by mobility resonances instead of anti-resonances allows increasing accuracy. Simpler method that allows the better accuracy and uncertainty than ISO 16940.
a r t i c l e
i n f o
Article history: Received 6 October 2017 Received in revised form 13 February 2018 Accepted 1 August 2018 Available online 9 August 2018 Keywords: Modulus of elasticity Young’s modulus Resonances Anti-resonances
a b s t r a c t The importance of an accurate determination of the modulus of elasticity has increased with the development, characterization and manufacturing processes of more complex materials as plasterboards or layered materials. This fact has led to the need of methods beyond the classical stress-strain slope. Several standards can be found describing such procedures in order to determine the Young’s modulus. This work presents an analysis of different methods, including the standard ISO 16940. Several of the methods presented are based on the response of the system to a harmonic force as this standard, but this work also presents a method based on the free response of the tested sample to an initial loading condition. This method involves a significant reduction of the experimental procedure and equipment investment. The methods formulation is presented and illustrated with a numerical case based on a finite element model. The methods presented are also used on an actual case: the determination of the modulus of elasticity of a plasterboard. From both cases, conclusions on the accuracy and precision of the methods are stated. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction Sound transmission of plain building elements depends, among other things, on its bending stiffness, which, in turn, depends on the Young’s modulus (or modulus of elasticity). Traditionally, general tabulated values of this last parameter have been used for ordinary building materials such as brick, concrete, plasterboard or glass. An accurate characterization of the elastic properties of these materials is increasingly important due to the necessity to develop materials that are stronger, stiffer, lighter and even more resistant to extreme environmental conditions in order to assure their performances as well as to guide further developments. This is the case for new products like laminated glasses or some types of plasterboard plates. Nowadays, the use of plasterboards for both structural and nonstructural elements in buildings construction has ⇑ Corresponding author. E-mail address: [email protected] (M. Chimeno Manguán). https://doi.org/10.1016/j.conbuildmat.2018.08.001 0950-0618/Ó 2018 Elsevier Ltd. All rights reserved.
been increased since the properties of these materials, particularly the gypsum, can be modified to meet specific requirements such as impact, fire or humidity resistance [1,2]. A significant effort has been made in the last years to improve specific characteristics, as thermal storage by means of incorporating phase change materials to the plasterboards [3], or mechanical properties using reinforcement materials as fibres of different materials [4,1] (glass, carbon, polypropylene and expanded vermiculite, between others). The accurate determination of the mechanical properties (i.e, Young’s modulus or tensile/compressive strength) of the plasterboards plays a key role in the whole performance of the structure or building, as well as in the previous mechanical studies. For example, numerical models [5] of building components which include plasterboard elements require the definition of plasterboards properties with a confidence level good enough. In addition, their mechanical properties also influence the acoustic performance and modelling of plasterboard components [6] such as internal partitions.
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For plasterboards, there are procedures for determining their mechanical properties [7,8] or particular performances [9] that concern both the plasterboard panels [1] or partitions [10,11]. Nevertheless, very limited studies are available in literature for these type of materials, despite their increasing importance in different areas of civil engineering. Some test procedures are included in references [3,12], however a lack of comprehensive test campaign on plasterboards in the current literature is noted. Numerical studies that are currently available employ values resulting from a small number of tests and the uncertainty related to this material is not usually taken into account. Significant advances have been made recently in the use of computer techniques to model new products manufacturing processes. The continuous development of methods to asses the elastic parameters denotes an actual effort to consider the mechanical behaviour of these materials [13,14]. Traditional static procedures for the determination of the modulus of elasticity [15] require specific equipment not usually found in acoustic laboratories. These methods are based on direct measurements of stresses and strains during mechanical tests, and Young’s moduli are determined from the slope of the linear region of the stress-strain curve. These techniques suffer from certain disadvantages, they are time consuming, expensive (because of their destructive nature) and considerable sample preparation is required [16,2]. On the other hand, the dynamical methods have the advantage of being non-destructive in nature, presenting an ease specimen preparation and being able to be often performed more rapidly, particularly with the availability of modern electronic test equipment. Nevertheless, they usually require sample sizes not big enough to be representative of the bulk building materials. In the last decade progress on digital processing has allowed developing dynamical procedures based on the flexural response of the specimen. As stated above, for acoustical applications, the Young’s modulus arises in the equation of the flexural stiffness modulus that, in turn, is a key quantity to determine the critical frequency by which the profile of the sound insulation curve of building elements is strongly influenced [17,18]. It also appears, for instance, in the equations of the vibration reduction indices of some types of wall junctions [19]. Thus, in the last decade, there have been published some measurement standards that, specifically, include guidelines for the determination of this quantity for building elements, as in [20] or [21, Appendix F]. In particular, for non isotropic materials as plasterboards, these standards establish also a procedure based on the average of the modulus of elasticity for several samples obtained in the longitudinal and transversal directions of a plate, thus obtaining an equivalent single value. The aim of these standards is to determine the modulus of elasticity analysing the response of the system in the same location where a harmonic force is applied to the sample. Such response can be studied in terms of mobility (vibration velocity divided by force) or its inverse, the impedance (force divided by vibration velocity). The characteristic values of such responses are their maximum and minimum values that can be theoretically related to the modulus of elasticity and so be used to its determination. The terms resonance and anti-resonance frequencies are usually used to name the mobility maxima and minima respectively, as it will be used in this work. Nevertheless, it is worth to highlight that mobility resonances are the impedance anti-resonances and that the mobility anti-resonances are the impedance resonances. The purpose of this study is to present a discussion about different methods for dynamic elastic properties determination of plasterboard elements taking as a reference the one included in [20] (together with [21]) emphasizing aspects such as its accuracy, simplicity, or equipment needed. The study is based on the determination of the modulus of elasticity from two different
sources: fundamental bending resonances or anti-resonances of the response to a harmonic excitation and the free response to an initial loading condition. Section 2 describes the fundamentals of the methods presented along with a description of [20]. A numerical application is studied in Section 3 to show the differences between the methods in terms of accuracy and precision. In Section 4 an experimental application with plasterboard specimens is considered. The conclusions from the work are summarised in Section 5. 2. Methods This section presents several methods for the determination of the bending modulus of materials. First, the ISO 16940 standard is summarised as this is one of the methodologies more widely accepted for the determination of the bending modulus. Later, three alternative methods are presented: two of them are based on the stationary response of a sample to a harmonic force (the same type of response studied in the ISO 16940) and the last one is based on the free response of the specimen. 2.1. The ISO 16940 procedure The specific aim of the ISO 16940 standard is the determination of the loss factor and bending stiffness modulus of glass beams. In laminated glass the interlayer plays a main role in the mechanical properties of the glazing and, as a consequence, in its acoustic properties. This way different interlayers can be compared, enabling the development of glazings with better sound insulation performances. Bending stiffness determination is based on the impedance response of a free-free beam excited by a driven force in its midpoint, where the vibration velocity is also measured. The standard establishes a relationship between each impedance resonance freISO
quency f ai and the bending stiffness as
ðkISO ai Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EI
2p
mðL=2Þ4
2
ISO f ai
¼
i ¼ 1; 2; 3 . . .
ð1Þ
where E is the modulus of elasticity, I is the area moment of the cross-section about the neutral axis, m is the mass per unit length and kISO ai is a parameter whose values are listed in Table 1. Solving for Ei :
0 12 m @2pðL=2Þ2 ISO A Ei ¼ f ai 2 I ðkISO Þ
ð2Þ
ai
The standard recommends using the first three resonance frequencies to obtain a mean E value. The values of parameter kISO ai shown in Table 1 come from an assumption in the standard that establishes that the vibration modes of the sample loaded in its middle point correspond to the bending modes of two ‘‘clamped-free” half length beams. Therefore it is established in the standard that impedance resonance frequencies—the mobility anti-resonances—of the free-free
Table 1 Values of the parameter kISO ai . i
kISO ai
1 2 3 4
1.87510 4.69410 7.85476 10.99554
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beam match the eigenfrequencies of a ‘‘clamped-free” half length beam.
Table 2 Values of parameter kFF i for eigenfrequencies of a free-free beam.
2.2. Anti-resonances of a free-free beam method This method is straightforwardly derived from the standard ISO 16940 [20] but unlike this one (that assimilates the response to that one of a half length ‘‘clamped-free” beam) it is derived from the analytical anti-resonances of the free-free beam. The displacement of a beam with free-free conditions between x ¼ 0 and x ¼ L by means of a continuous model can be found in textbooks as [22] and it can be expressed by modal superposition as
qðx; tÞ ¼
1 X
gi ðtÞ/i ðxÞ
kFF i
1 2 3 4 5
4.73004 7.85321 10.99561 14.13717 17.27876
Table 3 Values of the parameter kFF ar i for the anti-resonances of a free-free beam.
ð3Þ
i¼0
where gi ðtÞ are the modal coordinates and /i ðxÞ the normalized eigenmodes, considering i ¼ 0 as the rigid-body motion of the beam and modes i > 0 the flexural modes of the beam. The first flexural mode shapes (with unitary maximum) are illustrated in Fig. 1. The beam bending eigenfrequencies can be expressed as
sffiffiffiffiffiffiffiffiffi 2 xi ðkFF Þ EI fi ¼ ¼ i 2p 2p mL4
i
N
i
kFF ari
1
1
3.77003
6
1 2 3 4
3.75038 9.39740 15.73438 22.05536
7
1 2 3 4
3.75030 9.39504 15.72371 22.02538
ð4Þ
where parameter kFF i satisfies
cos kFF cosh kFF ¼1 i i
ð5Þ
First solutions are shown in Table 2. The response of the system to a harmonic load of intensity p0 and circular frequency X applied at x0 ; pðx; tÞ ¼ p0 dðx x0 ÞeiXt is determined by the response of the beam in normalized modal coordinates
g€ i þ x2i gi ¼ p0 /i ðx0 Þ
ð6Þ
For the case under analysis the load is applied in the centre of the beam (x0 ¼ L=2). As shown in Fig. 1, /i ðx0 Þ ¼ 0 for even i as they are antisymmetric functions respect to x ¼ L=2. Therefore the response of the system is
qðx; tÞ ¼ p0
" 1 X /2i1 ðx0 Þ/2i1 ðxÞ i¼1
x22i1 X2
1 mLX2
2 2 i¼1 x2i1 X
1 mLX2
f ari ¼
eiXt
ð7Þ
¼0
ðkFF ar i Þ
sffiffiffiffiffiffiffiffiffi EI
2
2p
mL4
ð9Þ
values of N. Then, from the experimentally determined anti-resonances of a beam at its middle point, the modulus of elasticity can be calculated from each anti-resonance as
EFF ar i
2 32 m4 2pL2 5 f ari ¼ 2 I ðkFF Þ
ð10Þ
ar i
ð8Þ
2.3. Resonances of a free – free beam method As shown in the previous section, the value of the numerical anti-resonance frequencies are dependant on the number of modes considered in Eq. (8). To avoid this dependency, a method based on the resonance frequencies of the response (that depends only on the corresponding eigenfrequency) is considered. Under the same conditions as above—free-free beam with harmonic load applied on the middle point—the response of the beam at the centre is due to odd i modes as even i modal shapes are null at that point as they are antisymmetric functions respect to that point as illustrated in Fig. 1. Therefore, the response of the beam at the centre will show resonance frequencies corresponding only to the odd i modes. Hence, the resonance frequencies are 2
FF f ri
Fig. 1. First four flexural mode shapes normalised to unitary maximum.
i ¼ 1; 2; 3 . . .
for parameter kFF ar i whose values are shown in Table 3 for different
#
where the term corresponding to the rigid-body motion of the beam has been excluded from the summation. Anti-resonances in the response of the system at the point of excitation are the solutions of qðx0 ; tÞ ¼ 0. Then, anti-resonances of the beam at its middle point are the circular frequencies X that satisfy 1 X /2i1 ðx0 Þ2
This equation can be solved for a given number of bending modes, N, determining a certain number of anti-resonances, f ari , that can be expressed in the same form as the eigenfrequencies
ðkFF Þ ¼ 2i1 2p
sffiffiffiffiffiffiffiffiffi EI mL4
i ¼ 1; 2; 3 . . .
ð11Þ
for the values of the parameter kFF i shown in Table 2. Then, from the experimentally determined resonances of the beam at its middle point, the modulus of elasticity can be calculated from the resonance frequencies as
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EFF ri
i
kCF i
1 2 3 4
1.87510 4.69409 7.85476 10.99554
" #2 m 2pL2 f ¼ I ri ðkFF Þ2 2i1
i ¼ 1; 2; 3 . . .
ECF i ¼
ð12Þ
Previous methods imply to perform a dynamic test on an actual specimen, that is, to apply a harmonic force on the specimen (usually through an electromagnetic shaker) and to measure both such force and the specimen response. Free response of a cantilever beam method is conceived to avoid the need of an externally applied force and it is based on the free response of a cantilever beam to a static load on its free end followed by a quick release. The displacement of a cantilever beam of mass per unit length m, modulus of elasticity, E, area moment of inertia of the crosssection about the neutral axis I and length L can be expressed by ~ superposition of the normalized modes /ðxÞ as 1 X
gi ðtÞ/~ i ðxÞ
ð13Þ
i¼1
The eigenfrequencies of this case can be expressed also in terms of a parameter kCF i that satisfies
cos kCF cosh kCF ¼ 1 i i
ð14Þ
whose first solutions are shown in Table 4. Modal coordinates are gi ðtÞ ¼ Ai cosð2pf i tÞ þ Bi sinð2pf i tÞ where constants Ai and Bi are determined by the initial conditions in dis_ 0Þ respectively. For an static placement qðx; 0Þ and velocity qðx; load of unitary amplitude applied on the free end of the beam fol_ 0Þ ¼ 0, then Bi ¼ 0. Ai can be defined lowed by a quick release, qðx; from the static response of the cantilever beam [23]
qðx; 0Þ ¼
1 Lx2 x3 EI 2 6
ð16Þ
3. Numerical application
2.4. Free response of a cantilever beam method
qðx; tÞ ¼
" #2 m 2pL2 fi 2 I ðkCF i Þ
!
The methods proposed are applied on a numerical case along with the methodology proposed in the standard ISO 16940 [20]. The different tests associated to each method are simulated numerically with a one-dimensional FE model solved with MSC Nastran [24] using beam elements (CBAR). Two beams are considered: a free-free beam of length 0.5 m and a cantilever beam of length 0.25 m, both 0.06 m wide and 0.005 m thick. They both are made up of aluminum alloy 5754 with a modulus of elasticity of 68 GPa, Poisson’s ratio 0.33 and a density of 2660 kg/m3. A modal damping of 5% of the critical one is assumed. The mesh size is set to ensure 10 elements per wavelength up to 4500 Hz. For the mechanical and area properties stated, the bending wavelength at said frequency is 0.1 m, therefore the element size is set to 0.01 m. 3.1. Simulated test results The mobility of the free-free beam in the middle point to a unitary force is calculated considering a frequency resolution of 0.1 Hz from 0 Hz to 4000 Hz as depicted in Fig. 2. The first resonance frequencies are 103.9 Hz, 561.0 Hz and 1382.3 Hz while the first antiresonance frequencies are 65.3 Hz, 408.8 Hz and 1141.9 Hz. The transitory response of the cantilever beam for the application of method described in Section 2.4 is also simulated assuming a perfect clamped boundary condition. The time response of the free end of the beam is shown in Fig. 3. The discrete Fourier transform [25] is calculated with a frequency resolution of 0.1 Hz and its root-mean-square values are shown in Fig. 4 whose first resonance peak corresponds to 65.3 Hz. As described in Section 2.4 and observed in Fig. 4 only one resonance frequency can be determined adequately. 3.2. Determination of the modulus of elasticity From the resonance and anti-resonance values obtained with the FE model, the methods proposed in Section 2 are applied along with the method described in standard ISO 16940 [20] as reference.
ð15Þ
Coefficients Ai decrease with i as Table 5 shows for the first coefficients (normalized to the one corresponding to the fundamental harmonic) that shows that the response of the system is mainly due to the fundamental mode as the contribution of the rest of modes is at least two orders of magnitude lower. For an arbitrary initial displacement (large enough to measure the response of the system but within the linear response regime), the Fourier Transform of the response can be used to determine the first eigenfrequencies of the beam, f i , and the corresponding modulus of elasticity.
Table 5 Coefficients of the modal displacements normalised with the first one. i
j Ai =A1 j
1 2 3 4 5
1 0.025461 0.003249 0.000834 0.000310
Fig. 2. FEM mobility of the middle point of the free-free beam to a unitary load in the same location.
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Table 6 Value of the modulus of elasticity obtained from each resonance or anti-resonance, mean value and standard deviation through each method: ISO 16940, anti-resonances of a free-free beam method, resonances of free-free beam method and free response of a cantilever beam method. i
EISO (GPa)
EFF ar (GPa)
EFF r (GPa)
ECF (GPa)
1 2 3
67.916 67.773 67.448
67.909 67.576 67.205
67.941 67.828 67.533
67.916
l r
67.712 0.240
67.563 0.352
67.768 0.210
67.916
Table 7 Error in percentage of the values of the modulus of elasticity obtained from each resonance or anti-resonance for each method: ISO 16940, anti-resonances of a freefree beam method, resonances of free-free beam method and free response of a cantilever beam method.
Fig. 3. FEM displacement of the cantilever beam free end after a static load followed by a quick release.
i
EISO
EFF ar
EFF r
ECF
1 2 3
0.12% 0.33% 0.81%
0.13% 0.62% 1.17%
0.09% 0.25% 0.69%
0.12%
l
0.42%
0.64%
0.34%
0.12%
Table 8 Values of the parameter kISO ai referred to the whole length of the free-free beam compared to parameter kFF ar i . kISO ai 2
kFF ar i
Diff. (%)
3.75020 9.38818 15.70952 21.99100
3.75030 9.39504 15.72371 22.02538
0.003 0.073 0.090 0.156
and anti-resonance frequencies of the simulated tests and therefore of the modulus of elasticity obtained from them. Standard ISO 16940 is based on considering equal the response of the tested free-free beam to the response of a half-length clamped-free beam. In fact, it approximates the anti-resonances of the free-free beam by the resonances of the clamped-free beam. This approximation can be analysed taking the Standard values kISO ai Fig. 4. Discrete Fourier transform of the FEM displacement of the cantilever beam free end.
Standard ISO 16940 and the method described in Section 2.2 take into account the mobility anti-resonances while method described in Section 2.3 takes into account the mobility resonance frequencies. Method described in Section 2.4 takes into account the eigenfrequency determined by means of the DFT of the free response. The value of the modulus of elasticity obtained by means of each method is shown in Table 6 for each resonance or antiresonance considered as well as its mean and standard deviation except for the free response of a cantilever beam as it has only one resonance, then it is not necessary to calculate the mean and standard deviation values. The error of the results respect to the nominal value (68 GPa) is shown in Table 7. 3.3. Discussion Results show that all methods provide values of the modulus of elasticity close to the actual value (maximum error of 1:17%) although a bias to lower values is found. This is due to not considering any damping in the methods formulation unlike in the numerical model. This fact leads to lower values of the resonance
(Table 1) that are defined for a beam of half length of the tested one and refer them to the full length (in practice it implies that the values are doubled). Now, these values can be compared to the parameter values of the anti-resonances determined in Section 2.2, kFF ari . Both magnitudes and their difference are shown in Table 8. The comparison shows that the assumption made in Standard ISO 16940 is adequate although it adds a bias to lower values in addition to the one related to the damping as stated above. Both the ISO 16940 standard and the method described in Section 2.2—anti-resonances of a free-free beam method—calculate the modulus of elasticity from the anti-resonance of the response and as discussed in Section 2.2, its formulation depends on the number of modes considered to contribute to the system response. Unlike these methods, method described in Section 2.3—resonances of a free-free beam method—determines the resonance frequency only from the eigenfrequencies of the system, providing a much lower error compared to both former ones as shown in Table 7. The method based on the free response of the clamped-free beam, Section 2.4, provides a lower error than the methods based on the anti-resonances of the response. This is especially important if the actual application of the methods is taken into account as the testing procedure and requirements of this method are much less cumbersome than the other ones. For instance, it does not require
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Table 9 Dimensions and density of the samples. Sample
Length (m)
Width (m)
Thickness (m)
Density (kg/m3)
1 2 3 4 5 6
0.599 0.599 0.599 0.600 0.600 0.599
0.0600 0.0609 0.0605 0.0613 0.0600 0.0600
0.0124 0.0125 0.0124 0.0124 0.0125 0.0126
646.2 626.3 640.6 654.8 652.4 651.1
to apply any load to the system so no electromagnetic shaker is needed, leading to a much lower cost in terms of equipment. 4. Experimental application Standard ISO 16940 [20] and the three methods proposed are applied to determine the modulus of elasticity of a plasterboard. As it is an anisotropic material, several samples in longitudinal and transversal directions of a plate are considered [21]: three samples in the longitudinal direction (samples 1–3) and three in the transversal direction (samples 4–6). The dimensions and density of the six samples are listed in Table 9. 4.1. Determination of resonances and anti-resonances A test campaign is performed following the measurement setup described in Standard ISO 16940, Annex A [20] to determine the resonance and anti-resonances of the free-free beam. A harmonic force is applied at the centre of the sample as shown in Fig. 5. The force applied and the acceleration at the excitation point are measured to determine the sample mobility. The load is applied by means of a LDS V406 shaker while the force and acceleration are measured through a PCB 208C02 force sensor and a PCB 352C33 accelerometer respectively. Transducer measurements are acquired through an OROS OR36 analyzer/recorder. The results obtained with the set of samples are depicted in Fig. 6, from which the first resonances and anti-resonances can be determined. Their values are shown in Tables 10 and 11 including their means by material direction, overall mean and standard uncertainty (u) [26]. The test required for the method proposed in Section 2.4, based on the transient response of the beams, is performed six times per each one of the six samples considered clamping one end as depicted in Fig. 7. The response at the free end is measured with a PCB 352C33 accelerometer acquired with an OROS OR36 analyzer/recorder. The test consists of shifting and then releasing the free end of the sample. The first resonance of each cantilever beam are determined. They are listed in Table 12.
Fig. 6. Mobility measured in the centre of the six samples of plasterboard.
Table 10 First three resonance frequencies determined from the mobility measured in the centre of the samples, overall and by material direction means and standard uncertainty. Sample
f r1 (Hz)
f r2 (Hz)
f r3 (Hz)
1 2 3
70.3 65.6 71.1 69.0
370.3 350.8 376.6 365.9
909 864 926 900
71.1 61.7 63.3 65.4
376.6 330.5 333.6 346.9
928 809 826 854
67.2 1.7
356.4 8.6
877 21
lL 4 5 6
lT l u
Table 11 First three anti-resonance frequencies determined from the mobility measured in the centre of the samples, overall and by material direction means and standard uncertainty.
1 2 3
lL 4 5 6
lT l u
Sample
f ar1 (Hz)
f ar2 (Hz)
41.4 38.3 42.2 40.6
289.8 271.1 293.0 284.6
806 754 810 790
42.2 35.2 36.7 38.0
293.8 257.8 264.1 271.9
816 714 734 755
39.3 1.2
278.3 6.5
772 18
4.2. Determination of the modulus of elasticity
Fig. 5. Measurement set up for measuring the mobilitiy in the center of the plasterboard samples.
The three methods proposed are applied considering antiresonances (ISO 1940 and the method described in Section 2.2), resonances (method described in Sections 2.3) and first eigenfrequency (method described in Section 2.4). Tables 13–16 list for each method the value of the modulus of elasticity determined from each frequency value Ei , along with the mean value per eigenfrequency and its standard uncertainty.
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Table 14 Modulus of elasticity determined by the anti-resonances of a free-free beam method for each anti-resonance, overall and by material direction means and standard uncertainty. Sample
E1 (GPa)
E2 (GPa)
E3 (GPa)
1 2 3
2.22 1.81 2.29 2.11 2.35 1.60 1.71 1.89 2.00 0.13
2.76 2.31 2.80 2.62 2.90 2.19 2.24 2.44 2.53 0.13
2.73 2.27 2.73 2.58 2.85 2.14 2.21 2.40 2.49 0.13
lL 4 5 6
lT l u
Table 15 Modulus of elasticity determined by the resonances of a free-free beam method for each resonance, overall and by material direction means and standard uncertainty.
Fig. 7. Measurement set up for measuring the free response of samples to a static load followed by a quick release.
Sample
E1 (GPa)
E2 (GPa)
E3 (GPa)
1 2 3
2.53 2.10 2.57 2.40 2.64 1.95 2.00 2.20 2.30 0.13
2.41 2.06 2.47 2.31 2.54 1.92 1.90 2.12 2.21 0.12
2.38 2.05 2.44 2.29 2.53 1.88 1.91 2.11 2.20 0.12
lL 4 5 6
Table 12 First eigenfrequencies determined from the discrete Fourier transform of the response to a static load followed by a quick release on the free end with the other end clamped, overall and by material direction means and standard uncertainty. Sample
f1
1 2 3
10.57 10.76 10.91 10.75
lL 4 5 6
lT l u
Table 16 Modulus of elasticity determined by the free response of a cantilever beam method, overall and by material direction means and standard uncertainty.
10.09 9.35 10.24 9.89
lT l
10.32 0.23
u
E1 (GPa)
E2 (GPa)
E3 (GPa)
1 2 3
2.22 1.81 2.29 2.11
2.77 2.31 2.81 2.63
2.74 2.28 2.74 2.59
2.35 1.60 1.71 1.89
2.91 2.19 2.25 2.45
2.86 2.15 2.22 2.41
2.00 0.13
2.54 0.13
2.50 0.13
lT l u
1 2 3
2.317 2.288 2.445 2.350 2.155 1.813 2.120 2.029 2.190 0.089
4 5 6
Sample
4 5 6
E1 (GPa)
lL
lT l
Table 13 Modulus of elasticity determined applying standard ISO 16940 for each antiresonance, overall and by material direction means and standard uncertainty.
lL
Sample
The final value of the modulus of elasticity determined by each method is the average of the value for each sample and eigenfrequency following [21]. These values and their standard uncertainty are listed in Table 17. 4.3. Discussion Experimental results obtained from the plasterboard show a clear dependency on the material direction as expected in this type
u
Table 17 Modulus of elasticity determined by means of each method: overall mean value and standard uncertainty. Method
E (GPa)
u (GPa)
Standard ISO 16940 (Section 2.1) Anti-resonances of a free-free beam method (Section 2.2) Resonances of a free-free beam method (Section 2.3) Free response of a cantilever beam method (Section 2.4)
2.34 2.34 2.24 2.190
0.13 0.13 0.12 0.089
of materials. Higher value of the modulus of elasticity is found for samples taken in the longitudinal direction (samples 1–3) than those for samples in the transversal direction (4–6). For these materials, usually, the overall mean is considered the modulus of elasticity of the material [21] although, due to their orthotropic and heterogeneous nature considering statistical properties [2] should be considered. Also, for final construction elements, as partitions, the directional values should be taken into account when considering the mounting direction and mounting conditions [10,11].
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For methods based in anti-resonances (Tables 13 and 14) a consistent lower value is found for the results related to the first anti-resonance. To analyse this result, the modal damping for each sample has been determined [27] from the mobility measured depicted in Fig. 6. For all the samples, a much higher modal damping is found for the first mode in free-free conditions. For instance, for sample 1, first mode damping is 15% of the critical damping compared to the second and third ones that are 3% and 2% of the critical damping, respectively. The high damping of the first mode has the highest influence on the first anti-resonance and the first resonance of the free-free response [28]. Given that the methods studied in this work are undamped, determination of the modulus of elasticity from highly damped modes should be done with caution, as they lead to underestimated values. Nevertheless, regarding the results between the different methods studied, values of the Young’s modulus shown in Tables 13–17 show the same trends identified in the results from the numerical case discussed in Section 3.3. Regarding the differences between methods based on anti-resonances and resonances, for this case it is also found that the method based on the resonances of a free-free beam method is characterised by a higher precision than those ones based on anti-resonances (ISO 16940 and the method based on the anti-resonances of the free-free beam). Since the accuracy of the method based on the free response of the clamped-free beam is similar to the ones of the other methods, it can be concluded that it is an efficient method to determine the modulus of elasticity without the need to have an electromagnetic shaker available or another means of applying a harmonic force as standard ISO 16940 requires. 5. Conclusions In this paper a study on methods to determine the Young’s modulus of materials has been presented. Methods are based on different approaches: standard ISO 16940 and some of the methods presented are based on the mobility of a specimen under a harmonic load and one of the methods proposed is based on the free response of the sample to an initial loading condition. Within the first group, the study shows that determining the modulus of elasticity from the resonance frequencies of the response leads to a higher accuracy and lower spread as it is shown by the study of a numerical and an experimental case. Also, it has been shown that analysing the mobility anti-resonances of a freefree beam by means of the eigenfrequencies of a half-length clamped-free beam adds a small bias to the determination of the modulus of elasticity towards higher values. In addition to methods based on the response to a harmonic load, that require in practice a device to apply such a force, a method based on the free response of the sample has been presented. The results of the numerical application show that this method allows determining the Young’s modulus with the same level of accuracy that those methods based on anti-resonances with a much simpler and affordable testing procedure. Regarding its uncertainty, results of the experimental application show that it is lower that the other methods one. Finally, it is worth mentioning that, according to these results, when the sample material has a high damping value, it would be advantageous to drop the first anti-resonance from the determination after procedures based on anti-resonances as the ISO procedure and the method based on the anti-resonances of a free-free beam.
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